Game Playing AI in Action: Mastering Game Strategies with Minimax and Beyond !!

Abstract:

Game Playing is an important domain of artificial intelligence. Games don’t require much

knowledge; the only knowledge we need to provide is the rules, legal moves and the conditions

of winning or losing the game.

Both players try to win the game. So, both of them try to make the best move possible at each turn. Searching techniques like BFS(Breadth First Search) are not accurate for this as the

branching factor is very high, so searching will take a lot of time. So, we need another search

procedures that improve –

• Generate procedure so that only good moves are generated.
• Test procedure so that the best move can be explored first.

The most common search technique in game playing is Minimax search procedure. It is depthfirst depth-limited search procedure. It is used for games like chess and tic-tac-toe.

The minimax algorithm is a recursive algorithm used in game theory to find the best move for a player in a game. It's based on the assumption that the other player is also playing optimally. The minimax algorithm is used in many two-player games, including: Tic-tac-toe, Go, Chess, Isola, and Checkers. 

 

Here are some things to know about the minimax algorithm: 

 

How it works

The minimax algorithm models a game as a tree, with different parts of the tree representing different elements of the game, such as the current state and all possible moves. It searches for the move that will minimize the opponent's options. 

 

Alpha-beta pruning

Alpha-beta pruning is an optimization technique that can be used with the minimax algorithm to reduce the amount of time needed to find the best move. It removes branches that won't affect the final part of the game. 

 

Parallel minimax algorithm

In 1988, Althöfer proved that it's possible to develop a parallel minimax algorithm that achieves linear speedup in the average case. 

Keywords:

Game Playing, Mini- Max  Algorithms, Alpha-Beta Pruning, Rule based Game Playing, chess, Minecraft and tic-tac-toe

Learning Outcomes

Upon completing this article, you will be equipped with a comprehensive understanding of the following key concepts:

1. What's exactly Game Playing in AI?

2. Why Game Playing is important ?

3. What's MINI-MAX Search?

4. What's Alpha- Beta (α-β) Pruning ?

5. How to  write algorithms of Game Playing in AI?

6. Game Playing Applications in Artificial Intelligence 

7. Types of Game Playing in AI

8. Characteristics of Game Playing in AI 

9. Rules of Game Playing in AI 

10. Examples of Games playing in AI

11. Advantages of Game Playing in AI

12. Disadvantages of Game Playing in Artificial Intelligence:

13. Conclusions

14. FAQs

References 

1. Game Playing in Artificial Intelligence

Game Playing is an important domain of artificial intelligence. Games don’t require much

knowledge; the only knowledge we need to provide is the rules, legal moves and the conditions

of winning or losing the game.

Both players try to win the game. So, both of them try to make the best move possible at each turn. 

Searching techniques like BFS(Breadth First Search) are not accurate for this as the

branching factor is very high, so searching will take a lot of time. So, we need another search

procedures that improve –

• Generate procedure so that only good moves are generated.
• Test procedure so that the best move can be explored first.

The most common search technique in game playing is Minimax search procedure. It is depthfirst depth-limited search procedure. It is used for games like chess and tic-tac-toe.

2. Why Game Playing is important ?

Game playing is important in artificial intelligence (AI) for several reasons, including: 

 

Testing AI

Games are a cost-effective way to test AI's ability, safety, and robustness in a controllable environment. 

 

Learning from AI

Comparing AI to human players can help researchers understand how AI improves over time and the difficulty of problems it can solve. 

 

Studying human behavior

AI-powered games can help researchers learn how humans interact with each other, including how we cooperate and compete. 

 

Motivating learning

Games can be a fun way to learn about AI programming, which can be challenging due to the math and troubleshooting involved. 

 

Solving real-world problems

Games can reflect real-world problems like scheduling, design, and decision making. 

 

Teaching AI

AI can be taught to solve problems by playing games, starting with simple games like Pong and progressing to more complex ones. 

 

Counteracting cheating

AI can be used to identify cheaters in games, such as in PlayerUnknown's Battlegrounds. 

 

3. What's MINI-MAX Search?

 MINI-MAX Search

Games have always been an important application area for heuristic algorithms. In playing

games whose state space may be exhaustively delineated, the primary difficulty is in accounting for the actions of the opponent. This can be handled easily by assuming that the opponent uses

the same knowledge of the state space as us and applies that knowledge in a consistent effort to win the game. Minmax implements game search under referred to as MIN and MAX.

imax algorithm, is a backtracking algorithm that's used in artificial intelligence (AI), game theory, and decision-making to determine the best move for a player in a game. It's commonly used in two-player games like chess, checkers, tic-tac-toe, and go. 

