Mini-Max Algorithm in AI: An Essential Tool for Strategic Decision-Making

Have you ever predicted your opponent’s move in rock-paper-scissors and came out on top? It seems almost impossible, but what if you could calculate every possible outcome and make the perfect move every time? This is precisely what the mini max algorithm does for AI—it evaluates all potential moves and selects the best possible choice from all possible outcomes.

The global AI market is expected to reach USD 826.7 billion by 2030. AI-driven decision-making tools like mini max are at the forefront of this revolution. Whether used in strategic games or real-world applications, the algorithm plays a pivotal role in shaping AI’s ability to make smart, calculated decisions.

In this blog, you will explore the mini max algorithm in AI and check its applications in real-world scenarios. So dive right in!

What is Mini Max Algorithm?

The mini max algorithm is a strategy used by computers or AI to make the best possible move in games where two players take turns. The idea behind the mini max algorithm is that one player tries to maximize their chances of winning while the other tries to minimize the first player’s chances. 

Here’s how an AI uses a mini max algorithm.

The mini max algorithm works by creating a game tree, which is a map of all possible scenarios in a game. At each level of the game tree:

• The AI (the maximizing player) looks for the move that gives the best possible outcome for itself.
• The opponent (the minimizing player) tries to make the move that hurts the AI the most.
• The algorithm follows all the possible moves, looks to predict the results, and then chooses the best path based on these predictions.

Here are some of the example scenarios of mini max algorithm.

• Tic-tac-toe

The algorithm analyzes all the possible moves on the entire board. If the AI can win in the next move, it will choose that. If not, it will block the opponent from winning.

• Chess

The algorithm will look at all the possible moves that the AI and its opponent can make. For each move, it checks the potential outcomes and scores them. The AI tries to pick the move that leads to the best possible result for it.

Also Read: Ensemble Methods in Machine Learning Algorithms Explained

Do you know the mini max algorithm in AI can help you make strategic decisions? Here’s how.

Mini-Max Algorithm in AI

The mini max is a fundamental algorithm in artificial intelligence allows agents to make optimal decisions in two-player, zero-sum games, where one player's gain is another's loss.

Let’s explore the critical features of the mini max algorithm in the subsequent sections.

Key Components of Mini Max Algorithm 

The mini max algorithm works by simulating all possible moves in a game, examining both the AI's and the opponent's moves. The algorithm is structured into two nodes: a max node and a min node.

• Max node

The max node represents the decision point for the player (or AI) whose goal is to maximize their score or advantage. At a max node, the algorithm evaluates all possible moves that the player can make and chooses the move that leads to the highest possible score. 

• Min mode

The min node represents the opponent's decision point, as the opponent is trying to minimize the player's score. At a min node, the algorithm assumes that the opponent is also playing optimally and will try to make the move that will reduce the player's potential score.

Also Read: What is Supervised Machine Learning? Algorithms, Examples

In essence, you can say that the max node focuses on maximizing your own score, while the min node focuses on minimizing your opponent’s score.

Mathematical Representation of Mini Max Algorithm of AI

You can mathematically represent the mini max algorithm by recursive formulas, where each node in the game tree is evaluated based on whether it's a max node (AI’s turn) or a min node (opponent's turn). Here’s the mathematical representation of mini max algorithm.

1. Max node:

AI tries to maximize the score at the max node. The score for this node is the maximum of the scores of its child nodes:

score(max_node) = max(score(child1​), score(child2​),…,score(childn​))

The AI will choose the move that leads to the highest score from the possible moves.

2. Min node:

The opponent tries to minimize the score of at min node. The score for this node is the minimum of the scores of its child nodes:

score(min_node) = min(score(child1​), score(child2​),…,score(childn​))

This means that the opponent will choose the move that reduces the AI’s score, looking to minimize the AI’s chances of winning.

Recursive formula

The mini max algorithm recursively evaluates the game tree from the root node, alternating between maximizing and minimizing at each node:

score(root) = max(score(child1​), score(child2​),…,score(childn​)) (if it’s AI’s turn)

or

score(root) = min(score(child1​), score(child2​),…,score(childn​)) (if it’s the opponent’s turn)

Now that you have an idea of how the mini max algorithm in AI works, let’s examine it in detail.

