From 9debf4de57aa042108f97e0775588af294846cca Mon Sep 17 00:00:00 2001
From: Mohammad Abolnejadian <mohammad1379_a@yahoo.com>
Date: Fri, 22 Oct 2021 18:23:22 +0330
Subject: [PATCH 01/62] initial creation done

---
 .../adversarial_search_fall_2021/index.ipynb  |  19 ++++++++
 .../matin_daghyani.jpeg                       | Bin 0 -> 56252 bytes
 .../adversarial_search_fall_2021/metadata.yml |  41 ++++++++++++++++++
 .../mohammad_abolnejadian.jpeg                | Bin 0 -> 93109 bytes
 .../mohammadali_mohammadkhani.jpeg            | Bin 0 -> 91800 bytes
 notebooks/index.yml                           |   2 +
 6 files changed, 62 insertions(+)
 create mode 100644 notebooks/adversarial_search_fall_2021/index.ipynb
 create mode 100644 notebooks/adversarial_search_fall_2021/matin_daghyani.jpeg
 create mode 100644 notebooks/adversarial_search_fall_2021/metadata.yml
 create mode 100644 notebooks/adversarial_search_fall_2021/mohammad_abolnejadian.jpeg
 create mode 100644 notebooks/adversarial_search_fall_2021/mohammadali_mohammadkhani.jpeg

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
new file mode 100644
index 00000000..a6100842
--- /dev/null
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -0,0 +1,19 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  },
+  "orig_nbformat": 4
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/notebooks/adversarial_search_fall_2021/matin_daghyani.jpeg b/notebooks/adversarial_search_fall_2021/matin_daghyani.jpeg
new file mode 100644
index 0000000000000000000000000000000000000000..a307675d3df133ea5ee76b9a2c094a002c1386ec
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diff --git a/notebooks/adversarial_search_fall_2021/metadata.yml b/notebooks/adversarial_search_fall_2021/metadata.yml
new file mode 100644
index 00000000..7d95830f
--- /dev/null
+++ b/notebooks/adversarial_search_fall_2021/metadata.yml
@@ -0,0 +1,41 @@
+title: Adversarial Search # shown on browser tab
+
+header:
+    title: Adversarial Search # title of your notebook
+    description: An intro into aversarial search # short description of your notebook
+
+authors:
+    label: 
+        position: top
+    content:
+    - name: Mohammad Abolnejadian 
+      role: Author 
+      contact:
+      - link: https://github.com/mothegoat
+        icon: fab fa-github
+      - link: mailto:mohammad.abolnejadian@gmail.com
+        icon: fas fa-envelope
+
+    - name: Mohammadali MohammadKhani 
+      role: Author 
+      contact:
+      - link: https://github.com/MoaliMkh
+        icon: fab fa-github
+      - link: mailto:mohammadali1379@gmail.com
+        icon: fas fa-envelope
+      # optionally add other contact information like  
+      # - link: <change this> # contact link
+      #   icon: <change this> # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons)
+
+    - name: Matin Daghyani 
+      role: Author 
+      contact:
+      - link: https://github.com/mtndaghyani
+        icon: fab fa-github
+       - link: mailto:matin.daghyani@gmail.com
+        icon: fas fa-envelope
+
+# comments:
+#     # enable comments for your post
+#     label: false
+#     kind: comments
\ No newline at end of file
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literal 0
HcmV?d00001

diff --git a/notebooks/index.yml b/notebooks/index.yml
index 1b2133c0..4dfca602 100644
--- a/notebooks/index.yml
+++ b/notebooks/index.yml
@@ -18,6 +18,8 @@ notebooks:
       kind: S2021, LN, PDF
     - notebook: notebooks/8_adversarial_search/
       kind: S2021, LN, Notebook
+    - notebook: notebooks/adversarial_search_fall_2021/
+      kind: F2021, LN, Notebook
     - notebook: notebooks/9_bayesian_networks/index1.ipynb
       kind: S2021, LN, Notebook
       metadata: notebooks/9_bayesian_networks/metadata.yml

From 82dca8a0b803a97c800163d3b9ffc2b898a5d3e9 Mon Sep 17 00:00:00 2001
From: Mohammad Abolnejadian <mohammad1379_a@yahoo.com>
Date: Fri, 22 Oct 2021 18:25:21 +0330
Subject: [PATCH 02/62] images added

---
 notebooks/adversarial_search_fall_2021/metadata.yml | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/notebooks/adversarial_search_fall_2021/metadata.yml b/notebooks/adversarial_search_fall_2021/metadata.yml
index 7d95830f..e52073fd 100644
--- a/notebooks/adversarial_search_fall_2021/metadata.yml
+++ b/notebooks/adversarial_search_fall_2021/metadata.yml
@@ -10,6 +10,7 @@ authors:
     content:
     - name: Mohammad Abolnejadian 
       role: Author 
+      image: mohammad_abolnejadian.jpg
       contact:
       - link: https://github.com/mothegoat
         icon: fab fa-github
@@ -18,6 +19,7 @@ authors:
 
     - name: Mohammadali MohammadKhani 
       role: Author 
+      image: mohammadali_mohammadkhani.jpg
       contact:
       - link: https://github.com/MoaliMkh
         icon: fab fa-github
@@ -29,6 +31,7 @@ authors:
 
     - name: Matin Daghyani 
       role: Author 
+      image: matin_daghyani.jpg
       contact:
       - link: https://github.com/mtndaghyani
         icon: fab fa-github

From 1060d86d82d8b2507bf516dbbcfaee9120d15f4c Mon Sep 17 00:00:00 2001
From: Mohammad Abolnejadian <mohammad1379_a@yahoo.com>
Date: Fri, 22 Oct 2021 18:57:42 +0330
Subject: [PATCH 03/62] image extensions fixed

---
 notebooks/adversarial_search_fall_2021/metadata.yml | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/metadata.yml b/notebooks/adversarial_search_fall_2021/metadata.yml
index e52073fd..0a8509a8 100644
--- a/notebooks/adversarial_search_fall_2021/metadata.yml
+++ b/notebooks/adversarial_search_fall_2021/metadata.yml
@@ -10,7 +10,7 @@ authors:
     content:
     - name: Mohammad Abolnejadian 
       role: Author 
-      image: mohammad_abolnejadian.jpg
+      image: mohammad_abolnejadian.jpeg
       contact:
       - link: https://github.com/mothegoat
         icon: fab fa-github
@@ -19,7 +19,7 @@ authors:
 
     - name: Mohammadali MohammadKhani 
       role: Author 
-      image: mohammadali_mohammadkhani.jpg
+      image: mohammadali_mohammadkhani.jpeg
       contact:
       - link: https://github.com/MoaliMkh
         icon: fab fa-github
@@ -31,7 +31,7 @@ authors:
 
     - name: Matin Daghyani 
       role: Author 
-      image: matin_daghyani.jpg
+      image: matin_daghyani.jpeg
       contact:
       - link: https://github.com/mtndaghyani
         icon: fab fa-github

From 7fec1fa36aa01ad195b6d6522abe18478f83c214 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sat, 23 Oct 2021 20:19:58 +0330
Subject: [PATCH 04/62] First part of Introduction

---
 notebooks/adversarial_search_fall_2021/index.ipynb | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index a6100842..9ced4471 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -1,11 +1,14 @@
 {
  "cells": [
   {
-   "cell_type": "code",
-   "execution_count": null,
+   "cell_type": "markdown",
    "metadata": {},
-   "outputs": [],
-   "source": []
+   "source": [
+    "# Introduction\n",
+    "### The concept of adversarial search and an introduction to Game Theory\n",
+    "In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search is introduced which is called **Adversarial Search**. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.\n",
+    "To gather more information about the concept of adversarial search, visit <a href=\"https://www.techslang.com/definition/what-is-adversarial-search/\">this link</a>."
+   ]
   }
  ],
  "metadata": {

From ee05b02b2ec99a30fb37bf3a30087ddbcbcacf3a Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sat, 23 Oct 2021 22:04:20 +0330
Subject: [PATCH 05/62] Second part of introduction done!

---
 .../adversarial_search_fall_2021/index.ipynb  | 25 +++++++++++++++++++
 1 file changed, 25 insertions(+)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 9ced4471..a1987306 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -9,6 +9,31 @@
     "In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search is introduced which is called **Adversarial Search**. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.\n",
     "To gather more information about the concept of adversarial search, visit <a href=\"https://www.techslang.com/definition/what-is-adversarial-search/\">this link</a>."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Game Theory Explanation\n",
+    "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers. To know the game theory better, you can stop by <a href=\"https://en.wikipedia.org/wiki/Game_theory\">here</a>.\n",
+    "To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:\n",
+    "<ul>\n",
+    "    <li>States of the game: $S_i$, starting with $S_0$</li>\n",
+    "Which means how the game will look after each action of competitors.\n",
+    "    <li>Players: P = {1,2,...,n}</li>\n",
+    "represents number of agents playing the game which varies from 1 to n.\n",
+    "    <li>Actions: $A_i$</li>\n",
+    "In every state, each player has a set of legitimate moves to change that state to another state by playing its turn.\n",
+    "    <li>Transition functions: S x A -> S</li>\n",
+    "Transition function is a function which takes a state S, and does the action A to that state, and returns another state which is built as a consequence of that action.\n",
+    "    <li>Terminal tests: S -> {True, False}</li>\n",
+    "It is a boolean function which determines whether our game has reached a final state or not.\n",
+    "    <li>Terminal utilities: S ✕ P -> R </li>\n",
+    "It takes a state S, and a player P, and returns a Real number which indicates the score of the player P (The utility of the P, to be more precise) till that moment.\n",
+    "    <li> Policy/Strategy: S -> A </li>\n",
+    "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
+    "</ul>"
+   ]
   }
  ],
  "metadata": {

From 9c2932b1d1e8586597d86c30dc8e3d41d275745a Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sun, 24 Oct 2021 12:14:03 +0330
Subject: [PATCH 06/62] Different categories of games added Vs modifications in
 Introduction

---
 notebooks/adversarial_search_fall_2021/index.ipynb | 13 ++++++++++++-
 1 file changed, 12 insertions(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index a1987306..3cca0a0b 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -14,7 +14,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Game Theory Explanation\n",
+    "# Game Theory Explanation and Deterministic Games\n",
     "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers. To know the game theory better, you can stop by <a href=\"https://en.wikipedia.org/wiki/Game_theory\">here</a>.\n",
     "To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:\n",
     "<ul>\n",
@@ -34,6 +34,17 @@
     "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
     "</ul>"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Different catergories of games\n",
+    "<ul>\n",
+    "    <li>Zero-Sum games</li>\n",
+    "    <li>General games</li>\n",
+    "</ul>"
+   ]
   }
  ],
  "metadata": {

From b057f5fdee6e2d473a5e28804a589b5bb3344553 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sun, 24 Oct 2021 15:48:53 +0330
Subject: [PATCH 07/62] Different types of games and Single Agent Trees
 Finished!

---
 .../adversarial_search_fall_2021/index.ipynb  | 19 ++++++++++---------
 1 file changed, 10 insertions(+), 9 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 3cca0a0b..b9731f16 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -32,18 +32,19 @@
     "It takes a state S, and a player P, and returns a Real number which indicates the score of the player P (The utility of the P, to be more precise) till that moment.\n",
     "    <li> Policy/Strategy: S -> A </li>\n",
     "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
-    "</ul>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Different catergories of games\n",
+    "</ul>\n",
+    "\n",
+    "## Different catergories of games\n",
+    "Before we discuss types of games, we need to define a concept named **Utility Function** first. Each player has some states that will be more willing in them. For example, if the player P has a State S which is more satisfied in it, this means that P has better moves in S toward the goal state and is able to win the game more easily. So each player has a utility function which indicates that in every state, how satisfied the player is and what is the best result that it can score, starting with that state. Naturally in goal states, which the game has been finished, the utility function is maximum for the player.\n",
     "<ul>\n",
     "    <li>Zero-Sum games</li>\n",
+    "    In Zero-Sum games, the sum of utility functions for all the agents is equal to zero. For instance, in a two-agent game, if the player P1 has the utility finction with the value of 10 in an specific state, the player P2 has a utility function with the value -10 in that state. So this game becomes totally adversarial and competetive in a way that we have a single value for utility function and one of the players wants to maximize that value and another one wants to minimize it.\n",
     "    <li>General games</li>\n",
-    "</ul>"
+    "    In this type of games, each of the agents has its own utility function. These utility functions may be independant, so in some stages of the game, the agents may even have to cooperate in addition to competing and indifferting to increase their own utility function value. This type of the game is more general and also complicated compared to Zero-Sum games.\n",
+    "</ul>\n",
+    "\n",
+    "## Single Agent Trees\n",
+    "Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.\n"
    ]
   }
  ],

From 41358914771cf0e4b642d6e83efcd53dd9280144 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sun, 24 Oct 2021 16:21:12 +0330
Subject: [PATCH 08/62] Differnt types of trees completed!

---
 notebooks/adversarial_search_fall_2021/index.ipynb | 14 ++++++++++++--
 1 file changed, 12 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index b9731f16..d9e4abe3 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -42,9 +42,19 @@
     "    <li>General games</li>\n",
     "    In this type of games, each of the agents has its own utility function. These utility functions may be independant, so in some stages of the game, the agents may even have to cooperate in addition to competing and indifferting to increase their own utility function value. This type of the game is more general and also complicated compared to Zero-Sum games.\n",
     "</ul>\n",
-    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Differnt Types of Trees for games\n",
     "## Single Agent Trees\n",
-    "Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.\n"
+    "Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.\n",
+    "## Adversarial Agent Trees\n",
+    "Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.\n",
+    "Like Single Agent Trees, we can form a tree to express the moves of the agents, but the differnce is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  "
    ]
   }
  ],

From b040650f20be2b25a56ab1c7a556e72698a9d597 Mon Sep 17 00:00:00 2001
From: mtndaghyani <matin.daghyani@gmail.com>
Date: Sun, 24 Oct 2021 16:52:23 +0330
Subject: [PATCH 09/62] Add resource limits, eval function and minimax pruning

---
 .../.ipynb_checkpoints/index-checkpoint.ipynb |  80 +++++++++++
 notebooks/adversarial_search_fall_2021/1.jpg  | Bin 0 -> 52683 bytes
 .../adversarial_search_fall_2021/images/1.png | Bin 0 -> 31620 bytes
 .../adversarial_search_fall_2021/images/2.png | Bin 0 -> 35693 bytes
 .../adversarial_search_fall_2021/images/3.png | Bin 0 -> 39003 bytes
 .../adversarial_search_fall_2021/images/4.png | Bin 0 -> 41177 bytes
 .../adversarial_search_fall_2021/images/5.png | Bin 0 -> 43179 bytes
 .../images/depth-limited.jpg                  | Bin 0 -> 34956 bytes
 .../adversarial_search_fall_2021/index.ipynb  | 130 +++++++++++++++++-
 9 files changed, 207 insertions(+), 3 deletions(-)
 create mode 100644 notebooks/adversarial_search_fall_2021/.ipynb_checkpoints/index-checkpoint.ipynb
 create mode 100644 notebooks/adversarial_search_fall_2021/1.jpg
 create mode 100644 notebooks/adversarial_search_fall_2021/images/1.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/2.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/3.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/4.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/5.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/depth-limited.jpg

diff --git a/notebooks/adversarial_search_fall_2021/.ipynb_checkpoints/index-checkpoint.ipynb b/notebooks/adversarial_search_fall_2021/.ipynb_checkpoints/index-checkpoint.ipynb
new file mode 100644
index 00000000..76492f75
--- /dev/null
+++ b/notebooks/adversarial_search_fall_2021/.ipynb_checkpoints/index-checkpoint.ipynb
@@ -0,0 +1,80 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Introduction\n",
+    "### The concept of adversarial search and an introduction to Game Theory\n",
+    "In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search is introduced which is called **Adversarial Search**. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.\n",
+    "To gather more information about the concept of adversarial search, visit <a href=\"https://www.techslang.com/definition/what-is-adversarial-search/\">this link</a>."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Game Theory Explanation\n",
+    "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers. To know the game theory better, you can stop by <a href=\"https://en.wikipedia.org/wiki/Game_theory\">here</a>.\n",
+    "To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:\n",
+    "<ul>\n",
+    "    <li>States of the game: $S_i$, starting with $S_0$</li>\n",
+    "Which means how the game will look after each action of competitors.\n",
+    "    <li>Players: P = {1,2,...,n}</li>\n",
+    "represents number of agents playing the game which varies from 1 to n.\n",
+    "    <li>Actions: $A_i$</li>\n",
+    "In every state, each player has a set of legitimate moves to change that state to another state by playing its turn.\n",
+    "    <li>Transition functions: S x A -> S</li>\n",
+    "Transition function is a function which takes a state S, and does the action A to that state, and returns another state which is built as a consequence of that action.\n",
+    "    <li>Terminal tests: S -> {True, False}</li>\n",
+    "It is a boolean function which determines whether our game has reached a final state or not.\n",
+    "    <li>Terminal utilities: S ✕ P -> R </li>\n",
+    "It takes a state S, and a player P, and returns a Real number which indicates the score of the player P (The utility of the P, to be more precise) till that moment.\n",
+    "    <li> Policy/Strategy: S -> A </li>\n",
+    "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
+    "</ul>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Resource Limits\n",
+    "Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine is exponential in the depth of the tree.\n",
+    "\n",
+    "Consider a _reasonable_ chess game with $b \\approx 35$ and $m \\approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Depth-limited search\n",
+    "One idea might be running the algorithm up to a specified depth instead of the searching the whole tree which(_depth-limited search_).\n",
+    "\n",
+    "![hello](1.jpg)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
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HcmV?d00001

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index a1987306..1c82dac8 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -34,13 +34,137 @@
     "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
     "</ul>"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Resource Limits\n",
+    "Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine is exponential in the depth of the tree.\n",
+    "\n",
+    "Consider a _reasonable_ chess game with $b \\approx 35$ and $m \\approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Depth-limited search\n",
+    "One idea might be running the algorithm up to a specified depth instead of the searching the whole tree which we call _depth-limited search_ .\n",
+    "\n",
+    "![depth-limited](./images/depth-limited.jpg)\n",
+    "\n",
+    "But using this technique is not much satisfying because it leads to antoher issue: _How can we find the minimax value while there is no solution at a limited depth?_\n",
+    "Recall how we find the minimax value of each node in the original form. We continue searching until we reach a final state and then use recurision to calculate the parents' values. But in the _limited_ format, there is no goal states at depth $k < m$.\n",
+    "\n",
+    "Now to deal with this issue, we'll introduce _*Evaluation function*_."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Evaluation Functions\n",
+    "The goal of an evaluation function is to answer to the question: _How good is the current position?_ Or _How probable is it to reach to winning terminal state from this position?_\n",
+    "\n",
+    "Let us make this idea more concrete. An evaluation function returns an _estimate_ of the utility in each game state, just like the _heuristic_ functions that estimate the remaining cost to a goal state which were discussed before.\n",
+    "\n",
+    "Obviously, defining an appropriate and precise evaluation function is strongly effective on the player's performance. Actually an inaccurate evaluation function may lead to a lost position in the game.\n",
+    "\n",
+    "Most evaluation functions calculate the value of the position by using a _weighted sum of features_:\n",
+    "\n",
+    "$Eval(s) = w_1f_1(s) + w_2f_2(s) + ... + w_nf_n(s)$\n",
+    "\n",
+    "Each of the $f_i$s is calculating a specific _feature_ of the state _s_.For instance, in chess, we would have features for the number of white pawns, black pawns, white queens, black queens, and etc. To better differentiate the effect of each feature, we multiply them by _weights_.\n",
+    "\n",
+    "Ideally, evaluation function returns exactly the minimax value of the current state.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Iterative Deepening Search\n",
+    "The accuracy of the evaluation function is critical in shallower states. And the deeper in the tree we search, the less the quality of the evaluation function matters.\n",
+    "So a good idea is to maximize the depth of our search as much as possible, considering the limits that we have. In other words, we are looking for an algorithm that can returns an acceptable solution whenever we ask. This kind of algorithms are called _anytime algorithms_. These algorithms are expected to find better and better solutions the longer they keeps running.\n",
+    "\n",
+    "So simply instead of running the depth-limited-search once, we start running the algorithm with an initial depth limit ($k$). Then after we found s policy, increase the depth limit ($k' > k$) and run the algorithm again with this new limit to find a better and more accurate solution. We continue the iteration until the time limit reaches.\n",
+    "This alogrithm is called _iterative-deepening search_.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Minimax Pruning\n",
+    "\n",
+    "If we take a look at how minimax works, it is noticeable that it explores some parts of the tree that are not necessary at all. In other words, exploring some branches are totally useless but expensive. If we can predict which branches do not worth exploring and _prune_, we can improve minimax algorithm significantly.\n",
+    "\n",
+    "To better understand better, consider the following example:\n",
+    "<div width=\"50%\" height=\"50%\">\n",
+    "    <img src=\"./images/1.png\" alt=\"prunning1\" width=\"50%\" height=\"50%\"/>\n",
+    "</div>\n",
+    "\n",
+    "We start searching with respect to the assumption that we visit leftmost nodes first. Finally when we reach the terminal node with utility 4, _we can deduce that the value of it's parent node(which is a *Min* node) is definitely less than or equal to 4_:\n",
+    "\n",
+    "<div width=\"50%\" height=\"50%\">\n",
+    "    <img src=\"./images/2.png\" alt=\"prunning2\" width=\"50%\" height=\"50%\"/>\n",
+    "</div>\n",
+    "\n",
+    "At this point, this information doesn't help us a lot. So let's continue searching. When we visit the terminal node with utility 3, we find that the value for the parent is also 3. An important observation here is that we can _predict_ that the value of the root _is more than or equal to 3_. That's because the root is a * Max * node.\n",
+    "\n",
+    "\n",
+    "<div width=\"50%\" height=\"50%\">\n",
+    "    <img src=\"./images/3.png\" alt=\"prunning3\" width=\"50%\" height=\"50%\"/>\n",
+    "</div>\n",
+    "\n",
+    "\n",
+    "After that, we continue searching, a terminal leaf with utility 2 will be discovered, and just like before, we can immediately notice that the value of it's parent is _less than or equal to 2_.\n",
+    "\n",
+    "<div width=\"50%\" height=\"50%\">\n",
+    "    <img src=\"./images/4.png\" alt=\"prunning4\" width=\"50%\" height=\"50%\"/>\n",
+    "</div>\n",
+    "\n",
+    "Now, let's ask ourselves: _Is it necessary that we explore the rightmost node?_. The answer is * No *. Because we already know that the value of the root is at least 3, while the maximum value of the right child of root is 2. So we can easily ignore exploring the last node and find that the minimax value of the root is 3:\n",
+    "\n",
+    "<div width=\"50%\" height=\"50%\">\n",
+    "    <img src=\"./images/5.png\" alt=\"prunning5\" width=\"50%\" height=\"50%\"/>\n",
+    "</div>\n",
+    "\n",
+    "This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called $\\alpha-\\beta $ _pruning_."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## References\n",
+    "\n",
+    "+ https://towardsdatascience.com/understanding-the-minimax-algorithm-726582e4f2c6\n",
+    "+ Russell, S. J., Norvig, P., &amp; Davis, E. (2022). Artificial Intelligence: A modern approach. Pearson Educación. \n",
+    "+ https://slideplayer.com/slide/4380560/\n",
+    "+ https://www.cs.cornell.edu/courses/cs312/2002sp/lectures/rec21.htm"
+   ]
   }
  ],
  "metadata": {
-  "language_info": {
-   "name": "python"
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
   },
-  "orig_nbformat": 4
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.5"
+  }
  },
  "nbformat": 4,
  "nbformat_minor": 2

From 3e55e002f8888274b3d2262450571a927ef22fd5 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sun, 24 Oct 2021 18:34:55 +0330
Subject: [PATCH 10/62] Minimax Startegy Completed

---
 .../adversarial_search_fall_2021/index.ipynb  | 19 +++++++++++++++++++
 1 file changed, 19 insertions(+)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 696bb67e..d23576dd 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -57,6 +57,25 @@
     "Like Single Agent Trees, we can form a tree to express the moves of the agents, but the differnce is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  "
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Minimax Strategy\n",
+    "To gain the value of every node in Adversarial tree, we classify the nodes as two groups:\n",
+    "<ol>\n",
+    "    <li>The nodes our agent does action on them. The utility value of these kind of nodes is assigned by the maximum of the utility values of its successors. Because we want the maximum utility value for our agent. (The max side of the Strategy) </li>\n",
+    "    <li>The nodes that the adversial agent does action on them. The utility value of these kind of nodes is assigned by the minimum of the utility values of its successors. Because we want the minimum utility value for the adversary agent. (The min side of the Strategy) </li>\n",
+    "</ol>\n",
+    "Now that we have every node's utlity value, we are able to find the best action for each state. The main idea is at each state, we choose the node with the greatest minimax value and move to it.\n",
+    "Why does this strategy work? Because while forming the tree and assigning the utility function, we've always considered the best move for the enemy in his/her point of view and the worst move in our point of view. So we assumed that the enemy always plays its best shot. Now Two situations are possible to emerge:\n",
+    "<ol>\n",
+    "    <li>The adversary agent is very smart. In this case, we are ready beacuse the strategy is based on this situation.</li>\n",
+    "    <li>The adversary agent is too naive and may do some inoptimal actions. In this case, we may even score more than what we gained as utillty value.</li>\n",
+    "</ol>\n",
+    "As a conclusion of what's said before, we can say that the utility value that we calculate for nodes in minimax strategy, is a ceiling value of what we may actually score."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},

From 56081683c829edf2ff5cb0505962935f4955c681 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sun, 24 Oct 2021 22:05:52 +0330
Subject: [PATCH 11/62] Minimax Strategy and properties completed!