 

Minimax search

How it works

Searches forward in a game tree to determine the best move by recursively computing values based on heuristic evaluations of different game positions

Goal

To choose the best move for the player currently making a move, assuming that the opposing player would also play optimally

How it's used

In AI, decision theory, game theory, statistics, and philosophy

The minimax algorithm works by:

Assigning two players to the game, one called MAX and one called MIN

Having MAX select the maximized value and MIN select the minimized value

Performing a depth-first search algorithm to explore the complete game tree

Proceeding all the way down to the terminal node of the tree, then backtracking the tree as the recursion 

4. What's Alpha- Beta (α-β) Pruning ?

Alpha- Beta (α-β) Pruning

Alpha-beta pruning is a search algorithm that helps reduce the number of nodes that a minimax algorithm needs to evaluate in a search tree. It's a modified version of the minimax algorithm, and is often used in machine-played two-player games like chess, tic-tac-toe, and Connect 4. 

 

Here are some key points about alpha-beta pruning: 

 

Pruning

The term "pruning" refers to removing branches or stems that are dead or overgrown. In artificial intelligence, pruning means removing useless branches from decision trees and random forests. 

 

Alpha and beta

Alpha-beta pruning is named after the two parameters it uses, alpha and beta. Alpha is the best value for the maximizer, while beta is the best value for the minimizer. 

 

Speed

Alpha-beta pruning helps models search faster, which allows them to explore deeper levels of a game tree. 

 

Depth

Alpha-beta pruning can be applied at any depth of a tree, and can sometimes prune entire sub-trees and tree leaves. 

 

5. How to  write algorithms of Game Playing in AI?

Writing algorithms for game playing in AI involves creating models that allow machines to play games intelligently by simulating human decision-making. Here's a general approach for writing algorithms for game playing in AI:

### 1. **Define the Game Rules**

   - Understand the game environment: types of moves, board state, win/lose conditions, and allowable actions.

   - Represent the game state: This could be in the form of matrices, arrays, graphs, or trees, depending on the game.

### 2. **State Representation**

   - Represent the current state of the game. For example, in a chess game, the state could be represented as a matrix of 8x8 squares, where each square contains either an empty space or a piece.

   - A common approach is to use **Game Trees**, where each node represents a state of the game, and edges represent transitions between these states.

### 3. **Minimax Algorithm**

   - **Minimax** is a decision-making algorithm often used in two-player games. It assumes that both players play optimally.

   - The goal is to minimize the possible loss for a worst-case scenario (hence, "minimax").

   - **Steps**:

     1. Generate all possible moves.

     2. Recursively calculate the value of each move.

     3. Choose the best move based on maximizing the player's gain and minimizing the opponent's potential gain.

   - **Algorithm**:

     ```

     function minimax(node, depth, maximizingPlayer):

         if depth == 0 or node is a terminal node:

             return evaluate(node)

         if maximizingPlayer:

             maxEval = -infinity

             for each child of node:

                 eval = minimax(child, depth - 1, false)

                 maxEval = max(maxEval, eval)

             return maxEval

         else:

             minEval = infinity

             for each child of node:

                 eval = minimax(child, depth - 1, true)

                 minEval = min(minEval, eval)

             return minEval

     ```

### 4. **Alpha-Beta Pruning**

   - Alpha-Beta pruning is an optimization technique for the minimax algorithm that eliminates the need to explore branches of the game tree that are not promising, reducing the number of nodes evaluated.

   - The idea is to prune branches where one player's best move is worse than the opponent’s best move.

   - **Algorithm**:

     ```

     function alphaBeta(node, depth, alpha, beta, maximizingPlayer):

         if depth == 0 or node is a terminal node:

             return evaluate(node)

         if maximizingPlayer:

             maxEval = -infinity

             for each child of node:

                 eval = alphaBeta(child, depth - 1, alpha, beta, false)

                 maxEval = max(maxEval, eval)

                 alpha = max(alpha, eval)

                 if beta <= alpha:

                     break

             return maxEval

         else:

             minEval = infinity

             for each child of node:

                 eval = alphaBeta(child, depth - 1, alpha, beta, true)

                 minEval = min(minEval, eval)

                 beta = min(beta, eval)

                 if beta <= alpha:

                     break

             return minEval

     ```

### 5. **Heuristic Evaluation Function**

   - For many games, exploring the entire game tree is computationally expensive. Instead, you can use a heuristic evaluation function to estimate the value of a game state (like in chess, evaluating material advantage, piece placement, etc.).