Want to learn more about Algorithms? Check out upGrad's free course on Data Structures and Algorithms.

Implementation of Mini Max Algorithm in AI

The Minimax algorithm provides a strategic approach to decision-making by exploring the game tree and evaluating potential moves. It enables players to make optimal choices from all available options. 

Here are the steps involved in implementing the Minimax algorithm in AI, focusing on the game tree representation and evaluating terminal states.

1. Tree Representation of Possible Moves

The game is represented in the form of a tree structure where:

• The root node represents the game’s current state. 
• Each branch in the tree represents a possible move the current player can make.
• Leaf nodes represent terminal states (win, loss, or draw), with associated values.

Example: Tic-Tac-Toe

The game tree for a simple  Tic-Tac-Toe can be visualized as follows.

• The top-level (root) is the current state of the board.
• The child nodes represent all possible moves the current player can make.
• Each of those child nodes can have further children representing the opponent's possible moves.
• This process continues until the game reaches a terminal state (win, loss, or draw).

2. Alternating Between Max and Min Levels

The Minimax algorithm works by alternating between maximizing and minimizing levels of the tree, where–

• Max level represents the current player (the one making the decision) whose aim is to maximize the score.
• Min level represents the opponents (the one responding) whose aim is to minimize the score.

At the max level, the algorithm chooses the maximum value of the child nodes, representing the best move for the player. 

At the min level, it chooses the minimum value of the child nodes, representing the worst move for the player (best for the opponent).

3. Evaluating Terminal States

The terminal states (leaf nodes) represent the end of the game, and a value is assigned based on the outcome.

• For win: +1 (indicating a favorable outcome).
• For loss: -1 (indicating an unfavorable outcome).
• For draw: 0 (indicating a neutral outcome).

The evaluation function gives values to these terminal states based on the game’s rules and the current player's perspective.

4. Backpropagating the Evaluation

Once the terminal states are evaluated, the algorithm backpropagates the values up the game tree:

• At max nodes, choose the maximum value from the child nodes (best move for the player).
• At min nodes, choose the minimum value from the child nodes (best move for the opponent).

This backpropagation continues until the root node is reached, which will then give the value of the optimal move for the current player.

Here is a breakdown of the mini max algorithm in AI.

Pseudocode of Mini Max Algorithm

The mini max algorithm analyzes all possible game states and evaluates them to determine the AI's best move. 

Here's the pseudocode of the mini max algorithm, along with symbolic logic for the scoring mechanism.

Pseudo code:

function minimax(node, depth, maximizingPlayer):
    if node is a terminal node or depth == max_depth:
        return evaluate(node)

    if maximizingPlayer:
        best = -infinity
        for each child of node:
            score = minimax(child, depth + 1, false)  // Minimize for the opponent
            best = max(best, score)
        return best
    else:
        best = +infinity
        for each child of node:
            score = minimax(child, depth + 1, true)   // Maximize for the AI
            best = min(best, score)
        return best

function evaluate(node):
    if node is a winning state for AI:
        return 1  // AI wins
    if node is a losing state for AI:
        return -1 // Opponent wins
    if node is a draw state:
        return 0  // Draw
    return 0  // Default for non-terminal nodes

Scoring mechanism:

• Maximizing Player: The AI tries to maximize the score.

score(max_node) = max(score(child1​), score(child2​),…,score(childn​))

• Minimizing Player: The opponent tries to minimize the AI's score.

score(min_node) = min(score(child1​), score(child2​),…,score(childn​))

• Evaluate: The evaluate function assigns a score to each terminal state.

Key Functions and Logic 

Here’s a brief explanation of the key functions and logic used in working of the mini max algorithm in AI.

1. evaluate(node)

This function checks the game’s current state and returns a score based on whether the game has ended (win, loss, or draw).

• If the AI wins, it returns +1.
• If the opponent wins, it returns -1.
• If the game results in a draw, it returns 0.

2. minimax(node, depth, maximizingPlayer)

This core function recursively explores all possible moves.

• Maximizing Player: During AI’s turn, it tries to maximize the score by choosing the move that gives the highest possible value from its children nodes.
• Minimizing Player: During the opponent's turn, it tries to minimize the AI's score by choosing the move that gives the lowest possible value from its children nodes.
• The function continues to alternate between maximizing and minimizing until it reaches a terminal state or the maximum depth.