---
 .../adversarial_search_fall_2021/index.ipynb  | 51 +++++++++++++++++--
 1 file changed, 48 insertions(+), 3 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index d23576dd..e27a40af 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -76,6 +76,49 @@
     "As a conclusion of what's said before, we can say that the utility value that we calculate for nodes in minimax strategy, is a ceiling value of what we may actually score."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Pseudocode of Minimax Strategy\n",
+    "\n",
+    "```python\n",
+    "def max_value(state):\n",
+    "    if leaf?(state), return U(state)\n",
+    "    initialize v = -∞\n",
+    "    for each c in children(state):\n",
+    "        v = max(v, min_value)\n",
+    "    return v\n",
+    "\n",
+    "\n",
+    "def min_value(state):\n",
+    "    if leaf?(state), return U(state)\n",
+    "    initialize v = ∞\n",
+    "    for each c in children(state):\n",
+    "        v = min(v, max_value)\n",
+    "    return v\n",
+    "    \n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Properties of Minimax Strategy\n",
+    "## Complete:\n",
+    "Yes, It is complete if the tree formed for the game is finite.\n",
+    "\n",
+    "## Optimal:\n",
+    "Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.\n",
+    "    \n",
+    "## Time Complexity:\n",
+    "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be great.\n",
+    "\n",
+    "## Space Complexity:\n",
+    "$O(bm)$, because it's like DFS and only stores the nodes on the current path.\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -189,9 +232,11 @@
   }
  ],
  "metadata": {
+  "interpreter": {
+   "hash": "bf0f1093e9a05e91d616174468df6ede651877b2c960cbb124380d685ad06d65"
+  },
   "kernelspec": {
-   "display_name": "Python 3",
-   "language": "python",
+   "display_name": "Python 3.8.3 64-bit",
    "name": "python3"
   },
   "language_info": {
@@ -204,7 +249,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.5"
+   "version": "3.8.3"
   }
  },
  "nbformat": 4,

From f1a852a87bf39e2eb7b362e87a5208823aa06eb3 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Sun, 24 Oct 2021 22:52:41 +0330
Subject: [PATCH 12/62] Initialize Other types of games

---
 notebooks/adversarial_search_fall_2021/index.ipynb | 10 ++++++++--
 1 file changed, 8 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index e27a40af..d4aa0461 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -81,7 +81,6 @@
    "metadata": {},
    "source": [
     "# Pseudocode of Minimax Strategy\n",
-    "\n",
     "```python\n",
     "def max_value(state):\n",
     "    if leaf?(state), return U(state)\n",
@@ -97,7 +96,6 @@
     "    for each c in children(state):\n",
     "        v = min(v, max_value)\n",
     "    return v\n",
-    "    \n",
     "```"
    ]
   },
@@ -218,6 +216,14 @@
     "This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called $\\alpha-\\beta $ _pruning_."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Other types of games:\n",
+    "On top of what's been discussed before, we can count another type of game called **Mixed Layer Types**. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},

From 1f89db57f747af4f7f74ac47d04df5e1b9d1d1cf Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Mon, 25 Oct 2021 07:01:55 +0330
Subject: [PATCH 13/62] First Version of summary

---
 .../adversarial_search_fall_2021/index.ipynb  | 43 ++++++++++++++++++-
 1 file changed, 42 insertions(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index d4aa0461..074f4205 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -221,7 +221,48 @@
    "metadata": {},
    "source": [
     "# Other types of games:\n",
-    "On top of what's been discussed before, we can count another type of game called **Mixed Layer Types**. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game."
+    "On top of what's been discussed before, we can count another type of game called **Mixed Layer Types**. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game.\n",
+    "\n",
+    "To do some calculations about the depth of the game tree, we can state:\n",
+    "\n",
+    "$b$ = $21$ because 21 dintinct states are created as an outcome of rolling 2 dices.\n",
+    "\n",
+    "$legal moves \\approx 20$ \n",
+    "\n",
+    "So if we want to build the tree till the depth of 2, we will have $20 * (21 * 20) ^ 3$ leaves and termainal states. as we go deeper in tree, this number increases.\n",
+    "\n",
+    "As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and prunning becomes trickier. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Multi-Agent Utilities\n",
+    "In games with multiple players, terminal states have utility n-tuples values and each player tries to maximize its own part. This can make cooperation strategies viable."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Summary\n",
+    "Adverserial Search is used when there are other agents in play, probably with conflicting objectives as in most games. Here, we are interested in finding a **Strategy of Playing** that determines which action we should take in different possible scenarios.\n",
+    "\n",
+    "Games can be classified into two main categories based on utility functions:\n",
+    "- Zero-sum Games: agents have opposite utilities (pure competition)\n",
+    "- General Games:  independent utilities (cooperation may be beneficial)\n",
+    "\n",
+    "We can represent possible game states using Game Trees with the initial state as the root node. Each node is followed by its children, representing different actions. Therefore terminal states are the leaf nodes.\n",
+    "\n",
+    "Optimal strategy against an opponent playing perfectly can be found using **Minimax** algorithm. We can also determine a minimum payoff that can be achieved in every game.\n",
+    "A complete search is often infeasible in minimax due to resource limits. So we can:\n",
+    "- Replace the utility function with an **Evaluation Function** that estimates utility.\n",
+    "- Do **Minimax Pruning**: While calculating minimax values, it is sometimes possible to detect and \"prune\" some suboptimal branches.\n",
+    "\n",
+    "Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search**.\n",
+    "\n",
+    "**Final Note**: While we are looking for a good strategy of playing, Optimism and Pessimism can both result in suboptimal behaviour so it's important to evaluate each problem realistically."
    ]
   },
   {

From 594ec02dc5210fd817c21ed021d3d6e86457cd45 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Mon, 25 Oct 2021 07:37:30 +0330
Subject: [PATCH 14/62] Summary Finished!

---
 .../adversarial_search_fall_2021/index.ipynb  | 30 ++++++++++---------
 1 file changed, 16 insertions(+), 14 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 074f4205..7cbc1ee7 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -229,9 +229,11 @@
     "\n",
     "$legal moves \\approx 20$ \n",
     "\n",
-    "So if we want to build the tree till the depth of 2, we will have $20 * (21 * 20) ^ 3$ leaves and termainal states. as we go deeper in tree, this number increases.\n",
+    "So if we want to build the tree till the depth of 2, we will have $20 * (21 * 20) ^ 3$ leaves and termainal states. As we go deeper in tree, this number increases.\n",
     "\n",
-    "As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and prunning becomes trickier. "
+    "As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and prunning becomes trickier. \n",
+    "\n",
+    "Td-Gammon is a computer backgammon program developed in 1992. It uses a 2-depth search, a very good evaluation function plus reinforcement learning and what we get, is a champion-class player! TD-Gammon was the first AI world champion in any game!"
    ]
   },
   {
@@ -239,7 +241,7 @@
    "metadata": {},
    "source": [
     "# Multi-Agent Utilities\n",
-    "In games with multiple players, terminal states have utility n-tuples values and each player tries to maximize its own part. This can make cooperation strategies viable."
+    "In games with multiple players, terminal states have utility in form of tuple values and each player tries to maximize its own part. This idea can raise the cooperation and competetion strategies dynamically."
    ]
   },
   {
@@ -247,22 +249,22 @@
    "metadata": {},
    "source": [
     "# Summary\n",
-    "Adverserial Search is used when there are other agents in play, probably with conflicting objectives as in most games. Here, we are interested in finding a **Strategy of Playing** that determines which action we should take in different possible scenarios.\n",
+    "Before this, we only discussed games with a single player. But we use Adversarial search when there are other players and agents in the game too, often with opposite objectives. With adversarial search, we are seeking a method called **Playing Strategy** that gives us correct directions and instructions at each state of game.\n",
     "\n",
-    "Games can be classified into two main categories based on utility functions:\n",
-    "- Zero-sum Games: agents have opposite utilities (pure competition)\n",
-    "- General Games:  independent utilities (cooperation may be beneficial)\n",
+    "We classified games in two main groups based on utility functions:\n",
+    "- Zero-Sum Games: The sum of agents' utility functions is equal to zero. (pure competition)\n",
+    "- General Games:  independent utility functions. (cooperation may be beneficial)\n",
     "\n",
-    "We can represent possible game states using Game Trees with the initial state as the root node. Each node is followed by its children, representing different actions. Therefore terminal states are the leaf nodes.\n",
+    "We formed trees for both single and adversarial agent games and we found a strategy to obtain the value of nodes in both situations.\n",
     "\n",
-    "Optimal strategy against an opponent playing perfectly can be found using **Minimax** algorithm. We can also determine a minimum payoff that can be achieved in every game.\n",
-    "A complete search is often infeasible in minimax due to resource limits. So we can:\n",
-    "- Replace the utility function with an **Evaluation Function** that estimates utility.\n",
-    "- Do **Minimax Pruning**: While calculating minimax values, it is sometimes possible to detect and \"prune\" some suboptimal branches.\n",
+    "Optimal strategy against an opponent playing perfectly can be found using **Minimax strategy**. We can also determine a floor value of the score that can be achieved in every game.\n",
+    "A complete search is often impossible in minimax because of resource limits. So There are two options:\n",
+    "- Replace the utility function with an **Evaluation Function** that estimates utility. This function must be defined very precise, because the whole solution depends on this.\n",
+    "- Do **Minimax Pruning**: While calculating minimax values, it's sometimes possible to detect and **prune** some suboptimal branches that we know we'll never visit them during the algorithm.\n",
     "\n",
-    "Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search**.\n",
+    "Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search** which was explained before.\n",
     "\n",
-    "**Final Note**: While we are looking for a good strategy of playing, Optimism and Pessimism can both result in suboptimal behaviour so it's important to evaluate each problem realistically."
+    "**Final Note**: While we are looking for a good strategy of playing, Optimism and Pessimism can both result in suboptimal behaviour so it's important to evaluate each problem realistically and choose our strategy in proportion of the adversarial agent's cleverness."
    ]
   },
   {

From ecca3b51b71271a33c0a76cf3f35d3f4691382ab Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Mon, 25 Oct 2021 07:51:36 +0330
Subject: [PATCH 15/62] Finished!

---
 .../adversarial_search_fall_2021/index.ipynb  | 19 ++++++++++++-------
 1 file changed, 12 insertions(+), 7 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 7cbc1ee7..94d9a0d3 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -15,7 +15,7 @@
    "metadata": {},
    "source": [
     "# Game Theory Explanation and Deterministic Games\n",
-    "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers. To know the game theory better, you can stop by <a href=\"https://en.wikipedia.org/wiki/Game_theory\">here</a>.\n",
+    "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers.\n",
     "To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:\n",
     "<ul>\n",
     "    <li>States of the game: $S_i$, starting with $S_0$</li>\n",
@@ -40,8 +40,9 @@
     "    <li>Zero-Sum games</li>\n",
     "    In Zero-Sum games, the sum of utility functions for all the agents is equal to zero. For instance, in a two-agent game, if the player P1 has the utility finction with the value of 10 in an specific state, the player P2 has a utility function with the value -10 in that state. So this game becomes totally adversarial and competetive in a way that we have a single value for utility function and one of the players wants to maximize that value and another one wants to minimize it.\n",
     "    <li>General games</li>\n",
-    "    In this type of games, each of the agents has its own utility function. These utility functions may be independant, so in some stages of the game, the agents may even have to cooperate in addition to competing and indifferting to increase their own utility function value. This type of the game is more general and also complicated compared to Zero-Sum games.\n",
+    "    In this type of games, each of the agents has its own utility function. These utility functions may be independant, so in some stages of the game, the agents may even have to cooperate in addition to competing and indifferenting to increase their own utility function value. This type of the game is more general and also complicated compared to Zero-Sum games.\n",
     "</ul>\n",
+    " To know the game theory better, you can stop by <a href=\"https://en.wikipedia.org/wiki/Game_theory\">here</a>.\n",
     "\n"
    ]
   },
@@ -73,7 +74,7 @@
     "    <li>The adversary agent is very smart. In this case, we are ready beacuse the strategy is based on this situation.</li>\n",
     "    <li>The adversary agent is too naive and may do some inoptimal actions. In this case, we may even score more than what we gained as utillty value.</li>\n",
     "</ol>\n",
-    "As a conclusion of what's said before, we can say that the utility value that we calculate for nodes in minimax strategy, is a ceiling value of what we may actually score."
+    "As a conclusion of what's said before, we can say that the utility value that we calculate for nodes in minimax strategy, is a floor value of what we may actually score."
    ]
   },
   {
@@ -111,7 +112,7 @@
     "Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.\n",
     "    \n",
     "## Time Complexity:\n",
-    "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be great.\n",
+    "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.\n",
     "\n",
     "## Space Complexity:\n",
     "$O(bm)$, because it's like DFS and only stores the nodes on the current path.\n"
@@ -262,9 +263,9 @@
     "- Replace the utility function with an **Evaluation Function** that estimates utility. This function must be defined very precise, because the whole solution depends on this.\n",
     "- Do **Minimax Pruning**: While calculating minimax values, it's sometimes possible to detect and **prune** some suboptimal branches that we know we'll never visit them during the algorithm.\n",
     "\n",
-    "Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search** which was explained before.\n",
+    "Another point that has to be considered is that While we are looking for a good strategy of playing, we must neighter act Optimistically nor Pessimismtically, because they can both result in suboptimal behaviour so it's important to evaluate each problem realistically and choose our strategy in proportion of the adversarial agent's cleverness.\n",
     "\n",
-    "**Final Note**: While we are looking for a good strategy of playing, Optimism and Pessimism can both result in suboptimal behaviour so it's important to evaluate each problem realistically and choose our strategy in proportion of the adversarial agent's cleverness."
+    "Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search** which was explained before."
    ]
   },
   {
@@ -276,7 +277,11 @@
     "+ https://towardsdatascience.com/understanding-the-minimax-algorithm-726582e4f2c6\n",
     "+ Russell, S. J., Norvig, P., &amp; Davis, E. (2022). Artificial Intelligence: A modern approach. Pearson Educación. \n",
     "+ https://slideplayer.com/slide/4380560/\n",
-    "+ https://www.cs.cornell.edu/courses/cs312/2002sp/lectures/rec21.htm"
+    "+ https://www.cs.cornell.edu/courses/cs312/2002sp/lectures/rec21.htm\n",
+    "+ https://www.techslang.com/definition/what-is-adversarial-search/\n",
+    "+ https://en.wikipedia.org/wiki/Game_theory\n",
+    "+ https://brilliant.org/wiki/minimax/#:~:text=In%20game%20theory%2C%20minimax%20is,payoff%20as%20large%20as%20possible.\n",
+    "+ https://en.wikipedia.org/wiki/Minimax"
    ]
   }
  ],

From c66f7be41b208835a76979ddb0c29d2717f00692 Mon Sep 17 00:00:00 2001
From: Mohammad Abolnejadian <mohammad1379_a@yahoo.com>
Date: Fri, 29 Oct 2021 22:18:39 +0330
Subject: [PATCH 16/62] alpha beta pruning

---
 notebooks/adversarial_search_fall_2021/1.jpg  | Bin 52683 -> 0 bytes
 .../images/Alpha_Beta_Intro.png               | Bin 0 -> 18069 bytes
 .../images/alpha_beta_run_1.png               | Bin 0 -> 18296 bytes
 .../images/alpha_beta_run_2.png               | Bin 0 -> 19588 bytes
 .../images/alpha_beta_run_3.png               | Bin 0 -> 22313 bytes
 .../images/alpha_beta_run_4.png               | Bin 0 -> 23571 bytes
 .../adversarial_search_fall_2021/index.ipynb  | 123 ++++++++++++++++--
 7 files changed, 114 insertions(+), 9 deletions(-)
 delete mode 100644 notebooks/adversarial_search_fall_2021/1.jpg
 create mode 100644 notebooks/adversarial_search_fall_2021/images/Alpha_Beta_Intro.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/alpha_beta_run_1.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/alpha_beta_run_2.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/alpha_beta_run_3.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/alpha_beta_run_4.png

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diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 94d9a0d3..90508aa1 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -81,7 +81,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Pseudocode of Minimax Strategy\n",
+    "## Pseudocode of Minimax Strategy\n",
     "```python\n",
     "def max_value(state):\n",
     "    if leaf?(state), return U(state)\n",
@@ -104,17 +104,17 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Properties of Minimax Strategy\n",
-    "## Complete:\n",
+    "## Properties of Minimax Strategy\n",
+    "### Complete:\n",
     "Yes, It is complete if the tree formed for the game is finite.\n",
     "\n",
-    "## Optimal:\n",
+    "### Optimal:\n",
     "Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.\n",
     "    \n",
-    "## Time Complexity:\n",
+    "### Time Complexity:\n",
     "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.\n",
     "\n",
-    "## Space Complexity:\n",
+    "### Space Complexity:\n",
     "$O(bm)$, because it's like DFS and only stores the nodes on the current path.\n"
    ]
   },
@@ -122,7 +122,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Resource Limits\n",
+    "# Resource Limits\n",
     "Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine is exponential in the depth of the tree.\n",
     "\n",
     "Consider a _reasonable_ chess game with $b \\approx 35$ and $m \\approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below."
@@ -217,6 +217,110 @@
     "This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called $\\alpha-\\beta $ _pruning_."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Alph-Beta Pruning\n",
+    "\n",
+    "As we are trying to improve our algorithm even more, there is another approach to prune branches that cannot possibly influence the final decision. So in the last section, we talked about Minimax Pruning, but how can we implement it? \n",
+    "\n",
+    "The approach we are trying to explain here as you can see in the title is called Alpha-Beta Pruning.\n",
+    "\n",
+    "The main idea here is that if a node (e.g. $m$) has a better choice than another node (e.g. $n$) or its parents, then that another node ($n$) will never be wanted to expanded. So we can prune it and never check its successor nodes."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "![depth-limited](./images/Alpha_Beta_Intro.png)</br>\n",
+    "*As $m$ can give the root node a higher value, $n$ will be pruned.* \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We do this by keeping track of two parameters:\n",
+    "\n",
+    "+  $\\alpha$: The value of best choice (i.e. highest value) we have found so far for a Maximizing Node (i.e. our player).\n",
+    "+  $\\beta$: The value of best choice  (i.e. lowest value) we have found so far for a Minimizing Node (i.e opponent player)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Sample Code:\n",
+    "\n",
+    "Here is a Python function to make you understand the concept even better."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "```python\n",
+    "def alpha_beta_pruning(position, depth, alpha, beta, maximizing_player):\n",
+    "    if depth == 0 or game_state(position) == GAMESTATE.GAMEOVER:\n",
+    "        if terminal_utility(position):\n",
+    "            return terminal_utility(position)\n",
+    "        return eval_func(position)\n",
+    "\n",
+    "    if maximizing_player: #for maximizing player (our player)\n",
+    "        max_eval = - math.inf\n",
+    "        for child in position.children:\n",
+    "            eval = alpha_beta_pruning(child, depth - 1, alpha, beta, false)\n",
+    "            max_eval = max(max_eval, eval)\n",
+    "            alpha = max(max_eval, eval)\n",
+    "            if beta <= alpha:\n",
+    "                break\n",
+    "        return max_eval\n",
+    "    else: #for minimizing player(opponent player)\n",
+    "        min_eval = + math.inf\n",
+    "        for child in position.children:\n",
+    "            eval = alpha_beta_pruning(child, depth - 1,alpha, beta, true)\n",
+    "            min_eval = min(min_eval, eval)\n",
+    "            beta = min(beta, eval)\n",
+    "            if beta <= alpha:\n",
+    "                break\n",
+    "        return min_eval\n",
+    "```     \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "So to initialize this function, first we need to give alpha the worst value for our player, which is $-inf$, and give beta the worst value for the opponent, which is $+inf$.\n",
+    "\n",
+    "```python\n",
+    "    #initialization\n",
+    "    alpha_beta_pruning(current_position, 0, - math.inf, + math.inf, true)\n",
+    "```\n",
+    "\n",
+    "Now let's look at an example while running this code."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<p align=\"center\">\n",
+    "    <img src=\"./images/alpha_beta_run_1.png\" alt=\"alpha_beta_run_1\" /></br>\n",
+    "    <em>We approach the first leaf and since the parent is a Minimizing Node, its beta value gets updated</em></br>\n",
+    "    <img src=\"./images/alpha_beta_run_2.png\" alt=\"alpha_beta_run_2\" /></br>\n",
+    "    <em>Second leaf has a bigger value, so since beta always updates at a lower value, nothing changes here</em></br>\n",
+    "    <img src=\"./images/alpha_beta_run_3.png\" alt=\"alpha_beta_run_3\" /></br>\n",
+    "    <em>The algorithm goes back to the root's second child and then goes to the third leaf. Keep in mind that the parent's alpha and beta values always gets passed to their children.</em></br>\n",
+    "    <img src=\"./images/alpha_beta_run_4.png\" alt=\"alpha_beta_run_4\" /></br>\n",
+    "    <em>Now since beta is less than alpha, we know that our root's best possible value from this child will be a 3 which is lower than its other child (i.e. 5), so we prune the other leaf.</em></br>\n",
+    "</p>\n",
+    "\n",
+    "\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -275,13 +379,14 @@
     "## References\n",
     "\n",
     "+ https://towardsdatascience.com/understanding-the-minimax-algorithm-726582e4f2c6\n",
-    "+ Russell, S. J., Norvig, P., &amp; Davis, E. (2022). Artificial Intelligence: A modern approach. Pearson Educación. \n",
+    "+ Russell, S. J., Norvig, P., &amp; Davis, E. (3rd. Ed). Artificial Intelligence: A modern approach. Pearson Educación. \n",
     "+ https://slideplayer.com/slide/4380560/\n",
     "+ https://www.cs.cornell.edu/courses/cs312/2002sp/lectures/rec21.htm\n",
     "+ https://www.techslang.com/definition/what-is-adversarial-search/\n",
     "+ https://en.wikipedia.org/wiki/Game_theory\n",
     "+ https://brilliant.org/wiki/minimax/#:~:text=In%20game%20theory%2C%20minimax%20is,payoff%20as%20large%20as%20possible.\n",
-    "+ https://en.wikipedia.org/wiki/Minimax"
+    "+ https://en.wikipedia.org/wiki/Minimax\n",
+    "+ Sebastian Lague's Youtube channel"
    ]
   }
  ],