   - The heuristic function should return a score that evaluates the desirability of the current game state.

   

   Example for Tic-Tac-Toe:

   ```

   function evaluate(board):

       if player wins:

           return +10

       if opponent wins:

           return -10

       return 0

   ```

### 6. **Monte Carlo Tree Search (MCTS)**

   - MCTS is a probabilistic algorithm for decision-making that is especially useful for games with large state spaces.

   - It involves four steps: selection, expansion, simulation, and backpropagation.

   - **Steps**:

     1. **Selection**: Traverse the game tree from the root to a leaf node.

     2. **Expansion**: Expand the tree by adding one or more child nodes.

     3. **Simulation**: Simulate a random game from the current state to a terminal state.

     4. **Backpropagation**: Update the nodes based on the result of the simulation.

   **Algorithm**:

   ```

   function MCTS(node):

       while within computational limits:

           leaf = select(node)

           child = expand(leaf)

           result = simulate(child)

           backpropagate(child, result)

       return bestChild(node)

   ```

### 7. **Q-Learning (Reinforcement Learning)**

   - In Q-learning, the AI learns the best actions through trial and error by interacting with the game environment and receiving feedback in the form of rewards or penalties.

   - **Steps**:

     1. Initialize Q-table with all state-action pairs.

     2. For each state, choose an action based on a policy (like epsilon-greedy).

     3. Update the Q-value for the state-action pair.

     4. Repeat until convergence.

   - **Algorithm**:

     ```

     function QLearning():

         for each episode:

             state = initial state

             while state is not terminal:

                 action = choose_action(state)

                 next_state, reward = take_action(state, action)

                 Q[state, action] = Q[state, action] + alpha * (reward + gamma * max(Q[next_state, a]) - Q[state, action])

                 state = next_state

     ```

### 8. **Training and Fine-Tuning**

   - Train your algorithm using the chosen technique and test it in various scenarios to improve its performance. For deep learning-based AI, you may use neural networks to predict the best moves based on training data.

### 9. **Optimization Techniques**

   - Use parallel computation, game-specific heuristics, or deep learning models (like convolutional neural networks for visual recognition tasks in games like Go or Chess).

Algorithm is a recursive algorithm used in artificial intelligence (AI) to help computers make decisions in games by considering all possible moves and responses from an opponent. It's often used in two-player games like chess, tic-tac-toe, and backgammon. 

 

Here's how the minimax algorithm works: 

 

Assign values to game states

The algorithm assigns a value to each position or state in the game, based on how good it would be for a player to reach that position. 

 

Simulate future moves

The algorithm simulates the game's future moves and outcomes by constructing a game tree where each node represents a possible future state. 

 

Choose a move

The algorithm chooses the move that maximizes the minimum value of the position resulting from the opponent's possible following moves. 

 

Repeat

The algorithm repeats the process for each move, working its way down to the terminal node of the tree and then backtracking. 

 

The minimax algorithm is optimal if both opponents are playing optimally. 

 

6. Game Playing Applications in Artificial Intelligence 

Game Playing has following applications, which include:

Graphics enhancement

Intelligent NPCs

Pathfinding

Sentiment analysis

Procedural content Generation

Analytics

Natural language processing

Game AI

Enemy AI

7. Types of Game Playing in AI 

Some types of games that use AI include: 

 

Alien: Isolation

A horror game based on the aesthetic of Ridley Scott's movie, this game uses AI techniques to design non-player characters (NPCs). 

 

AlphaGo Zero

An AI that specializes in Go, it learns to play by playing against itself, starting from random play. It quickly surpassed human level of play and defeated the human champion. 

 

Some common types of AI used in gaming include: 

 

Rule-based AI

Uses predetermined rules and conditions to dictate the behavior of NPCs. 

 

Finite State Machines

Uses pre-defined cues to determine NPC behavior. 

 

Pathfinding AI

Calculates the most efficient way for a character to reach its destination. 