Mini Max Algorithm in AI Example

One of the best examples of a mini max algorithm is the tic-tac-toe game. Here is a Python implementation of the game.

Code snippet:

# Tic-Tac-Toe Minimax Algorithm in Python

# Evaluate the board state (terminal state evaluation)
def evaluate(board):
    # Check for rows, columns, and diagonals for a win
    for row in range(3):
        if board[row][0] == board[row][1] == board[row][2] != None:
            return 1 if board[row][0] == 'X' else -1
    for col in range(3):
        if board[0][col] == board[1][col] == board[2][col] != None:
            return 1 if board[0][col] == 'X' else -1
    if board[0][0] == board[1][1] == board[2][2] != None:
        return 1 if board[0][0] == 'X' else -1
    if board[0][2] == board[1][1] == board[2][0] != None:
        return 1 if board[0][2] == 'X' else -1

    # Check for draw (no more empty spaces)
    if all(board[row][col] is not None for row in range(3) for col in range(3)):
        return 0  # Draw

    return None  # Game is still ongoing

# Minimax algorithm to find the best move for the AI
def minimax(board, depth, is_maximizing_player):
    score = evaluate(board)
    
    # If the game is over (win, loss, draw), return the score
    if score is not None:
        return score

    if is_maximizing_player:  # AI's turn ('X')
        best_score = -float('inf')
        for row in range(3):
            for col in range(3):
                if board[row][col] is None:
                    board[row][col] = 'X'  # Make the move
                    best_score = max(best_score, minimax(board, depth + 1, False))  # Minimize for opponent
                    board[row][col] = None  # Undo the move
        return best_score
    else:  # Opponent's turn ('O')
        best_score = float('inf')
        for row in range(3):
            for col in range(3):
                if board[row][col] is None:
                    board[row][col] = 'O'  # Make the move
                    best_score = min(best_score, minimax(board, depth + 1, True))  # Maximize for AI
                    board[row][col] = None  # Undo the move
        return best_score

# Function to find the best move for AI
def find_best_move(board):
    best_move = None
    best_score = -float('inf')
    for row in range(3):
        for col in range(3):
            if board[row][col] is None:
                board[row][col] = 'X'  # Make the move
                move_score = minimax(board, 0, False)  # Call minimax for the opponent's turn
                board[row][col] = None  # Undo the move
                if move_score > best_score:
                    best_score = move_score
                    best_move = (row, col)
    return best_move

# Example board (None represents empty spaces)
board = [
    ['X', 'O', 'X'],
    [None, 'X', None],
    ['O', None, None]
]

# Find and display the best move for AI ('X')
best_move = find_best_move(board)
print(f"The best move for AI is at row {best_move[0]}, column {best_move[1]}")

Explanation:

• evaluate(board): This function checks the current state of the tic-tac-toe board. It returns 1 if the AI ('X') wins, -1 if the opponent ('O') wins, and 0 if it’s a draw.
• minimax(board, depth, is_maximizing_player): This function implements the Minimax algorithm recursively. It evaluates all possible moves for both the AI (Maximizing Player) and the opponent (Minimizing Player) and returns the best score.
• find_best_move(board): This function runs the mini max algorithm for all possible moves and returns the AI's best move.

Output:

The best move for AI is at row 1, column 2

Also Read: 5 Types of Classification Algorithms in Machine Learning

What are the Variations of Mini Max Algorithm?

While the basic Minimax algorithm is effective for solving many problems, there are several variations and optimizations that can improve its performance, especially for complex games with large search spaces.

Here are the different variations of mini max algorithm.

Alpha-Beta Pruning (Optimization technique)

Alpha-beta pruning improves the efficiency of the mini max algorithm by reducing the number of nodes that need to be evaluated in the game tree. It skips over ‘branches’ of the game tree that don’t need to be explored, thus saving time and computational costs.

The basic idea behind alpha-beta pruning is to maintain two values:

• Alpha: The best score that the maximizing player can achieve (AI's best score).
• Beta: The best score that the minimizing player can achieve ( opponent's best score).

The alpha-beta technique evaluates the decision tree in the following ways.