From 4207d73e89fc7be44025db3846671683cdd9213f Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Fri, 29 Oct 2021 22:41:05 +0330
Subject: [PATCH 17/62] colors added

---
 .../adversarial_search_fall_2021/index.ipynb  | 116 +++++++++++++++---
 1 file changed, 96 insertions(+), 20 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 90508aa1..7d0cbf3a 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -4,8 +4,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Introduction\n",
-    "### The concept of adversarial search and an introduction to Game Theory\n",
+    "<font size = 7 color=red>\n",
+    "<b>Introduction</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>The concept of adversarial search and an introduction to Game Theory</b>\n",
+    "</font>\n",
+    "\n",
     "In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search is introduced which is called **Adversarial Search**. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.\n",
     "To gather more information about the concept of adversarial search, visit <a href=\"https://www.techslang.com/definition/what-is-adversarial-search/\">this link</a>."
    ]
@@ -14,7 +20,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Game Theory Explanation and Deterministic Games\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b> Game Theory Explanation and Deterministic Games</b>\n",
+    "</font>\n",
+    "<br>\n",
     "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers.\n",
     "To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:\n",
     "<ul>\n",
@@ -34,7 +43,11 @@
     "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
     "</ul>\n",
     "\n",
-    "## Different catergories of games\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b> Different catergories of games</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
     "Before we discuss types of games, we need to define a concept named **Utility Function** first. Each player has some states that will be more willing in them. For example, if the player P has a State S which is more satisfied in it, this means that P has better moves in S toward the goal state and is able to win the game more easily. So each player has a utility function which indicates that in every state, how satisfied the player is and what is the best result that it can score, starting with that state. Naturally in goal states, which the game has been finished, the utility function is maximum for the player.\n",
     "<ul>\n",
     "    <li>Zero-Sum games</li>\n",
@@ -50,10 +63,22 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Differnt Types of Trees for games\n",
-    "## Single Agent Trees\n",
+    "<font size = 7 color=red>\n",
+    "<b>Differnt Types of Trees for games</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Single Agent Trees</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
     "Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.\n",
-    "## Adversarial Agent Trees\n",
+    "<br>\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Adversarial Agent Trees</b>\n",
+    "</font>\n",
+    "<br>\n",
     "Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.\n",
     "Like Single Agent Trees, we can form a tree to express the moves of the agents, but the differnce is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  "
    ]
@@ -62,7 +87,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Minimax Strategy\n",
+    "<font size = 7 color=red>\n",
+    "<b>Minimax Strategy</b>\n",
+    "</font>\n",
+    "<br>\n",
     "To gain the value of every node in Adversarial tree, we classify the nodes as two groups:\n",
     "<ol>\n",
     "    <li>The nodes our agent does action on them. The utility value of these kind of nodes is assigned by the maximum of the utility values of its successors. Because we want the maximum utility value for our agent. (The max side of the Strategy) </li>\n",
@@ -81,7 +109,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Pseudocode of Minimax Strategy\n",
+    "\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Pseudocode of Minimax Strategy</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
     "```python\n",
     "def max_value(state):\n",
     "    if leaf?(state), return U(state)\n",
@@ -104,7 +137,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Properties of Minimax Strategy\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Properties of Minimax Strategy</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
+    "\n",
     "### Complete:\n",
     "Yes, It is complete if the tree formed for the game is finite.\n",
     "\n",
@@ -122,7 +160,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Resource Limits\n",
+    "<font size = 7 color=red>\n",
+    "<b>Resource Limits</b>\n",
+    "</font>\n",
+    "<br>\n",
     "Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine is exponential in the depth of the tree.\n",
     "\n",
     "Consider a _reasonable_ chess game with $b \\approx 35$ and $m \\approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below."
@@ -132,7 +173,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Depth-limited search\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Depth-limited search</b>\n",
+    "</font>\n",
+    "<br>\n",
     "One idea might be running the algorithm up to a specified depth instead of the searching the whole tree which we call _depth-limited search_ .\n",
     "\n",
     "![depth-limited](./images/depth-limited.jpg)\n",
@@ -147,7 +191,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Evaluation Functions\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Evaluation Functions</b>\n",
+    "</font>\n",
+    "<br>\n",
     "The goal of an evaluation function is to answer to the question: _How good is the current position?_ Or _How probable is it to reach to winning terminal state from this position?_\n",
     "\n",
     "Let us make this idea more concrete. An evaluation function returns an _estimate_ of the utility in each game state, just like the _heuristic_ functions that estimate the remaining cost to a goal state which were discussed before.\n",
@@ -167,7 +214,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Iterative Deepening Search\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Iterative Deepening Search</b>\n",
+    "</font>\n",
+    "<br>\n",
     "The accuracy of the evaluation function is critical in shallower states. And the deeper in the tree we search, the less the quality of the evaluation function matters.\n",
     "So a good idea is to maximize the depth of our search as much as possible, considering the limits that we have. In other words, we are looking for an algorithm that can returns an acceptable solution whenever we ask. This kind of algorithms are called _anytime algorithms_. These algorithms are expected to find better and better solutions the longer they keeps running.\n",
     "\n",
@@ -179,7 +229,15 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Minimax Pruning\n",
+    "<font size = 7 color=red>\n",
+    "<b>Pruning</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Minimax Pruning</b>\n",
+    "</font>\n",
+    "<br>\n",
     "\n",
     "If we take a look at how minimax works, it is noticeable that it explores some parts of the tree that are not necessary at all. In other words, exploring some branches are totally useless but expensive. If we can predict which branches do not worth exploring and _prune_, we can improve minimax algorithm significantly.\n",
     "\n",
@@ -221,7 +279,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Alph-Beta Pruning\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Alpha-Beta Pruning</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
     "\n",
     "As we are trying to improve our algorithm even more, there is another approach to prune branches that cannot possibly influence the final decision. So in the last section, we talked about Minimax Pruning, but how can we implement it? \n",
     "\n",
@@ -325,7 +387,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Other types of games:\n",
+    "<font size = 7 color=red>\n",
+    "<b>Other types of games</b>\n",
+    "</font>\n",
+    "<br>\n",
     "On top of what's been discussed before, we can count another type of game called **Mixed Layer Types**. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game.\n",
     "\n",
     "To do some calculations about the depth of the game tree, we can state:\n",
@@ -345,7 +410,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Multi-Agent Utilities\n",
+    "<font size = 5 color=#FF7500>\n",
+    "<b>Multi-Agent Utilities</b>\n",
+    "</font>\n",
+    "<br>\n",
     "In games with multiple players, terminal states have utility in form of tuple values and each player tries to maximize its own part. This idea can raise the cooperation and competetion strategies dynamically."
    ]
   },
@@ -353,7 +421,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Summary\n",
+    "<font size = 7 color=red>\n",
+    "<b>Summary</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
     "Before this, we only discussed games with a single player. But we use Adversarial search when there are other players and agents in the game too, often with opposite objectives. With adversarial search, we are seeking a method called **Playing Strategy** that gives us correct directions and instructions at each state of game.\n",
     "\n",
     "We classified games in two main groups based on utility functions:\n",
@@ -376,7 +448,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## References\n",
+    "<font size = 7 color=red>\n",
+    "<b>References</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
     "\n",
     "+ https://towardsdatascience.com/understanding-the-minimax-algorithm-726582e4f2c6\n",
     "+ Russell, S. J., Norvig, P., &amp; Davis, E. (3rd. Ed). Artificial Intelligence: A modern approach. Pearson Educación. \n",

From 84ff450ef609370ddfae2accc60d5803c74fc80a Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Fri, 29 Oct 2021 22:50:04 +0330
Subject: [PATCH 18/62] new colors

---
 .../adversarial_search_fall_2021/index.ipynb  | 29 ++++++++++++++-----
 1 file changed, 22 insertions(+), 7 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 7d0cbf3a..bbdbb18d 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -141,19 +141,34 @@
     "<b>Properties of Minimax Strategy</b>\n",
     "</font>\n",
     "<br>\n",
+    "<br>\n",
     "\n",
     "\n",
-    "### Complete:\n",
+    "<font size = 3 color=#green>\n",
+    "<b>Complete:</b>\n",
+    "</font>\n",
+    "<br>\n",
     "Yes, It is complete if the tree formed for the game is finite.\n",
+    "<br>\n",
     "\n",
-    "### Optimal:\n",
+    "<font size = 3 color=#green>\n",
+    "<b>Optimal:</b>\n",
+    "</font>\n",
+    "<br>\n",
     "Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.\n",
-    "    \n",
-    "### Time Complexity:\n",
+    "<br>\n",
+    "<font size = 3 color=#green>\n",
+    "<b>Time Complexity:</b>\n",
+    "</font>\n",
+    "<br>\n",
     "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.\n",
-    "\n",
-    "### Space Complexity:\n",
-    "$O(bm)$, because it's like DFS and only stores the nodes on the current path.\n"
+    "<br>\n",
+    "<font size = 3 color=#green>\n",
+    "<b>Space Complexity:</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "$O(bm)$, because it's like DFS and only stores the nodes on the current path.\n",
+    "\n"
    ]
   },
   {

From f7f93076dfdb82dd31b1966344d958121498d64b Mon Sep 17 00:00:00 2001
From: Mohammad Abolnejadian <mohammad1379_a@yahoo.com>
Date: Sat, 30 Oct 2021 00:06:31 +0330
Subject: [PATCH 19/62] expectimax search

---
 .../images/expectimax_tree.png                | Bin 0 -> 17632 bytes
 .../images/snakes_ladders_img.png             | Bin 0 -> 95644 bytes
 .../adversarial_search_fall_2021/index.ipynb  | 129 ++++++++++++++----
 3 files changed, 104 insertions(+), 25 deletions(-)
 create mode 100644 notebooks/adversarial_search_fall_2021/images/expectimax_tree.png
 create mode 100644 notebooks/adversarial_search_fall_2021/images/snakes_ladders_img.png

diff --git a/notebooks/adversarial_search_fall_2021/images/expectimax_tree.png b/notebooks/adversarial_search_fall_2021/images/expectimax_tree.png
new file mode 100644
index 0000000000000000000000000000000000000000..32cb57339bb51bf706eae47474b58481e7332838
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diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index bbdbb18d..f40c3d52 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -4,11 +4,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
     "<b>Introduction</b>\n",
     "</font>\n",
     "<br>\n",
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>The concept of adversarial search and an introduction to Game Theory</b>\n",
     "</font>\n",
     "\n",
@@ -20,7 +20,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b> Game Theory Explanation and Deterministic Games</b>\n",
     "</font>\n",
     "<br>\n",
@@ -43,7 +43,7 @@
     "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
     "</ul>\n",
     "\n",
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b> Different catergories of games</b>\n",
     "</font>\n",
     "<br>\n",
@@ -63,19 +63,19 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
     "<b>Differnt Types of Trees for games</b>\n",
     "</font>\n",
     "<br>\n",
     "\n",
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Single Agent Trees</b>\n",
     "</font>\n",
     "<br>\n",
     "\n",
     "Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.\n",
     "<br>\n",
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Adversarial Agent Trees</b>\n",
     "</font>\n",
     "<br>\n",
@@ -87,7 +87,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
     "<b>Minimax Strategy</b>\n",
     "</font>\n",
     "<br>\n",
@@ -110,7 +110,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Pseudocode of Minimax Strategy</b>\n",
     "</font>\n",
     "<br>\n",
@@ -137,33 +137,33 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Properties of Minimax Strategy</b>\n",
     "</font>\n",
     "<br>\n",
     "<br>\n",
     "\n",
     "\n",
-    "<font size = 3 color=#green>\n",
+    "<font size = 2 color=#green>\n",
     "<b>Complete:</b>\n",
     "</font>\n",
     "<br>\n",
     "Yes, It is complete if the tree formed for the game is finite.\n",
     "<br>\n",
     "\n",
-    "<font size = 3 color=#green>\n",
+    "<font size = 2 color=#green>\n",
     "<b>Optimal:</b>\n",
     "</font>\n",
     "<br>\n",
     "Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.\n",
     "<br>\n",
-    "<font size = 3 color=#green>\n",
+    "<font size = 2 color=#green>\n",
     "<b>Time Complexity:</b>\n",
     "</font>\n",
     "<br>\n",
     "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.\n",
     "<br>\n",
-    "<font size = 3 color=#green>\n",
+    "<font size = 2 color=#green>\n",
     "<b>Space Complexity:</b>\n",
     "</font>\n",
     "<br>\n",
@@ -175,7 +175,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
     "<b>Resource Limits</b>\n",
     "</font>\n",
     "<br>\n",
@@ -188,7 +188,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Depth-limited search</b>\n",
     "</font>\n",
     "<br>\n",
@@ -206,7 +206,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Evaluation Functions</b>\n",
     "</font>\n",
     "<br>\n",
@@ -229,7 +229,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Iterative Deepening Search</b>\n",
     "</font>\n",
     "<br>\n",
@@ -244,12 +244,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
     "<b>Pruning</b>\n",
     "</font>\n",
     "<br>\n",
     "\n",
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Minimax Pruning</b>\n",
     "</font>\n",
     "<br>\n",
@@ -294,7 +294,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Alpha-Beta Pruning</b>\n",
     "</font>\n",
     "<br>\n",
@@ -402,7 +402,86 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
+    "<b>Expectimax Search</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
+    "\n",
+    "As we can see in the real world, not all the games are played only based on players' skills. We have so many games with stochastic variables in them, like throwing a dice or picking a random card. These games are called **stochastic games**.\n",
+    "\n",
+    "Even when there is no element of luck, we can expect unpredicted behaviors from our agent or the opponent's agent, like robot slipping on the wet floor, or the opponent taking a worse route.\n",
+    "\n",
+    "So the problem we face here is that while we know what legal actions we can take, we can't afford to know what moves will the opponent be able to take as takes its random element into use.(i.e. The transition function will be variable as the game goes on.).\n",
+    "\n",
+    "Let's introduce a new definition called **chance nodes**. We represent these kind of nodes as a circle in the game tree.\n",
+    "\n",
+    "Let's look at this with a example from the epic, legendary game, Snakes And Ladders."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<p align=\"center\">\n",
+    "    <img src=\"./images/snakes_ladders_img.png\" alt=\"snakes_ladders\" width=\"50%\" height=\"50%\"/></br>\n",
+    "    <em>This games is played by two or more players. Each can throw a dice at its turn and should move as many as the dice number.</em></br>\n",
+    "</p>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "So in this game, there is an element of chance. if the opponent rolls the dice on 1, it can move one square, if it rolls the dice on 2, it can move two squares, and so on. So the main point here is that the action the opponent can make depends on a random number from dice which its probability is 1/6 at a time.\n",
+    "\n",
+    "So the new game tree will be something like this:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<p align=\"center\">\n",
+    "    <img src=\"./images/expectimax_tree.png\" alt=\"expectimax_tree\" width=\"50%\" height=\"50%\"/></br>\n",
+    "    <em>The minimal expectimax tree for snakes ladder game.</em></br>\n",
+    "</p>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "<font size = 3 color=#FF7500>\n",
+    "<b>Node values in Expectimax Search:</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
+    "As we deal with probability in this kind of search so much, we expect to calculate the values based on some kind of probabilty.\n",
+    "\n",
+    "Terminal nodes and MAX and MIN nodes (for which the dice roll is known) work exactly the same way as before. So how should we calculate the value of chance nodes?\n",
+    "\n",
+    "For every chance node, the value is calculated from the expected value of its childer. \n",
+    "For example the value of every chance node in Snake ladder will be:\n",
+    "\n",
+    "$1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 * 6$\n",
+    "\n",
+    "So the general formula for values will be as follows:\n",
+    "\n",
+    "| value   | condition |\n",
+    "|------|--------|\n",
+    "| UTILITY(s) | s = terminal node |\n",
+    "|max EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MAX|\n",
+    "|min EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MIN|\n",
+    "| $\\sum_{r}$ P(r)EXPECTIMAX(RESULT(s,r)) | PLAYER(s)= CHANCE|"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<font size = 5 color=red>\n",
     "<b>Other types of games</b>\n",
     "</font>\n",
     "<br>\n",
@@ -425,7 +504,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 5 color=#FF7500>\n",
+    "<font size = 3 color=#FF7500>\n",
     "<b>Multi-Agent Utilities</b>\n",
     "</font>\n",
     "<br>\n",
@@ -436,7 +515,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
     "<b>Summary</b>\n",
     "</font>\n",
     "<br>\n",
@@ -463,7 +542,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "<font size = 7 color=red>\n",
+    "<font size = 5 color=red>\n",
     "<b>References</b>\n",
     "</font>\n",
     "<br>\n",

From 84742bc588f7a651416882349b56764b257a8163 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Tue, 9 Nov 2021 21:32:38 +0330
Subject: [PATCH 20/62] Table of contents added

---
 .../adversarial_search_fall_2021/index.ipynb  | 134 +++++++++++++++++-
 1 file changed, 133 insertions(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index f40c3d52..8b2eb548 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -1,5 +1,137 @@
 {
  "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<font size = 5 color=red>\n",
+    "<b>Table of Contents</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
+    "---\n",
+    "\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Introduction</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>The concept of adversarial search and an introduction to Game Theory</b>\n",
+    "    </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Game Theory Explanation and Deterministic Games</b>\n",
+    "    </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Different catergories of games</b>\n",
+    "    </font>\n",
+    "  \n",
+    "  <br>\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Different Types of Trees for games</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Single-Agent Trees</b>\n",
+    "    </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Adverserial Agent Trees</b>\n",
+    "    </font>\n",
+    "\n",
+    "  <br>\n",
+    "\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Minimax Strategy</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Pseudocode of Minimax Strategy</b>\n",
+    "    </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Properties of Minimax Strategy</b>\n",
+    "    </font>\n",
+    "\n",
+    "  <br>\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Resource Limit</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Depth-limited Search</b>\n",
+    "    </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Evaluation Functions</b>\n",
+    "    </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Iterative Deepening Search</b>\n",
+    "    </font>\n",
+    "  \n",
+    "  <br>\n",
+    "\n",
+    "\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Pruning</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Minimax Pruning</b>\n",
+    "    </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Alpha-Beta Pruning</b>\n",
+    "    </font>\n",
+    "  \n",
+    "  <br>\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Expectimax Search</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Node Values in Expectimax Search</b>\n",
+    "    </font>\n",
+    "  \n",
+    "  <br>\n",
+    "\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Other Types of Games</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Multi-Agent Utilities</b>\n",
+    "    </font>\n",
+    "\n",
+    "  <br>\n",
+    "\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Summary</b>\n",
+    "  </font>\n",
+    "  \n",
+    "  <br>\n",
+    "\n",
+    "+ <font size = 4.5 color=Orange>\n",
+    "  <b>Reference</b>\n",
+    "  </font>\n",
+    "\n",
+    "  <br>\n",
+    "\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -192,7 +324,7 @@
     "<b>Depth-limited search</b>\n",
     "</font>\n",
     "<br>\n",
-    "One idea might be running the algorithm up to a specified depth instead of the searching the whole tree which we call _depth-limited search_ .\n",
+    "One idea might be running the algorithm up to a specified depth instead of searching the whole tree, which we call <b>depth-limited search</b> .\n",
     "\n",
     "![depth-limited](./images/depth-limited.jpg)\n",
     "\n",

From 94dc1fc57983d597e327230d1d0fe170b0602c78 Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Tue, 9 Nov 2021 22:24:34 +0330
Subject: [PATCH 21/62] Some Modifications

---
 .../adversarial_search_fall_2021/index.ipynb  | 71 +++++++++++++------
 1 file changed, 49 insertions(+), 22 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 8b2eb548..53e61f4a 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -144,7 +144,7 @@
     "<b>The concept of adversarial search and an introduction to Game Theory</b>\n",
     "</font>\n",
     "\n",
-    "In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search is introduced which is called **Adversarial Search**. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.\n",
+    "In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search called **Adversarial Search** is introduced. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.\n",
     "To gather more information about the concept of adversarial search, visit <a href=\"https://www.techslang.com/definition/what-is-adversarial-search/\">this link</a>."
    ]
   },
@@ -158,29 +158,50 @@
     "<br>\n",
     "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers.\n",
     "To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:\n",
-    "<ul>\n",
-    "    <li>States of the game: $S_i$, starting with $S_0$</li>\n",
-    "Which means how the game will look after each action of competitors.\n",
-    "    <li>Players: P = {1,2,...,n}</li>\n",
-    "represents number of agents playing the game which varies from 1 to n.\n",
-    "    <li>Actions: $A_i$</li>\n",
-    "In every state, each player has a set of legitimate moves to change that state to another state by playing its turn.\n",
-    "    <li>Transition functions: S x A -> S</li>\n",
-    "Transition function is a function which takes a state S, and does the action A to that state, and returns another state which is built as a consequence of that action.\n",
-    "    <li>Terminal tests: S -> {True, False}</li>\n",
-    "It is a boolean function which determines whether our game has reached a final state or not.\n",
-    "    <li>Terminal utilities: S ✕ P -> R </li>\n",
-    "It takes a state S, and a player P, and returns a Real number which indicates the score of the player P (The utility of the P, to be more precise) till that moment.\n",
-    "    <li> Policy/Strategy: S -> A </li>\n",
-    "The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does action A which is the best action for that situation. \n",
-    "</ul>\n",
+    "\n",
+    "\n",
+    "\n",
+    "1. States of the game: $S_i$, starting with $S_0$\\\n",
+    "    Which means how the game will look after each action of competitors.\n",
+    "    <br>\n",
+    "<br>\n",
+    "\n",
+    "2. Players: $P = ${$1,2,...,n$}\\\n",
+    "    represents the agents playing the game where the index varies from 1 to n.\n",
+    "    <br>\n",
+    "<br>\n",
+    "\n",
+    "3. Actions: $A_i$\\\n",
+    "    In every state, each player has a set of legitimate moves to change that state to another state by playing its turn.\n",
+    "    <br>\n",
+    "<br>\n",
+    "\n",
+    "4. Transition functions: S x A -> S\\\n",
+    "    Transition function is a function which takes a state S, and does the action A to that state, and returns another state which is built as a consequence of that action.\n",
+    "    <br>\n",
+    "<br>\n",
+    "\n",
+    "5. Terminal tests: S -> {True, False}\\\n",
+    "    It is a boolean function which determines whether our game has reached a final state or not.\n",
+    "    <br>\n",
+    "<br>\n",
+    "\n",
+    "6. Terminal utilities: S ✕ P -> R\\\n",
+    "    It takes a state S, and a player P, and returns a Real number which indicates the score of the player P (The utility of the P, to be more precise) till that moment.\n",
+    "    <br>\n",
+    "<br>\n",
+    "\n",
+    "7. Policy/Strategy: S -> A <br>\n",
+    "    The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does the action A which is the best action for that situation. \n",
+    "    <br>\n",
+    "<br>\n",
     "\n",
     "<font size = 3 color=#FF7500>\n",
     "<b> Different catergories of games</b>\n",
     "</font>\n",
     "<br>\n",
     "\n",
-    "Before we discuss types of games, we need to define a concept named **Utility Function** first. Each player has some states that will be more willing in them. For example, if the player P has a State S which is more satisfied in it, this means that P has better moves in S toward the goal state and is able to win the game more easily. So each player has a utility function which indicates that in every state, how satisfied the player is and what is the best result that it can score, starting with that state. Naturally in goal states, which the game has been finished, the utility function is maximum for the player.\n",
+    "Before we discuss types of games, we need to define a concept named **Utility Function** first. Each player has some states that will be more willing in them. For example, if the player P has a State S which is more satisfied in it, this means that P has better moves in S toward the goal state and is able to win the game more easily. So each player has a utility function which indicates that in every state, how satisfied the player is and what is the best result that it can score, starting with that state. Naturally in goal states, which the game has been finished, the utility function is maximum for that player.\n",
     "<ul>\n",
     "    <li>Zero-Sum games</li>\n",
     "    In Zero-Sum games, the sum of utility functions for all the agents is equal to zero. For instance, in a two-agent game, if the player P1 has the utility finction with the value of 10 in an specific state, the player P2 has a utility function with the value -10 in that state. So this game becomes totally adversarial and competetive in a way that we have a single value for utility function and one of the players wants to maximize that value and another one wants to minimize it.\n",
@@ -212,7 +233,7 @@
     "</font>\n",
     "<br>\n",
     "Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.\n",
-    "Like Single Agent Trees, we can form a tree to express the moves of the agents, but the differnce is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  "
+    "Like Single Agent Trees, we can form a tree to express the moves of the agents, but the difference is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  "
    ]
   },
   {
@@ -223,12 +244,12 @@
     "<b>Minimax Strategy</b>\n",
     "</font>\n",
     "<br>\n",
-    "To gain the value of every node in Adversarial tree, we classify the nodes as two groups:\n",
+    "To gain the value of every node in Adversarial tree, we classify the nodes into two groups:\n",
     "<ol>\n",
     "    <li>The nodes our agent does action on them. The utility value of these kind of nodes is assigned by the maximum of the utility values of its successors. Because we want the maximum utility value for our agent. (The max side of the Strategy) </li>\n",
     "    <li>The nodes that the adversial agent does action on them. The utility value of these kind of nodes is assigned by the minimum of the utility values of its successors. Because we want the minimum utility value for the adversary agent. (The min side of the Strategy) </li>\n",
     "</ol>\n",
-    "Now that we have every node's utlity value, we are able to find the best action for each state. The main idea is at each state, we choose the node with the greatest minimax value and move to it.\n",
+    "Now that we have every node's utility value, we are able to find the best action for each state. The main idea is at each state, we choose the node with the greatest minimax value and move to it.\n",
     "Why does this strategy work? Because while forming the tree and assigning the utility function, we've always considered the best move for the enemy in his/her point of view and the worst move in our point of view. So we assumed that the enemy always plays its best shot. Now Two situations are possible to emerge:\n",
     "<ol>\n",
     "    <li>The adversary agent is very smart. In this case, we are ready beacuse the strategy is based on this situation.</li>\n",
@@ -279,26 +300,32 @@
     "<font size = 2 color=#green>\n",
     "<b>Complete:</b>\n",
     "</font>\n",
+    "\n",
     "<br>\n",
+    "\n",
     "Yes, It is complete if the tree formed for the game is finite.\n",
-    "<br>\n",
     "\n",
     "<font size = 2 color=#green>\n",
     "<b>Optimal:</b>\n",
     "</font>\n",
     "<br>\n",
+    "\n",
     "Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.\n",
     "<br>\n",
+    "\n",
     "<font size = 2 color=#green>\n",
     "<b>Time Complexity:</b>\n",
     "</font>\n",
     "<br>\n",
+    "\n",
     "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.\n",
     "<br>\n",
+    "\n",
     "<font size = 2 color=#green>\n",
     "<b>Space Complexity:</b>\n",
     "</font>\n",
     "<br>\n",
+    "\n",
     "$O(bm)$, because it's like DFS and only stores the nodes on the current path.\n",
     "\n"
    ]