 

Machine Learning AI

Enables NPCs to learn and adapt based on experience and data within the game. 

 

Procedural generation

Uses AI to automatically generate content for games, such as levels, environments, and characters. 

 

Cloud-based gaming

Uses machine learning algorithms to analyze player data and behavior to optimize server resources. 

 8. Characteristics of Game Playing in AI 

Characteristics of Games offers a new way to understand games: by focusing on certain traits--including number of players, rules, degrees of luck and skill needed, and reward/effort ratio--and using these characteristics as basic points of comparison and analysis.

In video games, artificial intelligence (AI) is used to generate responsive, adaptive or intelligent behaviors primarily in non-playable characters (NPCs) similar to human-like intelligence.

9. Rules of Game Playing in AI 

The strategies, often referred to as game playing algorithms in AI, can be broadly classified into two main categories: 

1. Rule-based systems and 

2. Machine learning-based systems. 

Rule-based systems, a cornerstone of game playing in AI, rely on predefined sets of rules to govern the behavior of AI agents during gameplay.

Rule-based AI operates on a set of predetermined rules and conditions that dictate the behavior of non-player characters (NPCs) within the game. These rules are usually programmed by developers and define how NPCs should react in various situations. For example, in a stealth game, if the player is spotted by an NPC, the rule-based AI might instruct the NPC to alert nearby guards.

10. Examples of Games playing in AI

Here are some examples of games that use artificial intelligence (AI): 

Chess

Computers can analyze the board and calculate the best move based on their programming. 

Artificial intelligence (AI) plays a significant role in chess, helping players improve their game by analyzing positions and making intelligent moves. Chess engines use algorithms and techniques like machine learning to evaluate positions and make moves. Some of the algorithms used in chess engines include: 

 

Alpha-beta pruning: A process that reduces the search tree space by eliminating paths that neither player will take 

 

Min-max searching: An algorithm used in chess engines 

 

Null move heuristic: An algorithm used in chess engines 

 

Quiescence searching: An algorithm used in chess engines 

 

Static board evaluation functions: An algorithm used in chess engines 

 

Opening move database: An algorithm used in chess engines 

 

Some examples of chess engines include:

Stockfish

A chess engine that has a rating of 3438

Leela Chess Zero

An open-source chess engine that uses deep reinforcement learning to learn chess through self-play games

AlphaZero

A neural net-based digital automaton that beat Stockfish 28–0 in a 100-game match in 2017 

Minecraft

The game's mobs exhibit simple AI behaviors, such as zombies chasing players, villagers communicating, and animals wandering around. 

Minecraft is a popular environment for artificial intelligence (AI) research and development because of its open-ended nature and limitless possibilities: 

 

Project Malmo

A Minecraft mod that allows computer scientists to train AI agents in Minecraft worlds. 

 

Microsoft AI demo

A demo that lets users tell Minecraft what to do, such as moving their character or collecting materials. 

 

MineRL

A Python 3 library that provides an interface for interacting with Minecraft and datasets of human gameplay. 

 

MineDojo

A Minecraft environment with predesigned challenges that won an award at NeurIPS, a major AI conference. 

 

AI Lesson Sequence in Minecraft Education

A lesson sequence that introduces students to AI and coding activities, including: 

 

Recognizing patterns on ocelots' coats to track endangered animals 

 

Using machine learning algorithms to improve crop yields and soil efficiency 

 

Some say that Minecraft's open-endedness makes it a good environment for training AI because it allows for endless possibilities, from simple tasks to complex ones. 

 

 

The Last of Us: Part I

The AI in this game allows NPC allies to detect enemies and adapt their actions accordingly. 

 

AlphaGo Zero

This AI learns to play Go by playing games against itself, eventually surpassing human level of play. 

 

Halo: Combat Evolved

The game's AI learns from the player's actions and adapts their tactics to counteract the player's strategies. 

 

Red Dead Redemption 2

The game features characters that are animated using a complex system of motion capture and AI algorithms. 

 

Tic-tac-toe

AI can use a minimax algorithm to predict possible future outcomes, which can lead to drawn games regardless of the human's move. 

Here are some ways that artificial intelligence (AI) can be used to play tic-tac-toe: 

 

Minimax algorithm

This algorithm is considered unbeatable and is based on the idea of making the best response to what your opponent might do. When two Minimaxes play against each other, they will always make moves that result in a draw. 