• Maximizing Player's Node: If the value of the current node is greater than or equal to beta (the opponent’s best score), there is no need to continue exploring this node’s children as the opponent will avoid this branch.
• Minimizing Player's Node: If the value of the current node is less than or equal to alpha (the AI’s best score), then there is no need to explore further, as the AI will avoid this branch.

Iterative Deepening 

Iterative Deepening Minimax (ID Minimax) is a modification of the standard Minimax algorithm that combines the benefits of depth-first search (DFS) and breadth-first search (BFS). 

The logic is to progressively increase the search depth and run the mini max algorithm at each depth level. This allows the algorithm to return a solution quickly while continuing to explore deeper levels as time permits.

Here’s the working of interactive deepening.

• Initially, the algorithm evaluates the game tree with a shallow depth limit (say, depth 1).
• The algorithm then incrementally deepens the search by increasing the depth limit (depth 2, depth 3, and so on).
• The Minimax algorithm is applied at each depth level, with the search expanding to the specified depth for that iteration.

Here are the steps in its implementation.

1. Start with a depth limit of 1.
2. Perform mini max search at this depth.
3. Increase the depth limit and repeat the mini max search.
4. Continue this process until a terminal node is reached or the maximum time or depth limit is exceeded.
5. The best move found at the deepest searched level is returned as a result.

Negamax Algorithm

The negamax algorithm is a simplified version of mini max that exploits the symmetry between the maximizer and minimizer. 

Instead of maintaining separate logic for maximizing and minimizing, the algorithm uses a single function that handles both operations by removing the evaluation of the opposite player’s best move.

Here’s how the negamax algorithm works.

• The negamax algorithm uses the same process to calculate the score for both players. The main difference is that the negation of the opponent's evaluation is used to switch between maximizing and minimizing moves.
• In min max, the two players alternate between maximizing their score and minimizing the opponent’s score. However, in negmax, both players are considered to be trying to maximize their score, but the opponent’s score is negated.
• This process simplifies the code and removes the need for separate max and min logic. The evaluation function can be the same for both players, and the decision process remains symmetric.

Here are the steps in the Negamax algorithm.

1. Just like mini max, the negamax recursively explores the game tree. Instead of alternating between maximizing and minimizing, it uses negation of the opponent's score.
2. Each move is evaluated recursively, and the value of a node is calculated as the negation of the evaluation of its child node.
3. For the maximizer’s turn, you choose the move with the highest evaluation. For the minimizer, it’s the move with the lowest evaluation. 
4. Since you are using negation, the highest negation (which represents the lowest value for the opponent) is selected.

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Let’s check how the mini max algorithm fares compared to other algorithms.

Comparing Mini Max Algorithm in AI with Other Algorithms

While the Minimax algorithm is a foundational technique in AI, it's crucial to evaluate its performance relative to other algorithms like Monte Carlo Tree Search and Reinforcement Learning. Here’s a comparative analysis of these algorithms.

Mini Max Algorithm vs Monte Carlo Tree Search (MCTS) vs Reinforcement Learning

While the minimax algorithm explores the entire game tree or a portion of it up to a specified depth, the MCTS doesn't. Instead, it uses probabilistic sampling and simulation. 

Reinforcement Learning (RL) learns through trial and error. The algorithm explores actions and receives rewards or penalties based on the outcome of those actions, adjusting its strategy accordingly.

You can check the differences in the following table.

Also Read: Top Advantages and Disadvantages of Machine Learning in 2024

While the Minimax algorithm is a valuable tool in AI, it has its own strengths and weaknesses. Let's explore these in detail.

What are the Advantages and Disadvantages of Mini Max Algorithm in AI?

The mini max algorithm has proven to be a fundamental technique in artificial intelligence, which requires you to make optimal decisions in a short period. While the algorithm is beneficial in certain cases, it has its limitations in applications.

Here are the advantages and disadvantages of mini max algorithm.

What are the Advantages of Mini Max Algorithm in AI?

You can check the following advantages of mini max algorithm.

• Optimal decision-making

The mini max algorithm is designed to make the best possible decision by exhaustively analyzing all possible moves in a game. It ensures that the player makes the best move possible, considering all potential outcomes.