From 7979b87e67202949021386e78e3a1c1f172e6d1c Mon Sep 17 00:00:00 2001
From: MoaliMkh <mohammadali1379@gmail.com>
Date: Tue, 9 Nov 2021 23:46:32 +0330
Subject: [PATCH 22/62] Reviewed The grammar for the first time

---
 .../adversarial_search_fall_2021/index.ipynb  | 32 +++++++++----------
 1 file changed, 16 insertions(+), 16 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 53e61f4a..0c245b9f 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -338,7 +338,7 @@
     "<b>Resource Limits</b>\n",
     "</font>\n",
     "<br>\n",
-    "Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine is exponential in the depth of the tree.\n",
+    "Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine, is exponential in relation to the depth of the tree.\n",
     "\n",
     "Consider a _reasonable_ chess game with $b \\approx 35$ and $m \\approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below."
    ]
@@ -355,8 +355,8 @@
     "\n",
     "![depth-limited](./images/depth-limited.jpg)\n",
     "\n",
-    "But using this technique is not much satisfying because it leads to antoher issue: _How can we find the minimax value while there is no solution at a limited depth?_\n",
-    "Recall how we find the minimax value of each node in the original form. We continue searching until we reach a final state and then use recurision to calculate the parents' values. But in the _limited_ format, there is no goal states at depth $k < m$.\n",
+    "But using this technique is not much satisfying because it leads to another issue: _How can we find the minimax value while there is no solution at a limited depth?_\n",
+    "Recall how we find the minimax value of each node in the original form. We continue searching until we reach a final state and then use recursion to calculate the parents' values. But in the _limited_ format, there are no goal states at depth $k < m$.\n",
     "\n",
     "Now to deal with this issue, we'll introduce _*Evaluation function*_."
    ]
@@ -369,17 +369,17 @@
     "<b>Evaluation Functions</b>\n",
     "</font>\n",
     "<br>\n",
-    "The goal of an evaluation function is to answer to the question: _How good is the current position?_ Or _How probable is it to reach to winning terminal state from this position?_\n",
+    "The goal of an evaluation function is to answer to the question: <b>How good is the current position?</b> Or <b>How probable is it to reach to winning terminal state from this position?</b>\n",
     "\n",
     "Let us make this idea more concrete. An evaluation function returns an _estimate_ of the utility in each game state, just like the _heuristic_ functions that estimate the remaining cost to a goal state which were discussed before.\n",
     "\n",
-    "Obviously, defining an appropriate and precise evaluation function is strongly effective on the player's performance. Actually an inaccurate evaluation function may lead to a lost position in the game.\n",
+    "Obviously, defining an appropriate and precise evaluation function is strongly effective on the player's performance. Actually an inaccurate evaluation function may lead to a losing position in the game.\n",
     "\n",
     "Most evaluation functions calculate the value of the position by using a _weighted sum of features_:\n",
     "\n",
     "$Eval(s) = w_1f_1(s) + w_2f_2(s) + ... + w_nf_n(s)$\n",
     "\n",
-    "Each of the $f_i$s is calculating a specific _feature_ of the state _s_.For instance, in chess, we would have features for the number of white pawns, black pawns, white queens, black queens, and etc. To better differentiate the effect of each feature, we multiply them by _weights_.\n",
+    "Each of the $f_i s$  is calculating a specific _feature_ of the state _s_.For instance, in chess, we would have features for the number of white pawns, black pawns, white queens, black queens, and etc. To differentiate the effect of each feature better, we multiply them by _weights_.\n",
     "\n",
     "Ideally, evaluation function returns exactly the minimax value of the current state.\n"
    ]
@@ -393,10 +393,10 @@
     "</font>\n",
     "<br>\n",
     "The accuracy of the evaluation function is critical in shallower states. And the deeper in the tree we search, the less the quality of the evaluation function matters.\n",
-    "So a good idea is to maximize the depth of our search as much as possible, considering the limits that we have. In other words, we are looking for an algorithm that can returns an acceptable solution whenever we ask. This kind of algorithms are called _anytime algorithms_. These algorithms are expected to find better and better solutions the longer they keeps running.\n",
+    "So a good idea is to maximize the depth of our search as much as possible, considering the limits that we have. In other words, we are looking for an algorithm that can return an acceptable solution whenever we ask. This kind of algorithms are called _anytime algorithms_. These algorithms are expected to find better and better solutions the longer they keep running.\n",
     "\n",
-    "So simply instead of running the depth-limited-search once, we start running the algorithm with an initial depth limit ($k$). Then after we found s policy, increase the depth limit ($k' > k$) and run the algorithm again with this new limit to find a better and more accurate solution. We continue the iteration until the time limit reaches.\n",
-    "This alogrithm is called _iterative-deepening search_.\n"
+    "So simply instead of running the depth-limited-search once, we start running the algorithm with an initial depth limit ($k$). Then after we found s policy, increase the depth limit ($k' > k$) and run the algorithm again with this new limit to find a better and more accurate solution. We continue the iteration until the program reaches the time limit.\n",
+    "This algorithm is called _iterative-deepening search_.\n"
    ]
   },
   {
@@ -413,9 +413,9 @@
     "</font>\n",
     "<br>\n",
     "\n",
-    "If we take a look at how minimax works, it is noticeable that it explores some parts of the tree that are not necessary at all. In other words, exploring some branches are totally useless but expensive. If we can predict which branches do not worth exploring and _prune_, we can improve minimax algorithm significantly.\n",
+    "If we take a look at how minimax works, it is noticeable that it explores some parts of the tree that are not necessary at all. In other words, exploring some branches are totally useless but time-consuming. If we can predict which branches are not worthy to be explored so that we can  _prune_ them, we can improve minimax algorithm significantly.\n",
     "\n",
-    "To better understand better, consider the following example:\n",
+    "To understand better, consider the following example:\n",
     "<div width=\"50%\" height=\"50%\">\n",
     "    <img src=\"./images/1.png\" alt=\"prunning1\" width=\"50%\" height=\"50%\"/>\n",
     "</div>\n",
@@ -440,7 +440,7 @@
     "    <img src=\"./images/4.png\" alt=\"prunning4\" width=\"50%\" height=\"50%\"/>\n",
     "</div>\n",
     "\n",
-    "Now, let's ask ourselves: _Is it necessary that we explore the rightmost node?_. The answer is * No *. Because we already know that the value of the root is at least 3, while the maximum value of the right child of root is 2. So we can easily ignore exploring the last node and find that the minimax value of the root is 3:\n",
+    "Now, let's ask ourselves: _Is it necessary that we explore the rightmost node?_. The answer is <b>No</b>. Because we already know that the value of the root is at least 3, while the maximum value of the right child of root is 2. So we can easily ignore exploring the last node and find that the minimax value of the root is 3:\n",
     "\n",
     "<div width=\"50%\" height=\"50%\">\n",
     "    <img src=\"./images/5.png\" alt=\"prunning5\" width=\"50%\" height=\"50%\"/>\n",
@@ -490,7 +490,7 @@
    "source": [
     "### Sample Code:\n",
     "\n",
-    "Here is a Python function to make you understand the concept even better."
+    "Here is a Python function to make you understand the concept better."
    ]
   },
   {
@@ -592,7 +592,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "So in this game, there is an element of chance. if the opponent rolls the dice on 1, it can move one square, if it rolls the dice on 2, it can move two squares, and so on. So the main point here is that the action the opponent can make depends on a random number from dice which its probability is 1/6 at a time.\n",
+    "So in this game, there is an element of chance. if the opponent rolls the dice on 1, it can move one square, if it rolls the dice on 2, it can move two squares, and so on. So the main point here is that the action the opponent can make depends on a random number from dice which its probability is $1/6$ at a time.\n",
     "\n",
     "So the new game tree will be something like this:"
    ]
@@ -644,7 +644,7 @@
     "<b>Other types of games</b>\n",
     "</font>\n",
     "<br>\n",
-    "On top of what's been discussed before, we can count another type of game called **Mixed Layer Types**. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game.\n",
+    "On top of what's been discussed before, we can count another type of game called <b>Mixed Layer Types</b>. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game.\n",
     "\n",
     "To do some calculations about the depth of the game tree, we can state:\n",
     "\n",
@@ -654,7 +654,7 @@
     "\n",
     "So if we want to build the tree till the depth of 2, we will have $20 * (21 * 20) ^ 3$ leaves and termainal states. As we go deeper in tree, this number increases.\n",
     "\n",
-    "As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and prunning becomes trickier. \n",
+    "As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and pruning becomes trickier. \n",
     "\n",
     "Td-Gammon is a computer backgammon program developed in 1992. It uses a 2-depth search, a very good evaluation function plus reinforcement learning and what we get, is a champion-class player! TD-Gammon was the first AI world champion in any game!"
    ]

From 8bb4eb5e1b13bd6e725c728a1025f24721a7fe7d Mon Sep 17 00:00:00 2001
From: mtndaghyani <matin.daghyani@gmail.com>
Date: Thu, 11 Nov 2021 18:08:52 +0330
Subject: [PATCH 23/62] Add useful links section

---
 .../adversarial_search_fall_2021/index.ipynb    | 17 +++++++++++++++++
 1 file changed, 17 insertions(+)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 0c245b9f..62fc1fca 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -697,6 +697,23 @@
     "Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search** which was explained before."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<font size = 5 color=red>\n",
+    "<b>Useful links</b>\n",
+    "</font>\n",
+    "<br>\n",
+    "\n",
+    "+ [Step-by-Step alpha-betta example](https://www.youtube.com/watch?v=xBXHtz4Gbdo)\n",
+    "+ [A simple overview of adversarial search with some examples](https://www.baeldung.com/cs/expectimax-search)\n",
+    "+ [More about adversarial search in Chess game](https://medium.com/@SereneBiologist/the-anatomy-of-a-chess-ai-2087d0d565)\n",
+    "+ [About minimax time complexity](https://stackoverflow.com/questions/2080050/how-do-we-determine-the-time-and-space-complexity-of-minmax)\n",
+    "\n",
+    "\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},

From dc57c75dc4cbd1be7f4d54f66f7b022a5d80f0b2 Mon Sep 17 00:00:00 2001
From: Mohammad Abolnejadian <mohammad1379_a@yahoo.com>
Date: Fri, 12 Nov 2021 20:55:49 +0330
Subject: [PATCH 24/62] Better connection to minimax pruning.

---
 .../adversarial_search_fall_2021/index.ipynb  | 21 +++++++++++++++++++
 1 file changed, 21 insertions(+)

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index 62fc1fca..a295e1dd 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -82,6 +82,10 @@
     "+ <font size = 4.5 color=Orange>\n",
     "  <b>Pruning</b>\n",
     "  </font>\n",
+    "\n",
+    "  - <font size = 3 color=green>\n",
+    "    <b>Houston, We Have A Problem!</b>\n",
+    "    </font>\n",
     "  \n",
     "  - <font size = 3 color=green>\n",
     "    <b>Minimax Pruning</b>\n",
@@ -406,8 +410,25 @@
     "<font size = 5 color=red>\n",
     "<b>Pruning</b>\n",
     "</font>\n",
+    "<br>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<font size = 3 color=#FF7500>\n",
+    "<b>Houston, We have a problem!</b>\n",
+    "</font>\n",
     "<br>\n",
     "\n",
+    "So far, we have seen some imporvements upon our original Adversarial Search algorithm. But the problem with minimax search is that the number of game states it has to examine is exponential in the depth of the tree. The logical way to tackle this problem is to cut off some nodes and not to see them at all. This brings us to the idea of \"Pruning\" which we use a lot when we can in AI algorithms. In the following chapter, we see two ways to prune our tree so we can solve our problem in less time."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "<font size = 3 color=#FF7500>\n",
     "<b>Minimax Pruning</b>\n",
     "</font>\n",

From 449bc7bd4ca7d768e95e921a7015877c1041852a Mon Sep 17 00:00:00 2001
From: Mohammad Abolnejadian <mohammad1379_a@yahoo.com>
Date: Fri, 12 Nov 2021 21:14:52 +0330
Subject: [PATCH 25/62] added images to better comprehend the problem.

---
 .../images/backgammon.jpeg                     | Bin 0 -> 405986 bytes
 .../images/zero-sum-game.jpeg                  | Bin 0 -> 55245 bytes
 .../adversarial_search_fall_2021/index.ipynb   |  16 +++++++++++++++-
 3 files changed, 15 insertions(+), 1 deletion(-)
 create mode 100644 notebooks/adversarial_search_fall_2021/images/backgammon.jpeg
 create mode 100644 notebooks/adversarial_search_fall_2021/images/zero-sum-game.jpeg

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diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
index a295e1dd..95f38253 100644
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ b/notebooks/adversarial_search_fall_2021/index.ipynb
@@ -209,6 +209,10 @@
     "<ul>\n",
     "    <li>Zero-Sum games</li>\n",
     "    In Zero-Sum games, the sum of utility functions for all the agents is equal to zero. For instance, in a two-agent game, if the player P1 has the utility finction with the value of 10 in an specific state, the player P2 has a utility function with the value -10 in that state. So this game becomes totally adversarial and competetive in a way that we have a single value for utility function and one of the players wants to maximize that value and another one wants to minimize it.\n",
+    "    <p align=\"center\">\n",
+    "    <img src=\"./images/backgammon.jpeg\" alt=\"backgammon\" width=\"50%\" height=\"50%\"/></br>\n",
+    "    <em>Backgammon, a classic zero-sum game.</em></br>\n",
+    "</p>\n",
     "    <li>General games</li>\n",
     "    In this type of games, each of the agents has its own utility function. These utility functions may be independant, so in some stages of the game, the agents may even have to cooperate in addition to competing and indifferenting to increase their own utility function value. This type of the game is more general and also complicated compared to Zero-Sum games.\n",
     "</ul>\n",
@@ -216,6 +220,16 @@
     "\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<p align=\"center\">\n",
+    "    <img src=\"./images/zero-sum-game.jpeg\" alt=\"zero-sum-game\" width=\"50%\" height=\"50%\"/></br>\n",
+    "    <em>A good example on zero-sum vs. non-zero-sum games. (Credit: Market Business News)</em></br>\n",
+    "</p>"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -422,7 +436,7 @@
     "</font>\n",
     "<br>\n",
     "\n",
-    "So far, we have seen some imporvements upon our original Adversarial Search algorithm. But the problem with minimax search is that the number of game states it has to examine is exponential in the depth of the tree. The logical way to tackle this problem is to cut off some nodes and not to see them at all. This brings us to the idea of \"Pruning\" which we use a lot when we can in AI algorithms. In the following chapter, we see two ways to prune our tree so we can solve our problem in less time."
+    "So far, we have seen some imporvements upon our original Adversarial Search algorithm. But the problem with minimax search is that the number of game states it has to examine is **exponential** in the depth of the tree. The logical way to tackle this problem is to cut off some nodes and not to see them at all. This brings us to the idea of \"Pruning\" which we use a lot when we can in AI algorithms. In the following chapter, we see two ways to prune our tree so we can solve our problem in less time."
    ]
   },
   {

From 528909fca8ec68a965cf385c6210760fd5d2d3db Mon Sep 17 00:00:00 2001
From: mtndaghyani <matin.daghyani@gmail.com>
Date: Fri, 12 Nov 2021 22:42:46 +0330
Subject: [PATCH 26/62] Add index.md

---
 .../adversarial_search_fall_2021/index.md     | 618 ++++++++++++++++++
 1 file changed, 618 insertions(+)
 create mode 100644 notebooks/adversarial_search_fall_2021/index.md