 

Magic square

This algorithm uses a smart strategy that involves checking if the sum of two squares equals 15. The algorithm also considers the opponent's moves to block their chances of winning. 

 

Ternary number conversion

This algorithm converts the board to a ternary number and then to a decimal number. The algorithm then uses the decimal number as an index to access a vector in a move table. The vector indicates how the board will look after the next move. 

 

AI can also be used to detect cheating in multiplayer games. 

 

11. Advantages of Game Playing in AI

AI can enhance the gaming experience in many ways, including: 

 

Dynamic content generation

AI can create more complex and dynamic gaming environments, with non-player characters (NPCs) that react to player actions. 

 

Intelligent NPCs

AI can train NPCs to learn from interactions with players, adapting their behavior and responses. 

Virtual assistants

AI-powered virtual assistants can provide guidance and support to help players navigate gameplay. 

Voice recognition

Voice recognition allows players to control characters, navigate menus, and interact with other players without a controller or keyboard. 

Balancing

AI can analyze player behavior to adapt game mechanics and ensure balanced gameplay for all skill levels. 

 

Predictive analytics

AI can analyze large amounts of data to predict the outcome of a game, which can help coaches make better decisions. 

12. Disadvantages of Game Playing in Artificial Intelligence

There are several disadvantages to using artificial intelligence (AI) in games, including: 

 

Addiction

AI-driven games can be highly addictive, making it difficult for players to stop playing. 

 

Frustration

Players may become frustrated if they perceive the AI as unfair or overly difficult. 

 

Social isolation

Spending too much time playing AI-based games can lead to social isolation. 

 

Privacy concerns

Some AI-based games collect and analyze player data, which can raise privacy concerns. 

 

Health issues

Immersive AI games can lead to a sedentary lifestyle, which can contribute to health issues. 

 

Time consumption

AI-driven games can be time-consuming, which can lead to players neglecting other responsibilities. 

 

Unpredictable outcomes

AI systems can behave in ways that weren't originally intended by the creators. 

 

Technical issues

AI can introduce technical problems, such as bugs and breakdowns in gameplay. 

 

High development costs

Advanced AI systems can be expensive to develop. 

 

Loss of control

Developers may lose control over certain aspects of their game when using generative AI. 

 

Lack of emotions and creativity

AI systems can't possess genuine emotions or create with purpose. 

 

Repetition

AI can lead to repetition in topic and structure, making it difficult to create engaging stories. 

13. Conclusions

Game playing is an interesting part of artificial intelligence. The rules of the game are limited. Many human experts exist to assist in the developing of the programs. For the human expert, it is easy to explain in the rationale for a move unlike other domains.

The key to writing efficient game-playing AI is selecting the right algorithm based on the complexity and nature of the game. Minimax and Alpha-Beta pruning are excellent for simple, deterministic, turn-based games, while methods like Monte Carlo Tree Search and Reinforcement Learning are more suitable for complex, real-time, or partially observable environments.

14. FAQs

Q. What are the top 5 innovations of AI in gaming industry?

Ans. 

AI and Procedural Generation

AI in Game Content Creation - AI can generate diverse & dynamic game levels, quests, challenges, etc providing unique experiences to players.

Infinite Content Possibilities - Enhances replayability & extends the lifespan of games.

Efficiency & Scalability - Create larger, more intricate environments in less time.

Dynamic & Adaptive Environments - The game world can evolve, regenerate & adjust based on player interactions & behavior.

Endless Exploration & Discovery - Each level can unveil new areas, hidden treasures, unique encounters, etc allowing discovery of the game world in depth.

Q. What are the most popular AI Games?

Ans. 

Artificial Intelligence Games has definitely come a long way, some of the top video games that use artificial intelligence. Here are the 40 top AI based game that you need to start playing right now:

AI Dungeon

Screeps

CodeCombat

Cognitoy Dino

Hello Neighbor

Black & White

S.T.A.L.K.E.R. Shadow Of Chernobyl

Forza Motorsport Series

Civilization Series

FIFA Series

XCOM2

Total War Series

Rainbow Six Siege

Stellaris

Middle-Earth

Rocket League

Fallout 4

Hitman Series

The SIMS Series

F.E.A.R.