For example, the mini max algorithm in tic-tac-toe evaluates all possible moves and chooses the one that leads to a win or blocks the opponent's potential win.

• Simplicity in design

The algorithm's logic is easy to understand and implement. Its simple recursive approach makes it a great starting point for understanding game-playing AI.

For example, the logic for implementing tic-tac-toe is easy due to its small decision tree.

• Applicability in two-player games

The algorithm works well for two-player, zero-sum games in which players take turns and the game state changes predictably.

For example, the mini max strategy can analyze the chess game tree at a reasonable depth and suggest the best possible move.

• Non-randomized approach

The mini max algorithm is reliable in scenarios where consistent, predictable outcomes are needed. 

For example, in a checkers game, the AI’s moves are calculated based on the worst-case scenario of an opponent’s strategy.

Also Read: Best Machine Learning Courses in 2024

What are the Disadvantages of Mini Max Algorithm in AI?

While the mini max algorithm is a powerful tool for decision-making in two-player, zero-sum games, certain limitations can impact its performance and applicability, especially in complex scenarios.

Here are some major disadvantages of the mini max algorithm in AI.

• High computational cost

The algorithm requires evaluating all possible moves, which can be computationally expensive. For complex games, this leads to high processing power requirements.

For example, the mini max algorithm takes high computation powers for games like chess, which has millions of possible game states.

• Exponential growth in game trees

The rise in the number of possible moves exponentially increases the number of branches in the game tree. An increase in the game states makes the algorithm inefficient for large, complex games.

For example, the branching factor for chess is about 35, meaning each player has roughly 35 possible moves at every turn.

• Lack of practicality in complex games

Mini max algorithm is impractical for complex games with large state spaces, randomness, or incomplete information. 

For example, the mini max game cannot be easily applied in poker because it requires exhaustive exploration of all possibilities.

• Limited to perfect information games

The mini max algorithm works best in perfect information games, where both players have full knowledge of the game state. 

For example, in Poker, mini max's assumption of perfect information is violated, making it impractical.

Also Read: How to Implement Machine Learning Steps?

Now that we've explored the Minimax algorithm's strengths and weaknesses let's examine its practical applications in the real world.

What are the Applications of Mini Max Algorithm in AI?

The mini max algorithm has applications in areas such as robotics and self-driving cars, where optimal decision-making is required. Here are some important applications of mini max algorithm in AI.                                                                                              

• Game Theory

The algorithm is widely used in game theory, especially for two-player, zero-sum games in which one player’s gain is the other player’s loss.

For example, in chess, the algorithm simulates all possible moves for both players.

• Decision-making systems

The algorithms help decision-making systems make strategic choices in adversarial environments. For example, they are suitable in domains such as economics, negotiation, and military strategy, where opponents’ actions must be considered.

• Artificial intelligence in board games

The mini max algorithm calculates moves and makes decisions in board games such as Othello and Go.

For example, in Othello, the algorithm evaluates the board configurations and simulates the opponent's possible future moves. 

• Optimization problems

The algorithm is capable of solving optimization problems that involve resource allocation and scheduling. 

For example, the algorithm can be used in supply chain management to make optimal decisions based on factors such as market demand fluctuations.

• Robotics

The algorithms help robots to simulate all possible interactions with obstacles and make decisions accordingly. 

For example, mini max can model interactions between vehicles and pedestrians, helping vehicles take decisions to avoid collisions.

Also Read: 12 Best Robotics Projects Ideas

After going through the intricacies of the mini max algorithm, let's explore how you can leverage this knowledge to build a successful career in AI.

How Can UpGrad Help You Build a Career?

Just like the mini max algorithm helps you choose the best move in a game, you need to make calculated decisions in your career by evaluating all available opportunities. 

upGrad’s courses will support you at every step, helping you build the skills and knowledge to make informed and impactful decisions that propel your career forward.

Here are some popular courses by upGrad in AI and Machine Learning.

• Introduction to Generative AI
• Fundamentals of Deep Learning and Neural Networks
• Artificial Intelligence in the Real World

Do you need help deciding which course to choose for your career as an AI specialist? Contact upGrad for personalized counseling and valuable insights.

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References:
https://www.statista.com/outlook/tmo/artificial-intelligence/worldwide