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
new file mode 100644
index 00000000..8cebc9c6
--- /dev/null
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -0,0 +1,618 @@
+<font size = 5 color=red>
+<b>Table of Contents</b>
+</font>
+<br>
+
+---
+
+
++ <font size = 4.5 color=Orange>
+  <b>Introduction</b>
+  </font>
+  
+  - <font size = 3 color=green>
+    <b>The concept of adversarial search and an introduction to Game Theory</b>
+    </font>
+
+  - <font size = 3 color=green>
+    <b>Game Theory Explanation and Deterministic Games</b>
+    </font>
+
+  - <font size = 3 color=green>
+    <b>Different catergories of games</b>
+    </font>
+  
+  <br>
+
++ <font size = 4.5 color=Orange>
+  <b>Different Types of Trees for games</b>
+  </font>
+  
+  - <font size = 3 color=green>
+    <b>Single-Agent Trees</b>
+    </font>
+
+  - <font size = 3 color=green>
+    <b>Adverserial Agent Trees</b>
+    </font>
+
+  <br>
+
+
++ <font size = 4.5 color=Orange>
+  <b>Minimax Strategy</b>
+  </font>
+  
+  - <font size = 3 color=green>
+    <b>Pseudocode of Minimax Strategy</b>
+    </font>
+
+  - <font size = 3 color=green>
+    <b>Properties of Minimax Strategy</b>
+    </font>
+
+  <br>
+
++ <font size = 4.5 color=Orange>
+  <b>Resource Limit</b>
+  </font>
+  
+  - <font size = 3 color=green>
+    <b>Depth-limited Search</b>
+    </font>
+
+  - <font size = 3 color=green>
+    <b>Evaluation Functions</b>
+    </font>
+
+  - <font size = 3 color=green>
+    <b>Iterative Deepening Search</b>
+    </font>
+  
+  <br>
+
+
+
++ <font size = 4.5 color=Orange>
+  <b>Pruning</b>
+  </font>
+
+  - <font size = 3 color=green>
+    <b>Houston, We Have A Problem!</b>
+    </font>
+  
+  - <font size = 3 color=green>
+    <b>Minimax Pruning</b>
+    </font>
+
+  - <font size = 3 color=green>
+    <b>Alpha-Beta Pruning</b>
+    </font>
+  
+  <br>
+
+
+
+
++ <font size = 4.5 color=Orange>
+  <b>Expectimax Search</b>
+  </font>
+  
+  - <font size = 3 color=green>
+    <b>Node Values in Expectimax Search</b>
+    </font>
+  
+  <br>
+
+
++ <font size = 4.5 color=Orange>
+  <b>Other Types of Games</b>
+  </font>
+  
+  - <font size = 3 color=green>
+    <b>Multi-Agent Utilities</b>
+    </font>
+
+  <br>
+
+
++ <font size = 4.5 color=Orange>
+  <b>Summary</b>
+  </font>
+  
+  <br>
+
++ <font size = 4.5 color=Orange>
+  <b>Reference</b>
+  </font>
+
+  <br>
+
+
+
+<font size = 5 color=red>
+<b>Introduction</b>
+</font>
+<br>
+<font size = 3 color=#FF7500>
+<b>The concept of adversarial search and an introduction to Game Theory</b>
+</font>
+
+In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search called **Adversarial Search** is introduced. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.
+To gather more information about the concept of adversarial search, visit <a href="https://www.techslang.com/definition/what-is-adversarial-search/">this link</a>.
+
+<font size = 3 color=#FF7500>
+<b> Game Theory Explanation and Deterministic Games</b>
+</font>
+<br>
+Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers.
+To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:
+
+
+
+1. States of the game: $S_i$, starting with $S_0$\
+    Which means how the game will look after each action of competitors.
+    <br>
+<br>
+
+2. Players: $P = ${$1,2,...,n$}\
+    represents the agents playing the game where the index varies from 1 to n.
+    <br>
+<br>
+
+3. Actions: $A_i$\
+    In every state, each player has a set of legitimate moves to change that state to another state by playing its turn.
+    <br>
+<br>
+
+4. Transition functions: S x A -> S\
+    Transition function is a function which takes a state S, and does the action A to that state, and returns another state which is built as a consequence of that action.
+    <br>
+<br>
+
+5. Terminal tests: S -> {True, False}\
+    It is a boolean function which determines whether our game has reached a final state or not.
+    <br>
+<br>
+
+6. Terminal utilities: S ✕ P -> R\
+    It takes a state S, and a player P, and returns a Real number which indicates the score of the player P (The utility of the P, to be more precise) till that moment.
+    <br>
+<br>
+
+7. Policy/Strategy: S -> A <br>
+    The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does the action A which is the best action for that situation. 
+    <br>
+<br>
+
+<font size = 3 color=#FF7500>
+<b> Different catergories of games</b>
+</font>
+<br>
+
+Before we discuss types of games, we need to define a concept named **Utility Function** first. Each player has some states that will be more willing in them. For example, if the player P has a State S which is more satisfied in it, this means that P has better moves in S toward the goal state and is able to win the game more easily. So each player has a utility function which indicates that in every state, how satisfied the player is and what is the best result that it can score, starting with that state. Naturally in goal states, which the game has been finished, the utility function is maximum for that player.
+<ul>
+    <li>Zero-Sum games</li>
+    In Zero-Sum games, the sum of utility functions for all the agents is equal to zero. For instance, in a two-agent game, if the player P1 has the utility finction with the value of 10 in an specific state, the player P2 has a utility function with the value -10 in that state. So this game becomes totally adversarial and competetive in a way that we have a single value for utility function and one of the players wants to maximize that value and another one wants to minimize it.
+    <p align="center">
+    <img src="./images/backgammon.jpeg" alt="backgammon" width="50%" height="50%"/></br>
+    <em>Backgammon, a classic zero-sum game.</em></br>
+</p>
+    <li>General games</li>
+    In this type of games, each of the agents has its own utility function. These utility functions may be independant, so in some stages of the game, the agents may even have to cooperate in addition to competing and indifferenting to increase their own utility function value. This type of the game is more general and also complicated compared to Zero-Sum games.
+</ul>
+ To know the game theory better, you can stop by <a href="https://en.wikipedia.org/wiki/Game_theory">here</a>.
+
+
+
+<p align="center">
+    <img src="./images/zero-sum-game.jpeg" alt="zero-sum-game" width="50%" height="50%"/></br>
+    <em>A good example on zero-sum vs. non-zero-sum games. (Credit: Market Business News)</em></br>
+</p>
+
+<font size = 5 color=red>
+<b>Differnt Types of Trees for games</b>
+</font>
+<br>
+
+<font size = 3 color=#FF7500>
+<b>Single Agent Trees</b>
+</font>
+<br>
+
+Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.
+<br>
+<font size = 3 color=#FF7500>
+<b>Adversarial Agent Trees</b>
+</font>
+<br>
+Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.
+Like Single Agent Trees, we can form a tree to express the moves of the agents, but the difference is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  
+
+<font size = 5 color=red>
+<b>Minimax Strategy</b>
+</font>
+<br>
+To gain the value of every node in Adversarial tree, we classify the nodes into two groups:
+<ol>
+    <li>The nodes our agent does action on them. The utility value of these kind of nodes is assigned by the maximum of the utility values of its successors. Because we want the maximum utility value for our agent. (The max side of the Strategy) </li>
+    <li>The nodes that the adversial agent does action on them. The utility value of these kind of nodes is assigned by the minimum of the utility values of its successors. Because we want the minimum utility value for the adversary agent. (The min side of the Strategy) </li>
+</ol>
+Now that we have every node's utility value, we are able to find the best action for each state. The main idea is at each state, we choose the node with the greatest minimax value and move to it.
+Why does this strategy work? Because while forming the tree and assigning the utility function, we've always considered the best move for the enemy in his/her point of view and the worst move in our point of view. So we assumed that the enemy always plays its best shot. Now Two situations are possible to emerge:
+<ol>
+    <li>The adversary agent is very smart. In this case, we are ready beacuse the strategy is based on this situation.</li>
+    <li>The adversary agent is too naive and may do some inoptimal actions. In this case, we may even score more than what we gained as utillty value.</li>
+</ol>
+As a conclusion of what's said before, we can say that the utility value that we calculate for nodes in minimax strategy, is a floor value of what we may actually score.
+
+
+<font size = 3 color=#FF7500>
+<b>Pseudocode of Minimax Strategy</b>
+</font>
+<br>
+
+```python
+def max_value(state):
+    if leaf?(state), return U(state)
+    initialize v = -∞
+    for each c in children(state):
+        v = max(v, min_value)
+    return v
+
+
+def min_value(state):
+    if leaf?(state), return U(state)
+    initialize v = ∞
+    for each c in children(state):
+        v = min(v, max_value)
+    return v
+```
+
+<font size = 3 color=#FF7500>
+<b>Properties of Minimax Strategy</b>
+</font>
+<br>
+<br>
+
+
+<font size = 2 color=#green>
+<b>Complete:</b>
+</font>
+
+<br>
+
+Yes, It is complete if the tree formed for the game is finite.
+
+<font size = 2 color=#green>
+<b>Optimal:</b>
+</font>
+<br>
+
+Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.
+<br>
+
+<font size = 2 color=#green>
+<b>Time Complexity:</b>
+</font>
+<br>
+
+$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.
+<br>
+
+<font size = 2 color=#green>
+<b>Space Complexity:</b>
+</font>
+<br>
+
+$O(bm)$, because it's like DFS and only stores the nodes on the current path.
+
+
+
+<font size = 5 color=red>
+<b>Resource Limits</b>
+</font>
+<br>
+Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine, is exponential in relation to the depth of the tree.
+
+Consider a _reasonable_ chess game with $b \approx 35$ and $m \approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below.
+
+<font size = 3 color=#FF7500>
+<b>Depth-limited search</b>
+</font>
+<br>
+One idea might be running the algorithm up to a specified depth instead of searching the whole tree, which we call <b>depth-limited search</b> .
+
+![depth-limited](./images/depth-limited.jpg)
+
+But using this technique is not much satisfying because it leads to another issue: _How can we find the minimax value while there is no solution at a limited depth?_
+Recall how we find the minimax value of each node in the original form. We continue searching until we reach a final state and then use recursion to calculate the parents' values. But in the _limited_ format, there are no goal states at depth $k < m$.
+
+Now to deal with this issue, we'll introduce _*Evaluation function*_.
+
+<font size = 3 color=#FF7500>
+<b>Evaluation Functions</b>
+</font>
+<br>
+The goal of an evaluation function is to answer to the question: <b>How good is the current position?</b> Or <b>How probable is it to reach to winning terminal state from this position?</b>
+
+Let us make this idea more concrete. An evaluation function returns an _estimate_ of the utility in each game state, just like the _heuristic_ functions that estimate the remaining cost to a goal state which were discussed before.
+
+Obviously, defining an appropriate and precise evaluation function is strongly effective on the player's performance. Actually an inaccurate evaluation function may lead to a losing position in the game.
+
+Most evaluation functions calculate the value of the position by using a _weighted sum of features_:
+
+$Eval(s) = w_1f_1(s) + w_2f_2(s) + ... + w_nf_n(s)$
+
+Each of the $f_i s$  is calculating a specific _feature_ of the state _s_.For instance, in chess, we would have features for the number of white pawns, black pawns, white queens, black queens, and etc. To differentiate the effect of each feature better, we multiply them by _weights_.
+
+Ideally, evaluation function returns exactly the minimax value of the current state.
+
+
+<font size = 3 color=#FF7500>
+<b>Iterative Deepening Search</b>
+</font>
+<br>
+The accuracy of the evaluation function is critical in shallower states. And the deeper in the tree we search, the less the quality of the evaluation function matters.
+So a good idea is to maximize the depth of our search as much as possible, considering the limits that we have. In other words, we are looking for an algorithm that can return an acceptable solution whenever we ask. This kind of algorithms are called _anytime algorithms_. These algorithms are expected to find better and better solutions the longer they keep running.
+
+So simply instead of running the depth-limited-search once, we start running the algorithm with an initial depth limit ($k$). Then after we found s policy, increase the depth limit ($k' > k$) and run the algorithm again with this new limit to find a better and more accurate solution. We continue the iteration until the program reaches the time limit.
+This algorithm is called _iterative-deepening search_.
+
+
+<font size = 5 color=red>
+<b>Pruning</b>
+</font>
+<br>
+
+<font size = 3 color=#FF7500>
+<b>Houston, We have a problem!</b>
+</font>
+<br>
+
+So far, we have seen some imporvements upon our original Adversarial Search algorithm. But the problem with minimax search is that the number of game states it has to examine is **exponential** in the depth of the tree. The logical way to tackle this problem is to cut off some nodes and not to see them at all. This brings us to the idea of "Pruning" which we use a lot when we can in AI algorithms. In the following chapter, we see two ways to prune our tree so we can solve our problem in less time.
+
+<font size = 3 color=#FF7500>
+<b>Minimax Pruning</b>
+</font>
+<br>
+
+If we take a look at how minimax works, it is noticeable that it explores some parts of the tree that are not necessary at all. In other words, exploring some branches are totally useless but time-consuming. If we can predict which branches are not worthy to be explored so that we can  _prune_ them, we can improve minimax algorithm significantly.
+
+To understand better, consider the following example:
+<div width="50%" height="50%">
+    <img src="./images/1.png" alt="prunning1" width="50%" height="50%"/>
+</div>
+
+We start searching with respect to the assumption that we visit leftmost nodes first. Finally when we reach the terminal node with utility 4, _we can deduce that the value of it's parent node(which is a *Min* node) is definitely less than or equal to 4_:
+
+<div width="50%" height="50%">
+    <img src="./images/2.png" alt="prunning2" width="50%" height="50%"/>
+</div>
+
+At this point, this information doesn't help us a lot. So let's continue searching. When we visit the terminal node with utility 3, we find that the value for the parent is also 3. An important observation here is that we can _predict_ that the value of the root _is more than or equal to 3_. That's because the root is a * Max * node.
+
+
+<div width="50%" height="50%">
+    <img src="./images/3.png" alt="prunning3" width="50%" height="50%"/>
+</div>
+
+
+After that, we continue searching, a terminal leaf with utility 2 will be discovered, and just like before, we can immediately notice that the value of it's parent is _less than or equal to 2_.
+
+<div width="50%" height="50%">
+    <img src="./images/4.png" alt="prunning4" width="50%" height="50%"/>
+</div>
+
+Now, let's ask ourselves: _Is it necessary that we explore the rightmost node?_. The answer is <b>No</b>. Because we already know that the value of the root is at least 3, while the maximum value of the right child of root is 2. So we can easily ignore exploring the last node and find that the minimax value of the root is 3:
+
+<div width="50%" height="50%">
+    <img src="./images/5.png" alt="prunning5" width="50%" height="50%"/>
+</div>
+
+This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called $\alpha-\beta $ _pruning_.
+
+<font size = 3 color=#FF7500>
+<b>Alpha-Beta Pruning</b>
+</font>
+<br>
+
+
+As we are trying to improve our algorithm even more, there is another approach to prune branches that cannot possibly influence the final decision. So in the last section, we talked about Minimax Pruning, but how can we implement it? 
+
+The approach we are trying to explain here as you can see in the title is called Alpha-Beta Pruning.
+
+The main idea here is that if a node (e.g. $m$) has a better choice than another node (e.g. $n$) or its parents, then that another node ($n$) will never be wanted to expanded. So we can prune it and never check its successor nodes.
+
+![depth-limited](./images/Alpha_Beta_Intro.png)</br>
+*As $m$ can give the root node a higher value, $n$ will be pruned.* 
+
+
+We do this by keeping track of two parameters:
+
++  $\alpha$: The value of best choice (i.e. highest value) we have found so far for a Maximizing Node (i.e. our player).
++  $\beta$: The value of best choice  (i.e. lowest value) we have found so far for a Minimizing Node (i.e opponent player).
+
+### Sample Code:
+
+Here is a Python function to make you understand the concept better.
+
+```python
+def alpha_beta_pruning(position, depth, alpha, beta, maximizing_player):
+    if depth == 0 or game_state(position) == GAMESTATE.GAMEOVER:
+        if terminal_utility(position):
+            return terminal_utility(position)
+        return eval_func(position)
+
+    if maximizing_player: #for maximizing player (our player)
+        max_eval = - math.inf
+        for child in position.children:
+            eval = alpha_beta_pruning(child, depth - 1, alpha, beta, false)
+            max_eval = max(max_eval, eval)
+            alpha = max(max_eval, eval)
+            if beta <= alpha:
+                break
+        return max_eval
+    else: #for minimizing player(opponent player)
+        min_eval = + math.inf
+        for child in position.children:
+            eval = alpha_beta_pruning(child, depth - 1,alpha, beta, true)
+            min_eval = min(min_eval, eval)
+            beta = min(beta, eval)
+            if beta <= alpha:
+                break
+        return min_eval
+```     
+
+
+So to initialize this function, first we need to give alpha the worst value for our player, which is $-inf$, and give beta the worst value for the opponent, which is $+inf$.
+
+```python
+    #initialization
+    alpha_beta_pruning(current_position, 0, - math.inf, + math.inf, true)
+```
+
+Now let's look at an example while running this code.
+
+<p align="center">
+    <img src="./images/alpha_beta_run_1.png" alt="alpha_beta_run_1" /></br>
+    <em>We approach the first leaf and since the parent is a Minimizing Node, its beta value gets updated</em></br>
+    <img src="./images/alpha_beta_run_2.png" alt="alpha_beta_run_2" /></br>
+    <em>Second leaf has a bigger value, so since beta always updates at a lower value, nothing changes here</em></br>
+    <img src="./images/alpha_beta_run_3.png" alt="alpha_beta_run_3" /></br>
+    <em>The algorithm goes back to the root's second child and then goes to the third leaf. Keep in mind that the parent's alpha and beta values always gets passed to their children.</em></br>
+    <img src="./images/alpha_beta_run_4.png" alt="alpha_beta_run_4" /></br>
+    <em>Now since beta is less than alpha, we know that our root's best possible value from this child will be a 3 which is lower than its other child (i.e. 5), so we prune the other leaf.</em></br>
+</p>
+
+
+
+
+<font size = 5 color=red>
+<b>Expectimax Search</b>
+</font>
+<br>
+
+
+As we can see in the real world, not all the games are played only based on players' skills. We have so many games with stochastic variables in them, like throwing a dice or picking a random card. These games are called **stochastic games**.
+
+Even when there is no element of luck, we can expect unpredicted behaviors from our agent or the opponent's agent, like robot slipping on the wet floor, or the opponent taking a worse route.
+
+So the problem we face here is that while we know what legal actions we can take, we can't afford to know what moves will the opponent be able to take as takes its random element into use.(i.e. The transition function will be variable as the game goes on.).
+
+Let's introduce a new definition called **chance nodes**. We represent these kind of nodes as a circle in the game tree.
+
+Let's look at this with a example from the epic, legendary game, Snakes And Ladders.
+
+<p align="center">
+    <img src="./images/snakes_ladders_img.png" alt="snakes_ladders" width="50%" height="50%"/></br>
+    <em>This games is played by two or more players. Each can throw a dice at its turn and should move as many as the dice number.</em></br>
+</p>
+
+So in this game, there is an element of chance. if the opponent rolls the dice on 1, it can move one square, if it rolls the dice on 2, it can move two squares, and so on. So the main point here is that the action the opponent can make depends on a random number from dice which its probability is $1/6$ at a time.
+
+So the new game tree will be something like this:
+
+<p align="center">
+    <img src="./images/expectimax_tree.png" alt="expectimax_tree" width="50%" height="50%"/></br>
+    <em>The minimal expectimax tree for snakes ladder game.</em></br>
+</p>
+
+
+<font size = 3 color=#FF7500>
+<b>Node values in Expectimax Search:</b>
+</font>
+<br>
+
+As we deal with probability in this kind of search so much, we expect to calculate the values based on some kind of probabilty.
+
+Terminal nodes and MAX and MIN nodes (for which the dice roll is known) work exactly the same way as before. So how should we calculate the value of chance nodes?
+
+For every chance node, the value is calculated from the expected value of its childer. 
+For example the value of every chance node in Snake ladder will be:
+
+$1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 * 6$
+
+So the general formula for values will be as follows:
+
+| value   | condition |
+|------|--------|
+| UTILITY(s) | s = terminal node |
+|max EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MAX|
+|min EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MIN|
+| $\sum_{r}$ P(r)EXPECTIMAX(RESULT(s,r)) | PLAYER(s)= CHANCE|
+
+<font size = 5 color=red>
+<b>Other types of games</b>
+</font>
+<br>
+On top of what's been discussed before, we can count another type of game called <b>Mixed Layer Types</b>. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game.
+
+To do some calculations about the depth of the game tree, we can state:
+
+$b$ = $21$ because 21 dintinct states are created as an outcome of rolling 2 dices.
+
+$legal moves \approx 20$ 
+
+So if we want to build the tree till the depth of 2, we will have $20 * (21 * 20) ^ 3$ leaves and termainal states. As we go deeper in tree, this number increases.
+
+As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and pruning becomes trickier. 
+
+Td-Gammon is a computer backgammon program developed in 1992. It uses a 2-depth search, a very good evaluation function plus reinforcement learning and what we get, is a champion-class player! TD-Gammon was the first AI world champion in any game!
+
+<font size = 3 color=#FF7500>
+<b>Multi-Agent Utilities</b>
+</font>
+<br>
+In games with multiple players, terminal states have utility in form of tuple values and each player tries to maximize its own part. This idea can raise the cooperation and competetion strategies dynamically.
+
+<font size = 5 color=red>
+<b>Summary</b>
+</font>
+<br>
+
+Before this, we only discussed games with a single player. But we use Adversarial search when there are other players and agents in the game too, often with opposite objectives. With adversarial search, we are seeking a method called **Playing Strategy** that gives us correct directions and instructions at each state of game.
+
+We classified games in two main groups based on utility functions:
+- Zero-Sum Games: The sum of agents' utility functions is equal to zero. (pure competition)
+- General Games:  independent utility functions. (cooperation may be beneficial)
+
+We formed trees for both single and adversarial agent games and we found a strategy to obtain the value of nodes in both situations.
+
+Optimal strategy against an opponent playing perfectly can be found using **Minimax strategy**. We can also determine a floor value of the score that can be achieved in every game.
+A complete search is often impossible in minimax because of resource limits. So There are two options:
+- Replace the utility function with an **Evaluation Function** that estimates utility. This function must be defined very precise, because the whole solution depends on this.
+- Do **Minimax Pruning**: While calculating minimax values, it's sometimes possible to detect and **prune** some suboptimal branches that we know we'll never visit them during the algorithm.
+
+Another point that has to be considered is that While we are looking for a good strategy of playing, we must neighter act Optimistically nor Pessimismtically, because they can both result in suboptimal behaviour so it's important to evaluate each problem realistically and choose our strategy in proportion of the adversarial agent's cleverness.
+
+Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search** which was explained before.
+
+<font size = 5 color=red>
+<b>Useful links</b>
+</font>
+<br>
+
++ [Step-by-Step alpha-betta example](https://www.youtube.com/watch?v=xBXHtz4Gbdo)
++ [A simple overview of adversarial search with some examples](https://www.baeldung.com/cs/expectimax-search)
++ [More about adversarial search in Chess game](https://medium.com/@SereneBiologist/the-anatomy-of-a-chess-ai-2087d0d565)
++ [About minimax time complexity](https://stackoverflow.com/questions/2080050/how-do-we-determine-the-time-and-space-complexity-of-minmax)
+
+
+
+
+<font size = 5 color=red>
+<b>References</b>
+</font>
+<br>
+
+
++ https://towardsdatascience.com/understanding-the-minimax-algorithm-726582e4f2c6
++ Russell, S. J., Norvig, P., &amp; Davis, E. (3rd. Ed). Artificial Intelligence: A modern approach. Pearson Educación. 
++ https://slideplayer.com/slide/4380560/
++ https://www.cs.cornell.edu/courses/cs312/2002sp/lectures/rec21.htm
++ https://www.techslang.com/definition/what-is-adversarial-search/
++ https://en.wikipedia.org/wiki/Game_theory
++ https://brilliant.org/wiki/minimax/#:~:text=In%20game%20theory%2C%20minimax%20is,payoff%20as%20large%20as%20possible.
++ https://en.wikipedia.org/wiki/Minimax
++ Sebastian Lague's Youtube channel

From 096056cafc500df76471d770e77d839d4a5082c8 Mon Sep 17 00:00:00 2001
From: Matin Daghyani <59438691+mtndaghyani@users.noreply.github.com>
Date: Fri, 12 Nov 2021 23:46:46 +0330
Subject: [PATCH 27/62] Fix index.md fonts and headers

---
 .../adversarial_search_fall_2021/index.md     | 127 +++++++-----------
 1 file changed, 47 insertions(+), 80 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 8cebc9c6..ee4d4507 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -1,13 +1,11 @@
-<font size = 5 color=red>
-<b>Table of Contents</b>
-</font>
-<br>
+
+# Table of Contents
 
 ---
 
 
-+ <font size = 4.5 color=Orange>
-  <b>Introduction</b>
++ ### <font size = 4.5 color=Orange>
+  ### Introduction
   </font>
   
   - <font size = 3 color=green>
@@ -22,10 +20,8 @@
     <b>Different catergories of games</b>
     </font>
   
-  <br>
-
-+ <font size = 4.5 color=Orange>
-  <b>Different Types of Trees for games</b>
++ ### <font size = 4.5 color=Orange>
+  ### Different Types of Trees for games
   </font>
   
   - <font size = 3 color=green>
@@ -36,11 +32,8 @@
     <b>Adverserial Agent Trees</b>
     </font>
 
-  <br>
-
-
-+ <font size = 4.5 color=Orange>
-  <b>Minimax Strategy</b>
++ ### <font size = 4.5 color=Orange>
+  ### Minimax Strategy
   </font>
   
   - <font size = 3 color=green>
@@ -51,10 +44,8 @@
     <b>Properties of Minimax Strategy</b>
     </font>
 
-  <br>
-
-+ <font size = 4.5 color=Orange>
-  <b>Resource Limit</b>
++ ### <font size = 4.5 color=Orange>
+  ### Resource Limit
   </font>
   
   - <font size = 3 color=green>
@@ -69,12 +60,8 @@
     <b>Iterative Deepening Search</b>
     </font>
   
-  <br>
-
-
-
-+ <font size = 4.5 color=Orange>
-  <b>Pruning</b>
++ ### <font size = 4.5 color=Orange>
+  ### Pruning
   </font>
 
   - <font size = 3 color=green>
@@ -89,67 +76,48 @@
     <b>Alpha-Beta Pruning</b>
     </font>
   
-  <br>
-
-
-
-
-+ <font size = 4.5 color=Orange>
-  <b>Expectimax Search</b>
++ ### <font size = 4.5 color=Orange>
+  ### Expectimax Search
   </font>
   
   - <font size = 3 color=green>
     <b>Node Values in Expectimax Search</b>
     </font>
   
-  <br>
-
-
-+ <font size = 4.5 color=Orange>
-  <b>Other Types of Games</b>
++ ### <font size = 4.5 color=Orange>
+  ### Other Types of Games
   </font>
   
   - <font size = 3 color=green>
     <b>Multi-Agent Utilities</b>
     </font>
 
-  <br>
-
-
-+ <font size = 4.5 color=Orange>
-  <b>Summary</b>
++ ### <font size = 4.5 color=Orange>
+  ### Conclusion
   </font>
-  
-  <br>
-
-+ <font size = 4.5 color=Orange>
-  <b>Reference</b>
++ ### <font size = 4.5 color=Orange>
+  ### Reference
   </font>
 
-  <br>
-
-
-
-<font size = 5 color=red>
-<b>Introduction</b>
+## <font size = 5 color=red>
+## Introduction
 </font>
 <br>
-<font size = 3 color=#FF7500>
-<b>The concept of adversarial search and an introduction to Game Theory</b>
+
+### <font size = 3 color=#FF7500>
+### The concept of adversarial search and an introduction to Game Theory
 </font>
 
 In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search called **Adversarial Search** is introduced. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.
 To gather more information about the concept of adversarial search, visit <a href="https://www.techslang.com/definition/what-is-adversarial-search/">this link</a>.
 