First Cry Series

Dishonoured Series

Deus Ex Series

Watch Dogs Series

Meta Gear Solid Series

Alien & Alien: Isolation

Sniper Elite Series

The Last of Us Series

Ghost of Tsushima

Monster Hunter

Detroit

Red Dead Redemption 2

Half-Life

Bioshock Infinite

Grand Theft Auto 5

Halo: Combat Evolved

Minecraft

Darkforest

Tom Clancy’s Splinter Cell: Blacklist

AlphaGo Zero

Q. How Minimax algorithm works and briefly explain it?

Ans.

The **Minimax algorithm** is a decision-making algorithm commonly used in two-player, zero-sum games where one player's gain is the other player's loss. It is typically used in games like chess, tic-tac-toe, and checkers, where the objective is to find the optimal move by considering both the current player's moves and the opponent's counter-moves.

### Core Concepts

- **Maximizing Player**: This is the player for whom the algorithm is being executed, and they aim to maximize their score or utility.

- **Minimizing Player**: This is the opponent, who tries to minimize the maximizing player's score.

- **Game Tree**: A tree-like structure that represents all possible moves from the current state. Each node represents a state of the game, and edges represent the possible moves.

- **Terminal Node**: These are the nodes where the game ends (win, loss, or draw).

- **Utility Function**: A function used to evaluate terminal nodes, assigning a score (e.g., +1 for a win, -1 for a loss, and 0 for a draw).

### How Minimax Works

1. **Generate the game tree**: List all possible moves for the current player and the opponent, recursively.

2. **Evaluate the terminal nodes**: Use a utility function to assign scores to the terminal states (e.g., winning, losing, or drawing).

3. **Propagate the values**: Starting from the terminal nodes, propagate the scores back up the tree. If the node is the maximizing player's turn, pick the maximum score; if it's the minimizing player's turn, pick the minimum score.

4. **Choose the best move**: Once the scores are propagated back to the root of the tree, the algorithm selects the move corresponding to the highest (for the maximizing player) or lowest (for the minimizing player) score.

### Example Algorithm

#### 1. **Minimax Algorithm (Recursive Version)**

```python

def minimax(node, depth, maximizingPlayer):

    if depth == 0 or is_terminal(node):

        return evaluate(node)  # Terminal node or depth limit reached

    

    if maximizingPlayer:

        maxEval = -float('inf')

        for child in get_children(node):

            eval = minimax(child, depth - 1, False)

            maxEval = max(maxEval, eval)

        return maxEval

    else:

        minEval = float('inf')

        for child in get_children(node):

            eval = minimax(child, depth - 1, True)

            minEval = min(minEval, eval)

        return minEval

```

#### 2. **Supporting Functions**

- **is_terminal(node)**: Checks whether the node is a terminal node (game over).

- **evaluate(node)**: Evaluates the score of a terminal node (e.g., +1 for win, -1 for loss, 0 for draw).

- **get_children(node)**: Generates all possible next states (children) of the current game state (node).

### Example Walkthrough (Tic-Tac-Toe)

1. **Initial State**: The algorithm starts at the root, representing the current board configuration.

2. **Maximizing Player's Turn**: The maximizing player selects the best move by simulating all possible next moves (children).

3. **Minimizing Player's Turn**: After the maximizing player makes a move, the minimizing player responds by trying to minimize the maximizing player’s score.

4. **Evaluation**: The algorithm continues recursively until a terminal state is reached, where a win, loss, or draw is detected, and a score is assigned.

5. **Backpropagate Scores**: Scores are propagated back up the tree until the best move for the current player is determined.

### Alpha-Beta Pruning (Optimization for Minimax)

To improve the efficiency of Minimax, **Alpha-Beta pruning** can be applied to eliminate branches of the tree that don’t need to be explored. This helps reduce the computational complexity from \(O(b^d)\) to \(O(b^{d/2})\), where \(b\) is the branching factor and \(d\) is the depth of the tree.

**Key Concepts in Alpha-Beta Pruning**:

- **Alpha**: The best value the maximizing player can achieve so far.

- **Beta**: The best value the minimizing player can achieve so far.

- Prune branches if the current node's value is worse than the alpha or beta value, as these branches cannot affect the final decision.

### Final Thoughts

The **Minimax algorithm** is widely used in classical game-playing AI due to its ability to simulate all possible moves and counter-moves. However, it is computationally expensive for large game trees, making optimizations like **Alpha-Beta pruning** or other techniques necessary for efficiency in complex games.

References