-<font size = 3 color=#FF7500>
-<b> Game Theory Explanation and Deterministic Games</b>
+### <font size = 3 color=#FF7500>
+### Game Theory Explanation and Deterministic Games
 </font>
 <br>
 Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers.
 To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:
 
-
-
 1. States of the game: $S_i$, starting with $S_0$\
     Which means how the game will look after each action of competitors.
     <br>
@@ -185,8 +153,8 @@ To achieve this goal, we need to express our game and its rules and actions in a
     <br>
 <br>
 
-<font size = 3 color=#FF7500>
-<b> Different catergories of games</b>
+### <font size = 3 color=#FF7500>
+### Different catergories of games
 </font>
 <br>
 
@@ -210,8 +178,8 @@ Before we discuss types of games, we need to define a concept named **Utility Fu
     <em>A good example on zero-sum vs. non-zero-sum games. (Credit: Market Business News)</em></br>
 </p>
 
-<font size = 5 color=red>
-<b>Differnt Types of Trees for games</b>
+## <font size = 5 color=red>
+## Differnt Types of Trees for games</b>
 </font>
 <br>
 
@@ -229,8 +197,8 @@ Consider the game Pacman which has only an agent playing the game. Starting from
 Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.
 Like Single Agent Trees, we can form a tree to express the moves of the agents, but the difference is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  
 
-<font size = 5 color=red>
-<b>Minimax Strategy</b>
+## <font size = 5 color=red>
+## Minimax Strategy
 </font>
 <br>
 To gain the value of every node in Adversarial tree, we classify the nodes into two groups:
@@ -309,8 +277,8 @@ $O(bm)$, because it's like DFS and only stores the nodes on the current path.
 
 
 
-<font size = 5 color=red>
-<b>Resource Limits</b>
+## <font size = 5 color=red>
+## Resource Limits
 </font>
 <br>
 Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine, is exponential in relation to the depth of the tree.
@@ -360,8 +328,8 @@ So simply instead of running the depth-limited-search once, we start running the
 This algorithm is called _iterative-deepening search_.
 
 
-<font size = 5 color=red>
-<b>Pruning</b>
+## <font size = 5 color=red>
+## Pruning 
 </font>
 <br>
 
@@ -488,8 +456,8 @@ Now let's look at an example while running this code.
 
 
 
-<font size = 5 color=red>
-<b>Expectimax Search</b>
+## <font size = 5 color=red>
+## Expectimax Search
 </font>
 <br>
 
@@ -542,8 +510,8 @@ So the general formula for values will be as follows:
 |min EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MIN|
 | $\sum_{r}$ P(r)EXPECTIMAX(RESULT(s,r)) | PLAYER(s)= CHANCE|
 
-<font size = 5 color=red>
-<b>Other types of games</b>
+## <font size = 5 color=red>
+## Other types of games 
 </font>
 <br>
 On top of what's been discussed before, we can count another type of game called <b>Mixed Layer Types</b>. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game.
@@ -566,8 +534,8 @@ Td-Gammon is a computer backgammon program developed in 1992. It uses a 2-depth
 <br>
 In games with multiple players, terminal states have utility in form of tuple values and each player tries to maximize its own part. This idea can raise the cooperation and competetion strategies dynamically.
 
-<font size = 5 color=red>
-<b>Summary</b>
+## <font size = 5 color=red>
+## Conclusion
 </font>
 <br>
 
@@ -588,10 +556,9 @@ Another point that has to be considered is that While we are looking for a good
 
 Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search** which was explained before.
 
-<font size = 5 color=red>
-<b>Useful links</b>
+## <font size = 5 color=red>
+## Useful links
 </font>
-<br>
 
 + [Step-by-Step alpha-betta example](https://www.youtube.com/watch?v=xBXHtz4Gbdo)
 + [A simple overview of adversarial search with some examples](https://www.baeldung.com/cs/expectimax-search)
@@ -601,8 +568,8 @@ Uncertain factors (like rolling dice) are represented by Expectation Nodes in a
 
 
 
-<font size = 5 color=red>
-<b>References</b>
+## <font size = 5 color=red>
+## References
 </font>
 <br>
 

From 515d8d9d57afd08a550f90d5f5cbdb3f300b90c3 Mon Sep 17 00:00:00 2001
From: Matin Daghyani <59438691+mtndaghyani@users.noreply.github.com>
Date: Fri, 12 Nov 2021 23:50:55 +0330
Subject: [PATCH 28/62] Delete index.ipynb

---
 .../adversarial_search_fall_2021/index.ipynb  | 797 ------------------
 1 file changed, 797 deletions(-)
 delete mode 100644 notebooks/adversarial_search_fall_2021/index.ipynb

diff --git a/notebooks/adversarial_search_fall_2021/index.ipynb b/notebooks/adversarial_search_fall_2021/index.ipynb
deleted file mode 100644
index 95f38253..00000000
--- a/notebooks/adversarial_search_fall_2021/index.ipynb
+++ /dev/null
@@ -1,797 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Table of Contents</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "---\n",
-    "\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Introduction</b>\n",
-    "  </font>\n",
-    "  \n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>The concept of adversarial search and an introduction to Game Theory</b>\n",
-    "    </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Game Theory Explanation and Deterministic Games</b>\n",
-    "    </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Different catergories of games</b>\n",
-    "    </font>\n",
-    "  \n",
-    "  <br>\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Different Types of Trees for games</b>\n",
-    "  </font>\n",
-    "  \n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Single-Agent Trees</b>\n",
-    "    </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Adverserial Agent Trees</b>\n",
-    "    </font>\n",
-    "\n",
-    "  <br>\n",
-    "\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Minimax Strategy</b>\n",
-    "  </font>\n",
-    "  \n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Pseudocode of Minimax Strategy</b>\n",
-    "    </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Properties of Minimax Strategy</b>\n",
-    "    </font>\n",
-    "\n",
-    "  <br>\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Resource Limit</b>\n",
-    "  </font>\n",
-    "  \n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Depth-limited Search</b>\n",
-    "    </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Evaluation Functions</b>\n",
-    "    </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Iterative Deepening Search</b>\n",
-    "    </font>\n",
-    "  \n",
-    "  <br>\n",
-    "\n",
-    "\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Pruning</b>\n",
-    "  </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Houston, We Have A Problem!</b>\n",
-    "    </font>\n",
-    "  \n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Minimax Pruning</b>\n",
-    "    </font>\n",
-    "\n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Alpha-Beta Pruning</b>\n",
-    "    </font>\n",
-    "  \n",
-    "  <br>\n",
-    "\n",
-    "\n",
-    "\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Expectimax Search</b>\n",
-    "  </font>\n",
-    "  \n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Node Values in Expectimax Search</b>\n",
-    "    </font>\n",
-    "  \n",
-    "  <br>\n",
-    "\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Other Types of Games</b>\n",
-    "  </font>\n",
-    "  \n",
-    "  - <font size = 3 color=green>\n",
-    "    <b>Multi-Agent Utilities</b>\n",
-    "    </font>\n",
-    "\n",
-    "  <br>\n",
-    "\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Summary</b>\n",
-    "  </font>\n",
-    "  \n",
-    "  <br>\n",
-    "\n",
-    "+ <font size = 4.5 color=Orange>\n",
-    "  <b>Reference</b>\n",
-    "  </font>\n",
-    "\n",
-    "  <br>\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Introduction</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "<font size = 3 color=#FF7500>\n",
-    "<b>The concept of adversarial search and an introduction to Game Theory</b>\n",
-    "</font>\n",
-    "\n",
-    "In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search called **Adversarial Search** is introduced. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.\n",
-    "To gather more information about the concept of adversarial search, visit <a href=\"https://www.techslang.com/definition/what-is-adversarial-search/\">this link</a>."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b> Game Theory Explanation and Deterministic Games</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers.\n",
-    "To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:\n",
-    "\n",
-    "\n",
-    "\n",
-    "1. States of the game: $S_i$, starting with $S_0$\\\n",
-    "    Which means how the game will look after each action of competitors.\n",
-    "    <br>\n",
-    "<br>\n",
-    "\n",
-    "2. Players: $P = ${$1,2,...,n$}\\\n",
-    "    represents the agents playing the game where the index varies from 1 to n.\n",
-    "    <br>\n",
-    "<br>\n",
-    "\n",
-    "3. Actions: $A_i$\\\n",
-    "    In every state, each player has a set of legitimate moves to change that state to another state by playing its turn.\n",
-    "    <br>\n",
-    "<br>\n",
-    "\n",
-    "4. Transition functions: S x A -> S\\\n",
-    "    Transition function is a function which takes a state S, and does the action A to that state, and returns another state which is built as a consequence of that action.\n",
-    "    <br>\n",
-    "<br>\n",
-    "\n",
-    "5. Terminal tests: S -> {True, False}\\\n",
-    "    It is a boolean function which determines whether our game has reached a final state or not.\n",
-    "    <br>\n",
-    "<br>\n",
-    "\n",
-    "6. Terminal utilities: S ✕ P -> R\\\n",
-    "    It takes a state S, and a player P, and returns a Real number which indicates the score of the player P (The utility of the P, to be more precise) till that moment.\n",
-    "    <br>\n",
-    "<br>\n",
-    "\n",
-    "7. Policy/Strategy: S -> A <br>\n",
-    "    The Policy is the strategy defined by our AI program. It takes a state S, and according to the program, does the action A which is the best action for that situation. \n",
-    "    <br>\n",
-    "<br>\n",
-    "\n",
-    "<font size = 3 color=#FF7500>\n",
-    "<b> Different catergories of games</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "Before we discuss types of games, we need to define a concept named **Utility Function** first. Each player has some states that will be more willing in them. For example, if the player P has a State S which is more satisfied in it, this means that P has better moves in S toward the goal state and is able to win the game more easily. So each player has a utility function which indicates that in every state, how satisfied the player is and what is the best result that it can score, starting with that state. Naturally in goal states, which the game has been finished, the utility function is maximum for that player.\n",
-    "<ul>\n",
-    "    <li>Zero-Sum games</li>\n",
-    "    In Zero-Sum games, the sum of utility functions for all the agents is equal to zero. For instance, in a two-agent game, if the player P1 has the utility finction with the value of 10 in an specific state, the player P2 has a utility function with the value -10 in that state. So this game becomes totally adversarial and competetive in a way that we have a single value for utility function and one of the players wants to maximize that value and another one wants to minimize it.\n",
-    "    <p align=\"center\">\n",
-    "    <img src=\"./images/backgammon.jpeg\" alt=\"backgammon\" width=\"50%\" height=\"50%\"/></br>\n",
-    "    <em>Backgammon, a classic zero-sum game.</em></br>\n",
-    "</p>\n",
-    "    <li>General games</li>\n",
-    "    In this type of games, each of the agents has its own utility function. These utility functions may be independant, so in some stages of the game, the agents may even have to cooperate in addition to competing and indifferenting to increase their own utility function value. This type of the game is more general and also complicated compared to Zero-Sum games.\n",
-    "</ul>\n",
-    " To know the game theory better, you can stop by <a href=\"https://en.wikipedia.org/wiki/Game_theory\">here</a>.\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<p align=\"center\">\n",
-    "    <img src=\"./images/zero-sum-game.jpeg\" alt=\"zero-sum-game\" width=\"50%\" height=\"50%\"/></br>\n",
-    "    <em>A good example on zero-sum vs. non-zero-sum games. (Credit: Market Business News)</em></br>\n",
-    "</p>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Differnt Types of Trees for games</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Single Agent Trees</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.\n",
-    "<br>\n",
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Adversarial Agent Trees</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.\n",
-    "Like Single Agent Trees, we can form a tree to express the moves of the agents, but the difference is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Minimax Strategy</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "To gain the value of every node in Adversarial tree, we classify the nodes into two groups:\n",
-    "<ol>\n",
-    "    <li>The nodes our agent does action on them. The utility value of these kind of nodes is assigned by the maximum of the utility values of its successors. Because we want the maximum utility value for our agent. (The max side of the Strategy) </li>\n",
-    "    <li>The nodes that the adversial agent does action on them. The utility value of these kind of nodes is assigned by the minimum of the utility values of its successors. Because we want the minimum utility value for the adversary agent. (The min side of the Strategy) </li>\n",
-    "</ol>\n",
-    "Now that we have every node's utility value, we are able to find the best action for each state. The main idea is at each state, we choose the node with the greatest minimax value and move to it.\n",
-    "Why does this strategy work? Because while forming the tree and assigning the utility function, we've always considered the best move for the enemy in his/her point of view and the worst move in our point of view. So we assumed that the enemy always plays its best shot. Now Two situations are possible to emerge:\n",
-    "<ol>\n",
-    "    <li>The adversary agent is very smart. In this case, we are ready beacuse the strategy is based on this situation.</li>\n",
-    "    <li>The adversary agent is too naive and may do some inoptimal actions. In this case, we may even score more than what we gained as utillty value.</li>\n",
-    "</ol>\n",
-    "As a conclusion of what's said before, we can say that the utility value that we calculate for nodes in minimax strategy, is a floor value of what we may actually score."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "\n",
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Pseudocode of Minimax Strategy</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "```python\n",
-    "def max_value(state):\n",
-    "    if leaf?(state), return U(state)\n",
-    "    initialize v = -∞\n",
-    "    for each c in children(state):\n",
-    "        v = max(v, min_value)\n",
-    "    return v\n",
-    "\n",
-    "\n",
-    "def min_value(state):\n",
-    "    if leaf?(state), return U(state)\n",
-    "    initialize v = ∞\n",
-    "    for each c in children(state):\n",
-    "        v = min(v, max_value)\n",
-    "    return v\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Properties of Minimax Strategy</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "<br>\n",
-    "\n",
-    "\n",
-    "<font size = 2 color=#green>\n",
-    "<b>Complete:</b>\n",
-    "</font>\n",
-    "\n",
-    "<br>\n",
-    "\n",
-    "Yes, It is complete if the tree formed for the game is finite.\n",
-    "\n",
-    "<font size = 2 color=#green>\n",
-    "<b>Optimal:</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will surely win the game but not in an optimal way because we could've won the game sooner and with a higher score.\n",
-    "<br>\n",
-    "\n",
-    "<font size = 2 color=#green>\n",
-    "<b>Time Complexity:</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.\n",
-    "<br>\n",
-    "\n",
-    "<font size = 2 color=#green>\n",
-    "<b>Space Complexity:</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "$O(bm)$, because it's like DFS and only stores the nodes on the current path.\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Resource Limits</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine, is exponential in relation to the depth of the tree.\n",
-    "\n",
-    "Consider a _reasonable_ chess game with $b \\approx 35$ and $m \\approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Depth-limited search</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "One idea might be running the algorithm up to a specified depth instead of searching the whole tree, which we call <b>depth-limited search</b> .\n",
-    "\n",
-    "![depth-limited](./images/depth-limited.jpg)\n",
-    "\n",
-    "But using this technique is not much satisfying because it leads to another issue: _How can we find the minimax value while there is no solution at a limited depth?_\n",
-    "Recall how we find the minimax value of each node in the original form. We continue searching until we reach a final state and then use recursion to calculate the parents' values. But in the _limited_ format, there are no goal states at depth $k < m$.\n",
-    "\n",
-    "Now to deal with this issue, we'll introduce _*Evaluation function*_."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Evaluation Functions</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "The goal of an evaluation function is to answer to the question: <b>How good is the current position?</b> Or <b>How probable is it to reach to winning terminal state from this position?</b>\n",
-    "\n",
-    "Let us make this idea more concrete. An evaluation function returns an _estimate_ of the utility in each game state, just like the _heuristic_ functions that estimate the remaining cost to a goal state which were discussed before.\n",
-    "\n",
-    "Obviously, defining an appropriate and precise evaluation function is strongly effective on the player's performance. Actually an inaccurate evaluation function may lead to a losing position in the game.\n",
-    "\n",
-    "Most evaluation functions calculate the value of the position by using a _weighted sum of features_:\n",
-    "\n",
-    "$Eval(s) = w_1f_1(s) + w_2f_2(s) + ... + w_nf_n(s)$\n",
-    "\n",
-    "Each of the $f_i s$  is calculating a specific _feature_ of the state _s_.For instance, in chess, we would have features for the number of white pawns, black pawns, white queens, black queens, and etc. To differentiate the effect of each feature better, we multiply them by _weights_.\n",
-    "\n",
-    "Ideally, evaluation function returns exactly the minimax value of the current state.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Iterative Deepening Search</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "The accuracy of the evaluation function is critical in shallower states. And the deeper in the tree we search, the less the quality of the evaluation function matters.\n",
-    "So a good idea is to maximize the depth of our search as much as possible, considering the limits that we have. In other words, we are looking for an algorithm that can return an acceptable solution whenever we ask. This kind of algorithms are called _anytime algorithms_. These algorithms are expected to find better and better solutions the longer they keep running.\n",
-    "\n",
-    "So simply instead of running the depth-limited-search once, we start running the algorithm with an initial depth limit ($k$). Then after we found s policy, increase the depth limit ($k' > k$) and run the algorithm again with this new limit to find a better and more accurate solution. We continue the iteration until the program reaches the time limit.\n",
-    "This algorithm is called _iterative-deepening search_.\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Pruning</b>\n",
-    "</font>\n",
-    "<br>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Houston, We have a problem!</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "So far, we have seen some imporvements upon our original Adversarial Search algorithm. But the problem with minimax search is that the number of game states it has to examine is **exponential** in the depth of the tree. The logical way to tackle this problem is to cut off some nodes and not to see them at all. This brings us to the idea of \"Pruning\" which we use a lot when we can in AI algorithms. In the following chapter, we see two ways to prune our tree so we can solve our problem in less time."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Minimax Pruning</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "If we take a look at how minimax works, it is noticeable that it explores some parts of the tree that are not necessary at all. In other words, exploring some branches are totally useless but time-consuming. If we can predict which branches are not worthy to be explored so that we can  _prune_ them, we can improve minimax algorithm significantly.\n",
-    "\n",
-    "To understand better, consider the following example:\n",
-    "<div width=\"50%\" height=\"50%\">\n",
-    "    <img src=\"./images/1.png\" alt=\"prunning1\" width=\"50%\" height=\"50%\"/>\n",
-    "</div>\n",
-    "\n",
-    "We start searching with respect to the assumption that we visit leftmost nodes first. Finally when we reach the terminal node with utility 4, _we can deduce that the value of it's parent node(which is a *Min* node) is definitely less than or equal to 4_:\n",
-    "\n",
-    "<div width=\"50%\" height=\"50%\">\n",
-    "    <img src=\"./images/2.png\" alt=\"prunning2\" width=\"50%\" height=\"50%\"/>\n",
-    "</div>\n",
-    "\n",
-    "At this point, this information doesn't help us a lot. So let's continue searching. When we visit the terminal node with utility 3, we find that the value for the parent is also 3. An important observation here is that we can _predict_ that the value of the root _is more than or equal to 3_. That's because the root is a * Max * node.\n",
-    "\n",
-    "\n",
-    "<div width=\"50%\" height=\"50%\">\n",
-    "    <img src=\"./images/3.png\" alt=\"prunning3\" width=\"50%\" height=\"50%\"/>\n",
-    "</div>\n",
-    "\n",
-    "\n",
-    "After that, we continue searching, a terminal leaf with utility 2 will be discovered, and just like before, we can immediately notice that the value of it's parent is _less than or equal to 2_.\n",
-    "\n",
-    "<div width=\"50%\" height=\"50%\">\n",
-    "    <img src=\"./images/4.png\" alt=\"prunning4\" width=\"50%\" height=\"50%\"/>\n",
-    "</div>\n",
-    "\n",
-    "Now, let's ask ourselves: _Is it necessary that we explore the rightmost node?_. The answer is <b>No</b>. Because we already know that the value of the root is at least 3, while the maximum value of the right child of root is 2. So we can easily ignore exploring the last node and find that the minimax value of the root is 3:\n",
-    "\n",
-    "<div width=\"50%\" height=\"50%\">\n",
-    "    <img src=\"./images/5.png\" alt=\"prunning5\" width=\"50%\" height=\"50%\"/>\n",
-    "</div>\n",
-    "\n",
-    "This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called $\\alpha-\\beta $ _pruning_."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Alpha-Beta Pruning</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "\n",
-    "As we are trying to improve our algorithm even more, there is another approach to prune branches that cannot possibly influence the final decision. So in the last section, we talked about Minimax Pruning, but how can we implement it? \n",
-    "\n",
-    "The approach we are trying to explain here as you can see in the title is called Alpha-Beta Pruning.\n",
-    "\n",
-    "The main idea here is that if a node (e.g. $m$) has a better choice than another node (e.g. $n$) or its parents, then that another node ($n$) will never be wanted to expanded. So we can prune it and never check its successor nodes."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "![depth-limited](./images/Alpha_Beta_Intro.png)</br>\n",
-    "*As $m$ can give the root node a higher value, $n$ will be pruned.* \n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We do this by keeping track of two parameters:\n",
-    "\n",
-    "+  $\\alpha$: The value of best choice (i.e. highest value) we have found so far for a Maximizing Node (i.e. our player).\n",
-    "+  $\\beta$: The value of best choice  (i.e. lowest value) we have found so far for a Minimizing Node (i.e opponent player)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Sample Code:\n",
-    "\n",
-    "Here is a Python function to make you understand the concept better."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "```python\n",
-    "def alpha_beta_pruning(position, depth, alpha, beta, maximizing_player):\n",
-    "    if depth == 0 or game_state(position) == GAMESTATE.GAMEOVER:\n",
-    "        if terminal_utility(position):\n",
-    "            return terminal_utility(position)\n",
-    "        return eval_func(position)\n",
-    "\n",
-    "    if maximizing_player: #for maximizing player (our player)\n",
-    "        max_eval = - math.inf\n",
-    "        for child in position.children:\n",
-    "            eval = alpha_beta_pruning(child, depth - 1, alpha, beta, false)\n",
-    "            max_eval = max(max_eval, eval)\n",
-    "            alpha = max(max_eval, eval)\n",
-    "            if beta <= alpha:\n",
-    "                break\n",
-    "        return max_eval\n",
-    "    else: #for minimizing player(opponent player)\n",
-    "        min_eval = + math.inf\n",
-    "        for child in position.children:\n",
-    "            eval = alpha_beta_pruning(child, depth - 1,alpha, beta, true)\n",
-    "            min_eval = min(min_eval, eval)\n",
-    "            beta = min(beta, eval)\n",
-    "            if beta <= alpha:\n",
-    "                break\n",
-    "        return min_eval\n",
-    "```     \n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "So to initialize this function, first we need to give alpha the worst value for our player, which is $-inf$, and give beta the worst value for the opponent, which is $+inf$.\n",
-    "\n",
-    "```python\n",
-    "    #initialization\n",
-    "    alpha_beta_pruning(current_position, 0, - math.inf, + math.inf, true)\n",
-    "```\n",
-    "\n",
-    "Now let's look at an example while running this code."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<p align=\"center\">\n",
-    "    <img src=\"./images/alpha_beta_run_1.png\" alt=\"alpha_beta_run_1\" /></br>\n",
-    "    <em>We approach the first leaf and since the parent is a Minimizing Node, its beta value gets updated</em></br>\n",
-    "    <img src=\"./images/alpha_beta_run_2.png\" alt=\"alpha_beta_run_2\" /></br>\n",
-    "    <em>Second leaf has a bigger value, so since beta always updates at a lower value, nothing changes here</em></br>\n",
-    "    <img src=\"./images/alpha_beta_run_3.png\" alt=\"alpha_beta_run_3\" /></br>\n",
-    "    <em>The algorithm goes back to the root's second child and then goes to the third leaf. Keep in mind that the parent's alpha and beta values always gets passed to their children.</em></br>\n",
-    "    <img src=\"./images/alpha_beta_run_4.png\" alt=\"alpha_beta_run_4\" /></br>\n",
-    "    <em>Now since beta is less than alpha, we know that our root's best possible value from this child will be a 3 which is lower than its other child (i.e. 5), so we prune the other leaf.</em></br>\n",
-    "</p>\n",
-    "\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Expectimax Search</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "\n",
-    "As we can see in the real world, not all the games are played only based on players' skills. We have so many games with stochastic variables in them, like throwing a dice or picking a random card. These games are called **stochastic games**.\n",
-    "\n",
-    "Even when there is no element of luck, we can expect unpredicted behaviors from our agent or the opponent's agent, like robot slipping on the wet floor, or the opponent taking a worse route.\n",
-    "\n",
-    "So the problem we face here is that while we know what legal actions we can take, we can't afford to know what moves will the opponent be able to take as takes its random element into use.(i.e. The transition function will be variable as the game goes on.).\n",
-    "\n",
-    "Let's introduce a new definition called **chance nodes**. We represent these kind of nodes as a circle in the game tree.\n",
-    "\n",
-    "Let's look at this with a example from the epic, legendary game, Snakes And Ladders."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<p align=\"center\">\n",
-    "    <img src=\"./images/snakes_ladders_img.png\" alt=\"snakes_ladders\" width=\"50%\" height=\"50%\"/></br>\n",
-    "    <em>This games is played by two or more players. Each can throw a dice at its turn and should move as many as the dice number.</em></br>\n",
-    "</p>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "So in this game, there is an element of chance. if the opponent rolls the dice on 1, it can move one square, if it rolls the dice on 2, it can move two squares, and so on. So the main point here is that the action the opponent can make depends on a random number from dice which its probability is $1/6$ at a time.\n",
-    "\n",
-    "So the new game tree will be something like this:"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<p align=\"center\">\n",
-    "    <img src=\"./images/expectimax_tree.png\" alt=\"expectimax_tree\" width=\"50%\" height=\"50%\"/></br>\n",
-    "    <em>The minimal expectimax tree for snakes ladder game.</em></br>\n",
-    "</p>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "\n",
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Node values in Expectimax Search:</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "As we deal with probability in this kind of search so much, we expect to calculate the values based on some kind of probabilty.\n",
-    "\n",
-    "Terminal nodes and MAX and MIN nodes (for which the dice roll is known) work exactly the same way as before. So how should we calculate the value of chance nodes?\n",
-    "\n",
-    "For every chance node, the value is calculated from the expected value of its childer. \n",
-    "For example the value of every chance node in Snake ladder will be:\n",
-    "\n",
-    "$1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 * 6$\n",
-    "\n",
-    "So the general formula for values will be as follows:\n",
-    "\n",
-    "| value   | condition |\n",
-    "|------|--------|\n",
-    "| UTILITY(s) | s = terminal node |\n",
-    "|max EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MAX|\n",
-    "|min EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MIN|\n",
-    "| $\\sum_{r}$ P(r)EXPECTIMAX(RESULT(s,r)) | PLAYER(s)= CHANCE|"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Other types of games</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "On top of what's been discussed before, we can count another type of game called <b>Mixed Layer Types</b>. In this type, in addition to players' action, we have another random factor affecting the game. So in our computations we must consider this factor. Foe example, Backgammon is a Mixed layer types game, because rolling the dice will affect the states and result of the game.\n",
-    "\n",
-    "To do some calculations about the depth of the game tree, we can state:\n",
-    "\n",
-    "$b$ = $21$ because 21 dintinct states are created as an outcome of rolling 2 dices.\n",
-    "\n",
-    "$legal moves \\approx 20$ \n",
-    "\n",
-    "So if we want to build the tree till the depth of 2, we will have $20 * (21 * 20) ^ 3$ leaves and termainal states. As we go deeper in tree, this number increases.\n",
-    "\n",
-    "As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and pruning becomes trickier. \n",
-    "\n",
-    "Td-Gammon is a computer backgammon program developed in 1992. It uses a 2-depth search, a very good evaluation function plus reinforcement learning and what we get, is a champion-class player! TD-Gammon was the first AI world champion in any game!"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 3 color=#FF7500>\n",
-    "<b>Multi-Agent Utilities</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "In games with multiple players, terminal states have utility in form of tuple values and each player tries to maximize its own part. This idea can raise the cooperation and competetion strategies dynamically."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Summary</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "Before this, we only discussed games with a single player. But we use Adversarial search when there are other players and agents in the game too, often with opposite objectives. With adversarial search, we are seeking a method called **Playing Strategy** that gives us correct directions and instructions at each state of game.\n",
-    "\n",
-    "We classified games in two main groups based on utility functions:\n",
-    "- Zero-Sum Games: The sum of agents' utility functions is equal to zero. (pure competition)\n",
-    "- General Games:  independent utility functions. (cooperation may be beneficial)\n",
-    "\n",
-    "We formed trees for both single and adversarial agent games and we found a strategy to obtain the value of nodes in both situations.\n",
-    "\n",
-    "Optimal strategy against an opponent playing perfectly can be found using **Minimax strategy**. We can also determine a floor value of the score that can be achieved in every game.\n",
-    "A complete search is often impossible in minimax because of resource limits. So There are two options:\n",
-    "- Replace the utility function with an **Evaluation Function** that estimates utility. This function must be defined very precise, because the whole solution depends on this.\n",
-    "- Do **Minimax Pruning**: While calculating minimax values, it's sometimes possible to detect and **prune** some suboptimal branches that we know we'll never visit them during the algorithm.\n",
-    "\n",
-    "Another point that has to be considered is that While we are looking for a good strategy of playing, we must neighter act Optimistically nor Pessimismtically, because they can both result in suboptimal behaviour so it's important to evaluate each problem realistically and choose our strategy in proportion of the adversarial agent's cleverness.\n",
-    "\n",
-    "Uncertain factors (like rolling dice) are represented by Expectation Nodes in a game tree. Optimal strategy in such games can be found using **Expectimax Search** which was explained before."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>Useful links</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "+ [Step-by-Step alpha-betta example](https://www.youtube.com/watch?v=xBXHtz4Gbdo)\n",
-    "+ [A simple overview of adversarial search with some examples](https://www.baeldung.com/cs/expectimax-search)\n",
-    "+ [More about adversarial search in Chess game](https://medium.com/@SereneBiologist/the-anatomy-of-a-chess-ai-2087d0d565)\n",
-    "+ [About minimax time complexity](https://stackoverflow.com/questions/2080050/how-do-we-determine-the-time-and-space-complexity-of-minmax)\n",
-    "\n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<font size = 5 color=red>\n",
-    "<b>References</b>\n",
-    "</font>\n",
-    "<br>\n",
-    "\n",
-    "\n",
-    "+ https://towardsdatascience.com/understanding-the-minimax-algorithm-726582e4f2c6\n",
-    "+ Russell, S. J., Norvig, P., &amp; Davis, E. (3rd. Ed). Artificial Intelligence: A modern approach. Pearson Educación. \n",
-    "+ https://slideplayer.com/slide/4380560/\n",
-    "+ https://www.cs.cornell.edu/courses/cs312/2002sp/lectures/rec21.htm\n",
-    "+ https://www.techslang.com/definition/what-is-adversarial-search/\n",
-    "+ https://en.wikipedia.org/wiki/Game_theory\n",
-    "+ https://brilliant.org/wiki/minimax/#:~:text=In%20game%20theory%2C%20minimax%20is,payoff%20as%20large%20as%20possible.\n",
-    "+ https://en.wikipedia.org/wiki/Minimax\n",
-    "+ Sebastian Lague's Youtube channel"
-   ]
-  }
- ],
- "metadata": {
-  "interpreter": {
-   "hash": "bf0f1093e9a05e91d616174468df6ede651877b2c960cbb124380d685ad06d65"
-  },
-  "kernelspec": {
-   "display_name": "Python 3.8.3 64-bit",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.8.3"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}

From 7087e98878e81a590978556d396d4ff945c02c88 Mon Sep 17 00:00:00 2001
From: Matin Daghyani <59438691+mtndaghyani@users.noreply.github.com>
Date: Sat, 13 Nov 2021 11:45:08 +0330
Subject: [PATCH 29/62] Add introduction link

---
 notebooks/adversarial_search_fall_2021/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index ee4d4507..52f534bd 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -5,7 +5,7 @@
 
 
 + ### <font size = 4.5 color=Orange>
-  ### Introduction
+  ### [Introduction](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#introduction-1)
   </font>
   
   - <font size = 3 color=green>

From a8c417e509dbbc58e7c5c249879c74eedd294122 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 11:51:52 +0330
Subject: [PATCH 30/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 52f534bd..866fd3fa 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -9,7 +9,8 @@
   </font>
   
   - <font size = 3 color=green>
-    <b>The concept of adversarial search and an introduction to Game Theory</b>
+  ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-      concept-of-adversarial-search-and-an-introduction-to-game-theory)
+  
     </font>
 
   - <font size = 3 color=green>

From 827f5f4da7ba7edb7dd4338b62697f756d56c6f5 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 11:52:42 +0330
Subject: [PATCH 31/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 866fd3fa..af1bf73f 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -9,8 +9,7 @@
   </font>
   
   - <font size = 3 color=green>
-  ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-      concept-of-adversarial-search-and-an-introduction-to-game-theory)
-  
+  ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory)
     </font>
 
   - <font size = 3 color=green>

From 962092bb7ca5440e1e94f4a636e6416f11dc1d35 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 11:53:32 +0330
Subject: [PATCH 32/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index af1bf73f..3ffd7b22 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -9,7 +9,7 @@
   </font>
   
   - <font size = 3 color=green>
-  ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory)
+  ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory-1)
     </font>
 
   - <font size = 3 color=green>

From 9288c0554622b970875b0d5c0b571ac6dbfeeea0 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 11:56:46 +0330
Subject: [PATCH 33/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 3ffd7b22..ef88b348 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -8,7 +8,7 @@
   ### [Introduction](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#introduction-1)
   </font>
   
-  - <font size = 3 color=green>
+  - ### <font size = 3 color=green>
   ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory-1)
     </font>
 

From 26ba2c120d55a716dc7078b37ae29845b8f15c1a Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 11:58:44 +0330
Subject: [PATCH 34/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index ef88b348..767440f3 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -8,10 +8,10 @@
   ### [Introduction](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#introduction-1)
   </font>
   
-  - ### <font size = 3 color=green>
+  ### <font size = 3 color=green>
   ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory-1)
     </font>
-
+    
   - <font size = 3 color=green>
     <b>Game Theory Explanation and Deterministic Games</b>
     </font>

From 578d9488018a50e8564c9a2ed9c4af855449fd34 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:08:41 +0330
Subject: [PATCH 35/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 767440f3..57a5ddee 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -12,14 +12,15 @@
   ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory-1)
     </font>
     
-  - <font size = 3 color=green>
-    <b>Game Theory Explanation and Deterministic Games</b>
+  ### <font size = 3 color=green>
+  ### [Game Theory Explanation and Deterministic Games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#game-theory-explanation-and-deterministic-games)
     </font>
 
-  - <font size = 3 color=green>
-    <b>Different catergories of games</b>
+  ### <font size = 3 color=green>
+  ### [Different catergories of games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#game-theory-explanation-and-deterministic-games)
     </font>
-  
+    
+    
 + ### <font size = 4.5 color=Orange>
   ### Different Types of Trees for games
   </font>

From 057cd3a00a415687cbb06203a0e7d36de9ec73c5 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:09:18 +0330
Subject: [PATCH 36/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 57a5ddee..61b9872f 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -17,7 +17,7 @@
     </font>
 
   ### <font size = 3 color=green>
-  ### [Different catergories of games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#game-theory-explanation-and-deterministic-games)
+  ### [Different catergories of games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#different-catergories-of-games-1)
     </font>
     
     

From e80f1e801b661e802c5aa33a236f6e4122dd9e50 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:12:18 +0330
Subject: [PATCH 37/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 61b9872f..6dd0ab72 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -21,8 +21,10 @@
     </font>
     
     
+    
+    
 + ### <font size = 4.5 color=Orange>
-  ### Different Types of Trees for games
+  ### [Different Types of trees for games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#differnt-types-of-trees-for-games)
   </font>
   
   - <font size = 3 color=green>

From 906afd5ba8594d9a84f3ed485317a7e679b5e69f Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:13:58 +0330
Subject: [PATCH 38/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 6dd0ab72..53e1736e 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -187,14 +187,14 @@ Before we discuss types of games, we need to define a concept named **Utility Fu
 <br>
 
 <font size = 3 color=#FF7500>
-<b>Single Agent Trees</b>
+### <b>Single Agent Trees</b>
 </font>
 <br>
 
 Consider the game Pacman which has only an agent playing the game. Starting from an initial state, our agent can either move to left or right. based on this, and its following moves, we can form a tree called **Single Agent Tree**. Each node of this tree indicates an state of the game and the utility function has an specific value in every one of them. We call the leaves of the tree **The Terminal Nodes**. In these terminal nodes, the game is finshed and the utility values are defined. The main question is how can we obtain the utility value of a node? According on what's been said, we have the utility values of the leaves. Recursively, to get the utility value of a node N, we calculate the utility value of every successor of N and we choose the maximum of those numbers as the utility value of the node N.
 <br>
 <font size = 3 color=#FF7500>
-<b>Adversarial Agent Trees</b>
+### <b>Adversarial Agent Trees</b>
 </font>
 <br>
 Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.

From 5584980e4caa6af0fc3d93ab2f82ed5cafa684c3 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:14:48 +0330
Subject: [PATCH 39/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 1 +
 1 file changed, 1 insertion(+)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 53e1736e..2747e6a6 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -187,6 +187,7 @@ Before we discuss types of games, we need to define a concept named **Utility Fu
 <br>
 
 <font size = 3 color=#FF7500>
+  
 ### <b>Single Agent Trees</b>
 </font>
 <br>

From 6abd4a8dc263e8e84ab1753f7362ef45cae21819 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:16:38 +0330
Subject: [PATCH 40/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 13 ++++++++-----
 1 file changed, 8 insertions(+), 5 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 2747e6a6..d69d7c8f 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -27,14 +27,17 @@
   ### [Different Types of trees for games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#differnt-types-of-trees-for-games)
   </font>
   
-  - <font size = 3 color=green>
-    <b>Single-Agent Trees</b>
-    </font>
 
-  - <font size = 3 color=green>
-    <b>Adverserial Agent Trees</b>
+  ### <font size = 3 color=green>
+  ### [Single-Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#single-agent-trees)
     </font>
 
+  ### <font size = 3 color=green>
+  ### [Adversarial Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#single-agent-trees)
+    </font>
+    
+    
+   
 + ### <font size = 4.5 color=Orange>
   ### Minimax Strategy
   </font>

From b6dbfc95b5a6528370309adae5f7b3f244cb61c8 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:17:08 +0330
Subject: [PATCH 41/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index d69d7c8f..52f05861 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -33,7 +33,7 @@
     </font>
 
   ### <font size = 3 color=green>
-  ### [Adversarial Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#single-agent-trees)
+  ### [Adversarial Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#adversarial-agent-trees-1)
     </font>
     
     

From 66176b1f9378bd53cc3feaaf41898cc9f3d298fb Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:20:34 +0330
Subject: [PATCH 42/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 52f05861..4fbd5bb8 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -223,7 +223,8 @@ As a conclusion of what's said before, we can say that the utility value that we
 
 
 <font size = 3 color=#FF7500>
-<b>Pseudocode of Minimax Strategy</b>
+
+### <b>Pseudocode of Minimax Strategy</b>
 </font>
 <br>
 
@@ -245,7 +246,8 @@ def min_value(state):
 ```
 
 <font size = 3 color=#FF7500>
-<b>Properties of Minimax Strategy</b>
+  
+### <b>Properties of Minimax Strategy</b>
 </font>
 <br>
 <br>

From d7610d04e9a75d7bfdf81302f1d84c48dc5b39c2 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:20:42 +0330
Subject: [PATCH 43/62] Create index.md


From 933cc875dbef400225d98f70992f6742383f8f2b Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:24:26 +0330
Subject: [PATCH 44/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 13 ++++++++-----
 1 file changed, 8 insertions(+), 5 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 4fbd5bb8..ae96e245 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -39,16 +39,19 @@
     
    
 + ### <font size = 4.5 color=Orange>
-  ### Minimax Strategy
+  ### [Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#minimax-strategy-1)
   </font>
   
-  - <font size = 3 color=green>
-    <b>Pseudocode of Minimax Strategy</b>
+
+  ### <font size = 3 color=green>
+  ### [Pseudocode of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#pseudocode-of-minimax-strategy)
     </font>
 
-  - <font size = 3 color=green>
-    <b>Properties of Minimax Strategy</b>
+  ### <font size = 3 color=green>
+  ### [Properties of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#properties-of-minimax-strategy)
     </font>
+    
+    
 
 + ### <font size = 4.5 color=Orange>
   ### Resource Limit

From 4d0ac794ad82e95051049a08d67572271d3823a6 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 13:25:48 +0330
Subject: [PATCH 45/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index ae96e245..9d764539 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -44,11 +44,11 @@
   
 
   ### <font size = 3 color=green>
-  ### [Pseudocode of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#pseudocode-of-minimax-strategy)
+  ### [Pseudocode of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#pseudocode-of-minimax-strategy-1)
     </font>
 
   ### <font size = 3 color=green>
-  ### [Properties of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#properties-of-minimax-strategy)
+  ### [Properties of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#properties-of-minimax-strategy-1)
     </font>
     
     

From 5a176f7e5fcdb0a06276adfefd1966996f9ef02d Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 14:44:50 +0330
Subject: [PATCH 46/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 9 ++++++---
 1 file changed, 6 insertions(+), 3 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 9d764539..4de92403 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -298,7 +298,8 @@ Although the minimax algorithm would be proper for problems with relatively smal
 Consider a _reasonable_ chess game with $b \approx 35$ and $m \approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below.
 
 <font size = 3 color=#FF7500>
-<b>Depth-limited search</b>
+  
+### <b>Depth-limited search</b>
 </font>
 <br>
 One idea might be running the algorithm up to a specified depth instead of searching the whole tree, which we call <b>depth-limited search</b> .
@@ -311,7 +312,8 @@ Recall how we find the minimax value of each node in the original form. We conti
 Now to deal with this issue, we'll introduce _*Evaluation function*_.
 
 <font size = 3 color=#FF7500>
-<b>Evaluation Functions</b>
+  
+### <b>Evaluation Functions</b>
 </font>
 <br>
 The goal of an evaluation function is to answer to the question: <b>How good is the current position?</b> Or <b>How probable is it to reach to winning terminal state from this position?</b>
@@ -330,7 +332,8 @@ Ideally, evaluation function returns exactly the minimax value of the current st
 
 
 <font size = 3 color=#FF7500>
-<b>Iterative Deepening Search</b>
+  
+### <b>Iterative Deepening Search</b>
 </font>
 <br>
 The accuracy of the evaluation function is critical in shallower states. And the deeper in the tree we search, the less the quality of the evaluation function matters.

From 88d9bacdd30a37949e6d2b95ebbb061129c3fa09 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 14:48:44 +0330
Subject: [PATCH 47/62] Update index.md

---
 .../adversarial_search_fall_2021/index.md      | 18 ++++++++++--------
 1 file changed, 10 insertions(+), 8 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 4de92403..c1464c23 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -54,20 +54,22 @@
     
 
 + ### <font size = 4.5 color=Orange>
-  ### Resource Limit
+  ### [Resource limit](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#resource-limits)
   </font>
   
-  - <font size = 3 color=green>
-    <b>Depth-limited Search</b>
-    </font>
 
-  - <font size = 3 color=green>
-    <b>Evaluation Functions</b>
+  ### <font size = 3 color=green>
+  ### [Depth-limited Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#depth-limited-search)
     </font>
 
-  - <font size = 3 color=green>
-    <b>Iterative Deepening Search</b>
+  ### <font size = 3 color=green>
+  ### [Evaluation Function](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#evaluation-functions)
     </font>
+    
+  ### <font size = 3 color=green>
+  ### [Iterative Deepening Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#iterative-deepening-search)
+    </font>
+    
   
 + ### <font size = 4.5 color=Orange>
   ### Pruning

From 954482ace17313d1eb89145ce92c6cf589b695a9 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 14:50:57 +0330
Subject: [PATCH 48/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 9 ++++++---
 1 file changed, 6 insertions(+), 3 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index c1464c23..58f67637 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -351,14 +351,16 @@ This algorithm is called _iterative-deepening search_.
 <br>
 
 <font size = 3 color=#FF7500>
-<b>Houston, We have a problem!</b>
+
+### <b>Houston, We have a problem!</b>
 </font>
 <br>
 
 So far, we have seen some imporvements upon our original Adversarial Search algorithm. But the problem with minimax search is that the number of game states it has to examine is **exponential** in the depth of the tree. The logical way to tackle this problem is to cut off some nodes and not to see them at all. This brings us to the idea of "Pruning" which we use a lot when we can in AI algorithms. In the following chapter, we see two ways to prune our tree so we can solve our problem in less time.
 
 <font size = 3 color=#FF7500>
-<b>Minimax Pruning</b>
+
+### <b>Minimax Pruning</b>
 </font>
 <br>
 
@@ -398,7 +400,8 @@ Now, let's ask ourselves: _Is it necessary that we explore the rightmost node?_.
 This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called $\alpha-\beta $ _pruning_.
 
 <font size = 3 color=#FF7500>
-<b>Alpha-Beta Pruning</b>
+
+### <b>Alpha-Beta Pruning</b>
 </font>
 <br>
 

From 8dc4c2739eac8b906acade943db11f9829e55f14 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 14:53:40 +0330
Subject: [PATCH 49/62] Update index.md

---
 .../adversarial_search_fall_2021/index.md     | 21 +++++++++++--------
 1 file changed, 12 insertions(+), 9 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 58f67637..d7f2ba87 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -71,20 +71,23 @@
     </font>
     
   
+  
+  
 + ### <font size = 4.5 color=Orange>
-  ### Pruning
+  ### [Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#pruning-1)
   </font>
-
-  - <font size = 3 color=green>
-    <b>Houston, We Have A Problem!</b>
-    </font>
   
-  - <font size = 3 color=green>
-    <b>Minimax Pruning</b>
+
+  ### <font size = 3 color=green>
+  ### [Houston, We Have A Problem!](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#houston-we-have-a-problem)
     </font>
 
-  - <font size = 3 color=green>
-    <b>Alpha-Beta Pruning</b>
+  ### <font size = 3 color=green>
+  ### [Minimax Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#minimax-pruning)
+    </font>
+    
+  ### <font size = 3 color=green>
+  ### [Alpha-Beta Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#alpha-beta-pruning)
     </font>
   
 + ### <font size = 4.5 color=Orange>

From 74441c094313487fe504098b5dc36935c9aaf30c Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 14:54:37 +0330
Subject: [PATCH 50/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index d7f2ba87..b1027054 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -79,15 +79,15 @@
   
 
   ### <font size = 3 color=green>
-  ### [Houston, We Have A Problem!](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#houston-we-have-a-problem)
+  ### [Houston, We Have A Problem!](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#houston-we-have-a-problem-1)
     </font>
 
   ### <font size = 3 color=green>
-  ### [Minimax Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#minimax-pruning)
+  ### [Minimax Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#minimax-pruning-1)
     </font>
     
   ### <font size = 3 color=green>
-  ### [Alpha-Beta Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#alpha-beta-pruning)
+  ### [Alpha-Beta Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#alpha-beta-pruning-1)
     </font>
   
 + ### <font size = 4.5 color=Orange>

From 1dedaf3648392f1962dee8d1e51294e03d70a2df Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 14:55:47 +0330
Subject: [PATCH 51/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index b1027054..37205ef9 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -29,7 +29,7 @@
   
 
   ### <font size = 3 color=green>
-  ### [Single-Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#single-agent-trees)
+  ### [Single-Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#single-agent-trees-1)
     </font>
 
   ### <font size = 3 color=green>
@@ -59,7 +59,7 @@
   
 
   ### <font size = 3 color=green>
-  ### [Depth-limited Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#depth-limited-search)
+  ### [Depth-limited Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#depth-limited-search-1)
     </font>
 
   ### <font size = 3 color=green>

From 85845d538e34bb6b5dc3e8262f248e0303228b4a Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 14:57:14 +0330
Subject: [PATCH 52/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 6 +-----
 1 file changed, 1 insertion(+), 5 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 37205ef9..15c55d07 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -91,13 +91,9 @@
     </font>
   
 + ### <font size = 4.5 color=Orange>
-  ### Expectimax Search
+  ### [Expectimax Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#expectimax-search-1)
   </font>
   
-  - <font size = 3 color=green>
-    <b>Node Values in Expectimax Search</b>
-    </font>
-  
 + ### <font size = 4.5 color=Orange>
   ### Other Types of Games
   </font>

From 51b4a6490ba3d73cdb7f08a7af63c6717c851191 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 15:00:15 +0330
Subject: [PATCH 53/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 15c55d07..c2e26d47 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -548,7 +548,8 @@ As the deepness increases, the probability to reach a goal state gets smaller an
 Td-Gammon is a computer backgammon program developed in 1992. It uses a 2-depth search, a very good evaluation function plus reinforcement learning and what we get, is a champion-class player! TD-Gammon was the first AI world champion in any game!
 
 <font size = 3 color=#FF7500>
-<b>Multi-Agent Utilities</b>
+
+### <b>Multi-Agent Utilities</b>
 </font>
 <br>
 In games with multiple players, terminal states have utility in form of tuple values and each player tries to maximize its own part. This idea can raise the cooperation and competetion strategies dynamically.

From 13bc5459358fa3d2cbd3df7c2ae5fecb1a598b01 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 15:02:16 +0330
Subject: [PATCH 54/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index c2e26d47..a46d5195 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -95,11 +95,12 @@
   </font>
   
 + ### <font size = 4.5 color=Orange>
-  ### Other Types of Games
+  ### [Other types of games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#other-types-of-games-1)
   </font>
   
-  - <font size = 3 color=green>
-    <b>Multi-Agent Utilities</b>
+
+  ### <font size = 3 color=green>
+  ### [Multi-Agent Utilities](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#multi-agent-utilities-1)
     </font>
 
 + ### <font size = 4.5 color=Orange>

From 743c1d5dd14de40b6932ac6f93328fac862d7817 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 15:03:49 +0330
Subject: [PATCH 55/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index a46d5195..d80ca627 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -104,10 +104,10 @@
     </font>
 
 + ### <font size = 4.5 color=Orange>
-  ### Conclusion
+  ### [Conclusion](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#conclusion-1)
   </font>
 + ### <font size = 4.5 color=Orange>
-  ### Reference
+  ### [Reference](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#reference-1)
   </font>
 
 ## <font size = 5 color=red>

From 7e59b0249b787b6e44b3065922e2d511d09d6313 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 15:05:14 +0330
Subject: [PATCH 56/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 7 ++++++-
 1 file changed, 6 insertions(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index d80ca627..4c0117e0 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -106,8 +106,13 @@
 + ### <font size = 4.5 color=Orange>
   ### [Conclusion](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#conclusion-1)
   </font>
+  
++ ### <font size = 4.5 color=Orange>
+  ### [Useful Linkes](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#useful-links)
+  </font>
+  
 + ### <font size = 4.5 color=Orange>
-  ### [Reference](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#reference-1)
+  ### [Reference](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#references)
   </font>
 
 ## <font size = 5 color=red>

From c515ddfe3c50090912286683a366fd83449124b6 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 15:07:28 +0330
Subject: [PATCH 57/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 4c0117e0..35707ca8 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -13,7 +13,7 @@
     </font>
     
   ### <font size = 3 color=green>
-  ### [Game Theory Explanation and Deterministic Games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#game-theory-explanation-and-deterministic-games)
+  ### [Game Theory Explanation and Deterministic Games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#game-theory-explanation-and-deterministic-games-1)
     </font>
 
   ### <font size = 3 color=green>
@@ -67,7 +67,7 @@
     </font>
     
   ### <font size = 3 color=green>
-  ### [Iterative Deepening Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#iterative-deepening-search)
+  ### [Iterative Deepening Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#iterative-deepening-search-1)
     </font>
     
   

From dfe7de08b8e0ceaf8575d5db256192dd2152f6ce Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Sat, 13 Nov 2021 15:09:30 +0330
Subject: [PATCH 58/62] Update index.md

---
 .../adversarial_search_fall_2021/index.md     | 28 +++++++++----------
 1 file changed, 14 insertions(+), 14 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 35707ca8..78d1a3cf 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -9,15 +9,15 @@
   </font>
   
   ### <font size = 3 color=green>
-  ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory-1)
+  - ### [The concept of adversarial search and an introduction to Game Theory](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#the-concept-of-adversarial-search-and-an-introduction-to-game-theory-1)
     </font>
     
   ### <font size = 3 color=green>
-  ### [Game Theory Explanation and Deterministic Games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#game-theory-explanation-and-deterministic-games-1)
+  - ### [Game Theory Explanation and Deterministic Games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#game-theory-explanation-and-deterministic-games-1)
     </font>
 
   ### <font size = 3 color=green>
-  ### [Different catergories of games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#different-catergories-of-games-1)
+  - ### [Different catergories of games](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#different-catergories-of-games-1)
     </font>
     
     
@@ -29,11 +29,11 @@
   
 
   ### <font size = 3 color=green>
-  ### [Single-Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#single-agent-trees-1)
+  - ### [Single-Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#single-agent-trees-1)
     </font>
 
   ### <font size = 3 color=green>
-  ### [Adversarial Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#adversarial-agent-trees-1)
+  - ### [Adversarial Agent Trees](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#adversarial-agent-trees-1)
     </font>
     
     
@@ -44,11 +44,11 @@
   
 
   ### <font size = 3 color=green>
-  ### [Pseudocode of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#pseudocode-of-minimax-strategy-1)
+  - ### [Pseudocode of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#pseudocode-of-minimax-strategy-1)
     </font>
 
   ### <font size = 3 color=green>
-  ### [Properties of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#properties-of-minimax-strategy-1)
+  - ### [Properties of Minimax Strategy](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#properties-of-minimax-strategy-1)
     </font>
     
     
@@ -59,15 +59,15 @@
   
 
   ### <font size = 3 color=green>
-  ### [Depth-limited Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#depth-limited-search-1)
+  - ### [Depth-limited Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#depth-limited-search-1)
     </font>
 
   ### <font size = 3 color=green>
-  ### [Evaluation Function](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#evaluation-functions)
+  - ### [Evaluation Function](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#evaluation-functions)
     </font>
     
   ### <font size = 3 color=green>
-  ### [Iterative Deepening Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#iterative-deepening-search-1)
+  - ### [Iterative Deepening Search](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#iterative-deepening-search-1)
     </font>
     
   
@@ -79,15 +79,15 @@
   
 
   ### <font size = 3 color=green>
-  ### [Houston, We Have A Problem!](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#houston-we-have-a-problem-1)
+  - ### [Houston, We Have A Problem!](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#houston-we-have-a-problem-1)
     </font>
 
   ### <font size = 3 color=green>
-  ### [Minimax Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#minimax-pruning-1)
+  - ### [Minimax Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#minimax-pruning-1)
     </font>
     
   ### <font size = 3 color=green>
-  ### [Alpha-Beta Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#alpha-beta-pruning-1)
+  - ### [Alpha-Beta Pruning](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#alpha-beta-pruning-1)
     </font>
   
 + ### <font size = 4.5 color=Orange>
@@ -100,7 +100,7 @@
   
 
   ### <font size = 3 color=green>
-  ### [Multi-Agent Utilities](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#multi-agent-utilities-1)
+  - ### [Multi-Agent Utilities](https://github.com/mothegoat/notes/blob/master/notebooks/adversarial_search_fall_2021/index.md#multi-agent-utilities-1)
     </font>
 
 + ### <font size = 4.5 color=Orange>

From b9279a220a46ada790e3264186c1dcb0b3140f6d Mon Sep 17 00:00:00 2001
From: Matin Daghyani <59438691+mtndaghyani@users.noreply.github.com>
Date: Fri, 17 Dec 2021 19:39:33 +0330
Subject: [PATCH 59/62] Fix formulas

---
 .../adversarial_search_fall_2021/index.md     | 44 +++++++++----------
 1 file changed, 22 insertions(+), 22 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 78d1a3cf..d66b59b3 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -134,17 +134,17 @@ To gather more information about the concept of adversarial search, visit <a hre
 Briefly, Game Theory is designing the strategies and the steps of our player to interact in the best way, according to the rival's steps and strategies. In other words, Game theory is the study of mathematical models of strategic interactions among rational decision-makers.
 To achieve this goal, we need to express our game and its rules and actions in a mathematical form. The common model used to represent games is the model below:
 
-1. States of the game: $S_i$, starting with $S_0$\
+1. States of the game: <img src="https://render.githubusercontent.com/render/math?math=S_i">, starting with <img src="https://render.githubusercontent.com/render/math?math=S_0">\
     Which means how the game will look after each action of competitors.
     <br>
 <br>
 
-2. Players: $P = ${$1,2,...,n$}\
+2. Players: <img src="https://render.githubusercontent.com/render/math?math=P = {1,2,...,n}">
     represents the agents playing the game where the index varies from 1 to n.
     <br>
 <br>
 
-3. Actions: $A_i$\
+3. Actions: <img src="https://render.githubusercontent.com/render/math?math=A_i">\
     In every state, each player has a set of legitimate moves to change that state to another state by playing its turn.
     <br>
 <br>
@@ -284,7 +284,7 @@ Yes, It is optimal if the opponent acts optimally not naivly. otherwise, we will
 </font>
 <br>
 
-$O(b^m)$, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with $b$ = $35$ and $m$ = $100$, we can't use this strategy because the time complexity will be huge.
+<img src="https://render.githubusercontent.com/render/math?math=O(b^m)">, as it's obvious, it's dependant on the branching factor and the level the tree goes down. So on a game like chess with <img src="https://render.githubusercontent.com/render/math?math=b = 35"> and<img src="https://render.githubusercontent.com/render/math?math=m = 100">, we can't use this strategy because the time complexity will be huge.
 <br>
 
 <font size = 2 color=#green>
@@ -292,7 +292,7 @@ $O(b^m)$, as it's obvious, it's dependant on the branching factor and the level
 </font>
 <br>
 
-$O(bm)$, because it's like DFS and only stores the nodes on the current path.
+<img src="https://render.githubusercontent.com/render/math?math=O(b^m)">, because it's like DFS and only stores the nodes on the current path.
 
 
 
@@ -302,7 +302,7 @@ $O(bm)$, because it's like DFS and only stores the nodes on the current path.
 <br>
 Although the minimax algorithm would be proper for problems with relatively small state space, it isn't an efficient and feasible one for problems with more complicated and larger state space. Since the number of game states it has to examine, is exponential in relation to the depth of the tree.
 
-Consider a _reasonable_ chess game with $b \approx 35$ and $m \approx 100$ . Due to the time complexity of this algorithm which is $O(b^m)$, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below.
+Consider a _reasonable_ chess game with <img src="https://render.githubusercontent.com/render/math?math=b \approx 35"> and <img src="https://render.githubusercontent.com/render/math?math=m \approx 100"> . Due to the time complexity of this algorithm which is <img src="https://render.githubusercontent.com/render/math?math=O(b^m)">, solving the game is not possible at all. We'll discuss some ideas to solve the problem in details below.
 
 <font size = 3 color=#FF7500>
   
@@ -314,7 +314,7 @@ One idea might be running the algorithm up to a specified depth instead of searc
 ![depth-limited](./images/depth-limited.jpg)
 
 But using this technique is not much satisfying because it leads to another issue: _How can we find the minimax value while there is no solution at a limited depth?_
-Recall how we find the minimax value of each node in the original form. We continue searching until we reach a final state and then use recursion to calculate the parents' values. But in the _limited_ format, there are no goal states at depth $k < m$.
+Recall how we find the minimax value of each node in the original form. We continue searching until we reach a final state and then use recursion to calculate the parents' values. But in the _limited_ format, there are no goal states at depth <img src="https://render.githubusercontent.com/render/math?math=k < m">.
 
 Now to deal with this issue, we'll introduce _*Evaluation function*_.
 
@@ -331,9 +331,9 @@ Obviously, defining an appropriate and precise evaluation function is strongly e
 
 Most evaluation functions calculate the value of the position by using a _weighted sum of features_:
 
-$Eval(s) = w_1f_1(s) + w_2f_2(s) + ... + w_nf_n(s)$
+<img src="https://render.githubusercontent.com/render/math?math=Eval(s) = w_1f_1(s) + w_2f_2(s) + ... + w_nf_n(s)">
 
-Each of the $f_i s$  is calculating a specific _feature_ of the state _s_.For instance, in chess, we would have features for the number of white pawns, black pawns, white queens, black queens, and etc. To differentiate the effect of each feature better, we multiply them by _weights_.
+Each of the <img src="https://render.githubusercontent.com/render/math?math=f_i s">  is calculating a specific _feature_ of the state _s_.For instance, in chess, we would have features for the number of white pawns, black pawns, white queens, black queens, and etc. To differentiate the effect of each feature better, we multiply them by _weights_.
 
 Ideally, evaluation function returns exactly the minimax value of the current state.
 
@@ -346,7 +346,7 @@ Ideally, evaluation function returns exactly the minimax value of the current st
 The accuracy of the evaluation function is critical in shallower states. And the deeper in the tree we search, the less the quality of the evaluation function matters.
 So a good idea is to maximize the depth of our search as much as possible, considering the limits that we have. In other words, we are looking for an algorithm that can return an acceptable solution whenever we ask. This kind of algorithms are called _anytime algorithms_. These algorithms are expected to find better and better solutions the longer they keep running.
 
-So simply instead of running the depth-limited-search once, we start running the algorithm with an initial depth limit ($k$). Then after we found s policy, increase the depth limit ($k' > k$) and run the algorithm again with this new limit to find a better and more accurate solution. We continue the iteration until the program reaches the time limit.
+So simply instead of running the depth-limited-search once, we start running the algorithm with an initial depth limit (k). Then after we found s policy, increase the depth limit (k' > k) and run the algorithm again with this new limit to find a better and more accurate solution. We continue the iteration until the program reaches the time limit.
 This algorithm is called _iterative-deepening search_.
 
 
@@ -402,7 +402,7 @@ Now, let's ask ourselves: _Is it necessary that we explore the rightmost node?_.
     <img src="./images/5.png" alt="prunning5" width="50%" height="50%"/>
 </div>
 
-This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called $\alpha-\beta $ _pruning_.
+This idea helps us finding a general and systematic algorithm to predict the _bad branches_ and stop exploring them, which is called <img src="https://render.githubusercontent.com/render/math?math=\alpha-\beta"> _pruning_.
 
 <font size = 3 color=#FF7500>
 
@@ -415,16 +415,16 @@ As we are trying to improve our algorithm even more, there is another approach t
 
 The approach we are trying to explain here as you can see in the title is called Alpha-Beta Pruning.
 
-The main idea here is that if a node (e.g. $m$) has a better choice than another node (e.g. $n$) or its parents, then that another node ($n$) will never be wanted to expanded. So we can prune it and never check its successor nodes.
+The main idea here is that if a node (e.g. m) has a better choice than another node (e.g. n) or its parents, then that another node (n) will never be wanted to expanded. So we can prune it and never check its successor nodes.
 
 ![depth-limited](./images/Alpha_Beta_Intro.png)</br>
-*As $m$ can give the root node a higher value, $n$ will be pruned.* 
+*As m can give the root node a higher value, n will be pruned.* 
 
 
 We do this by keeping track of two parameters:
 
-+  $\alpha$: The value of best choice (i.e. highest value) we have found so far for a Maximizing Node (i.e. our player).
-+  $\beta$: The value of best choice  (i.e. lowest value) we have found so far for a Minimizing Node (i.e opponent player).
++  <img src="https://render.githubusercontent.com/render/math?math=\alpha">: The value of best choice (i.e. highest value) we have found so far for a Maximizing Node (i.e. our player).
++  <img src="https://render.githubusercontent.com/render/math?math=\beta">: The value of best choice  (i.e. lowest value) we have found so far for a Minimizing Node (i.e opponent player).
 
 ### Sample Code:
 
@@ -458,7 +458,7 @@ def alpha_beta_pruning(position, depth, alpha, beta, maximizing_player):
 ```     
 
 
-So to initialize this function, first we need to give alpha the worst value for our player, which is $-inf$, and give beta the worst value for the opponent, which is $+inf$.
+So to initialize this function, first we need to give alpha the worst value for our player, which is <img src="https://render.githubusercontent.com/render/math?math=-inf">, and give beta the worst value for the opponent, which is <img src="https://render.githubusercontent.com/render/math?math=+inf">.
 
 ```python
     #initialization
@@ -502,7 +502,7 @@ Let's look at this with a example from the epic, legendary game, Snakes And Ladd
     <em>This games is played by two or more players. Each can throw a dice at its turn and should move as many as the dice number.</em></br>
 </p>
 
-So in this game, there is an element of chance. if the opponent rolls the dice on 1, it can move one square, if it rolls the dice on 2, it can move two squares, and so on. So the main point here is that the action the opponent can make depends on a random number from dice which its probability is $1/6$ at a time.
+So in this game, there is an element of chance. if the opponent rolls the dice on 1, it can move one square, if it rolls the dice on 2, it can move two squares, and so on. So the main point here is that the action the opponent can make depends on a random number from dice which its probability is <img src="https://render.githubusercontent.com/render/math?math=1/6"> at a time.
 
 So the new game tree will be something like this:
 
@@ -524,7 +524,7 @@ Terminal nodes and MAX and MIN nodes (for which the dice roll is known) work exa
 For every chance node, the value is calculated from the expected value of its childer. 
 For example the value of every chance node in Snake ladder will be:
 
-$1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 * 6$
+<img src="https://render.githubusercontent.com/render/math?math=1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 * 6">
 
 So the general formula for values will be as follows:
 
@@ -533,7 +533,7 @@ So the general formula for values will be as follows:
 | UTILITY(s) | s = terminal node |
 |max EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MAX|
 |min EXPECTIMAX(Eval(s, a)) | PLAYER(s) = MIN|
-| $\sum_{r}$ P(r)EXPECTIMAX(RESULT(s,r)) | PLAYER(s)= CHANCE|
+| <img src="https://render.githubusercontent.com/render/math?math=\sum_{r}"> P(r)EXPECTIMAX(RESULT(s,r)) | PLAYER(s)= CHANCE|
 
 ## <font size = 5 color=red>
 ## Other types of games 
@@ -543,11 +543,11 @@ On top of what's been discussed before, we can count another type of game called
 
 To do some calculations about the depth of the game tree, we can state:
 
-$b$ = $21$ because 21 dintinct states are created as an outcome of rolling 2 dices.
+b = 1 because 21 dintinct states are created as an outcome of rolling 2 dices.
 
-$legal moves \approx 20$ 
+<img src="https://render.githubusercontent.com/render/math?math=legal moves \approx 20">
 
-So if we want to build the tree till the depth of 2, we will have $20 * (21 * 20) ^ 3$ leaves and termainal states. As we go deeper in tree, this number increases.
+So if we want to build the tree till the depth of 2, we will have <img src="https://render.githubusercontent.com/render/math?math=20 * (21 * 20) ^ 3"> leaves and termainal states. As we go deeper in tree, this number increases.
 
 As the deepness increases, the probability to reach a goal state gets smaller and limiting the depth gets less damaging. So the usefulness of search will be threatened and pruning becomes trickier. 
 

From 79588d1c3cf1226ef9e594a014b89b1ffcf5a573 Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Fri, 17 Dec 2021 20:47:17 +0330
Subject: [PATCH 60/62] Adversarial Agent bold fix

---
 notebooks/adversarial_search_fall_2021/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index d66b59b3..69a85d94 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -211,7 +211,7 @@ Consider the game Pacman which has only an agent playing the game. Starting from
 ### <b>Adversarial Agent Trees</b>
 </font>
 <br>
-Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, **The Ghost**.
+Like the previous Example, we explain this concept with the game Pacman but this time, the game has another agent called the adversarial agent or specific to this game, <b>The Ghost</b>.
 Like Single Agent Trees, we can form a tree to express the moves of the agents, but the difference is that at each level of the tree, only one agent can play. For example if we start with the player P1 at the root of the tree, based on its move, we gain 2 successors and we go to the first level of the tree. At this level of tree, only the other player called P2 can move and each of the 2 nodes will have 2 successors based on P2's move. So we have 4 nodes in the second level and the game will continue this way till the end. In this Configuration, how can we calculate the utility value of nodes? This will be our next topic.  
 
 ## <font size = 5 color=red>

From 4cf50bd89a22ab4f796b382b81c2c74ad9f0cd1a Mon Sep 17 00:00:00 2001
From: MoaliMkh <59166192+MoaliMkh@users.noreply.github.com>
Date: Fri, 17 Dec 2021 20:53:27 +0330
Subject: [PATCH 61/62] Update index.md

---
 notebooks/adversarial_search_fall_2021/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 69a85d94..56c8d0c3 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -382,7 +382,7 @@ We start searching with respect to the assumption that we visit leftmost nodes f
     <img src="./images/2.png" alt="prunning2" width="50%" height="50%"/>
 </div>
 
-At this point, this information doesn't help us a lot. So let's continue searching. When we visit the terminal node with utility 3, we find that the value for the parent is also 3. An important observation here is that we can _predict_ that the value of the root _is more than or equal to 3_. That's because the root is a * Max * node.
+At this point, this information doesn't help us a lot. So let's continue searching. When we visit the terminal node with utility 3, we find that the value for the parent is also 3. An important observation here is that we can _predict_ that the value of the root _is more than or equal to 3_. That's because the root is a *Max* node.
 
 
 <div width="50%" height="50%">

From 529897fa144215d2234b7306f1ebb7a47745949a Mon Sep 17 00:00:00 2001
From: Matin Daghyani <59438691+mtndaghyani@users.noreply.github.com>
Date: Sat, 18 Dec 2021 00:01:42 +0330
Subject: [PATCH 62/62] Add title and fix +inf

---
 notebooks/adversarial_search_fall_2021/index.md | 9 ++++++---
 1 file changed, 6 insertions(+), 3 deletions(-)

diff --git a/notebooks/adversarial_search_fall_2021/index.md b/notebooks/adversarial_search_fall_2021/index.md
index 56c8d0c3..c5b95fcb 100644
--- a/notebooks/adversarial_search_fall_2021/index.md
+++ b/notebooks/adversarial_search_fall_2021/index.md
@@ -1,5 +1,7 @@
+# Adversarial Search
 
-# Table of Contents
+
+## Table of Contents
 
 ---
 
@@ -125,7 +127,8 @@
 </font>
 
 In previous lectures, we discussed situations in which we had only a single agent. We didn't consider other parameters affecting our environment but in this chapter, a new type of search called **Adversarial Search** is introduced. In adversarial search, we define our problem in a multi-agent context. For instance, while playing a game, our agent has to consider the other agent's moves (adversarial moves) to play in an efficient way. Even in some games we can define a winning stategy which means we can guarantee that in every state of our game, no matter how bad it is, our agent is able to win the game.
-To gather more information about the concept of adversarial search, visit <a href="https://www.techslang.com/definition/what-is-adversarial-search/">this link</a>.
+To gather more 
+ormation about the concept of adversarial search, visit <a href="https://www.techslang.com/definition/what-is-adversarial-search/">this link</a>.
 
 ### <font size = 3 color=#FF7500>
 ### Game Theory Explanation and Deterministic Games
@@ -458,7 +461,7 @@ def alpha_beta_pruning(position, depth, alpha, beta, maximizing_player):
 ```     
 
 
-So to initialize this function, first we need to give alpha the worst value for our player, which is <img src="https://render.githubusercontent.com/render/math?math=-inf">, and give beta the worst value for the opponent, which is <img src="https://render.githubusercontent.com/render/math?math=+inf">.
+So to initialize this function, first we need to give alpha the worst value for our player, which is <img src="https://render.githubusercontent.com/render/math?math=-inf">, and give beta the worst value for the opponent, which is +inf.
 
 ```python
     #initialization