Minimax Algorithm in Artificial Intelligence: Use Cases & Examples Explained

The minimax algorithm in AI​ is a decision-making tool used in two-player, turn-based games like chess, tic-tac-toe, and checkers. It helps artificial intelligence (AI) determine the best move by predicting the opponent’s responses and minimizing potential losses. 

In 2025, as Artificial Intelligence extends beyond gaming into areas like strategic planning and automated negotiations, Minimax remains a key technique for creating smart, competitive agents.

In this blog, you’ll explore the core concepts of the minimax algorithm in AI​, see clear examples in action, and discover practical use cases where Minimax is applied!

Let’s explore the way the Minimax algorithm helps AI make strategic, outcome-driven decisions.

How the Minimax Algorithm in AI​ Helps in Decision-Making?

The Minimax algorithm is a classic AI strategy used in two-player turn-based games to decide the best possible move. One player tries to maximize their chances of winning, while the other tries to minimize those chances, hence the name "Minimax."

How Minimax Works?

The minimax algorithm in AI​ builds a game tree, a structured map of all possible moves and their outcomes. At each level of the tree:

• The AI (maximizing player) selects the move that leads to the highest possible reward.
• The opponent (minimizing player) chooses the move that would minimize the AI’s potential gain.

The algorithm continues until it reaches terminal states, positions where the game ends (win, loss, or draw). It then assigns scores to these outcomes and backtracks through the tree, choosing the move that leads to the most favorable result.

In more complex scenarios, this process can be enhanced with Alpha-Beta Pruning, which eliminates branches of the tree that don’t need to be explored, making the algorithm more efficient.

If you’re looking to become an expert in AI and create intelligent systems that automate tasks and minimize manual effort, upGrad offers some excellent courses to help you achieve your goals:

• Generative AI Foundations Certificate Program
• Advanced Generative AI Certification Course
• Master’s Degree in Artificial Intelligence and Data Science

Here are some of the example scenarios of the minimax algorithm.

• Tic-tac-toe

Minimax analyzes all possible moves on the board. If the AI can win on the next turn, it will do so. If not, it will block the opponent’s winning move. The algorithm continues to simulate moves until the best possible outcome is identified.

• Chess

The algorithm evaluates a set number of future moves for both players. It scores potential outcomes based on predefined evaluation metrics (e.g., material value, position control), and chooses the move that maximizes the AI's advantage while minimizing the opponent's.

Also Read: What Is Artificial Intelligence? Its Uses, Types and Examples

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

Key Components of Minimax Algorithm

The minimax algorithm works by stimulating 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

Represents the decision point for the player (or AI) aiming to maximize their score. The algorithm evaluates all possible moves and chooses the one with the highest potential score.

• Min mode

Represents the opponent’s decision point, who tries to minimize the player’s score. It assumes the opponent plays optimally and will select moves that reduce the player’s potential advantage.

Also Read: Actor Critic Model in Reinforcement Learning

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.

How the Minimax Algorithm in AI​Works: A Mathematical View

The minimax algorithm in AI​ can be expressed mathematically using recursive formulas that evaluate each node in the game tree based on whether it is a max node (AI’s turn) or a min node (opponent’s turn). Each node’s score represents the utility or value of the game state from the AI’s perspective.

1. Max Node: At a max node, the AI tries to maximize the score. The score of this node is the maximum of the scores of its child nodes, which represent possible next moves:

The AI will select the move leading to the highest score.

2. Min Node: At a min node, the opponent tries to minimize the AI’s score. The score of this node is the minimum of the scores of its child nodes:

The opponent chooses the move that reduces the AI’s chances of winning.

3. Recursive Formula: The minimax algorithm in AI​ recursively evaluates the game tree, alternating between maximizing and minimizing at each node, starting from the root node:

This process continues until terminal nodes (end game states) are reached, at which point scores are assigned based on the game outcome (win, loss, or draw). The algorithm then backtracks, choosing moves that maximize the AI’s chances of success.

Also Read: How to Build Your Own AI System: Step-by-Step Guide

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

Implementation of Minimax Algorithm in AI

The minimax algorithm in AI​ is a powerful strategy for decision-making in games, enabling players (or AI agents) to choose optimal moves by exploring all possible future game states. It systematically evaluates potential outcomes and selects the best course of action to maximize chances of winning while minimizing losses.

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 in AI 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.

In modern AI, advanced agents like AlphaGo combine the Minimax approach with deep learning techniques. For example, AlphaGo uses neural networks to evaluate game positions and Monte Carlo Tree Search (MCTS) to explore possible moves. This hybrid approach enhances decision-making by effectively exploring complex game spaces, going beyond the traditional Minimax algorithm's capabilities.

Here is a breakdown of the minimax algorithm in AI​.

Pseudocode of Minimax Algorithm

The minimax algorithm explores all possible game states to determine the best possible move for the AI by simulating both the AI (maximizing player) and the opponent (minimizing player). 

Here's the pseudocode of the minimax algorithm in AI​, 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

Output:

# Sample output call
best_score = minimax(current_node, 0, True)
print("Best score:", best_score)

# Expected output for a winning move scenario
# Output:
# Best score: 1
 

Output Explanation:

The minimax() function recursively explores all possible game outcomes, simulating both the AI and the opponent’s moves. It evaluates terminal nodes using the evaluate() function and backpropagates the scores to choose the move that maximizes the AI’s chance of winning.

Scoring Mechanism Explained:

Minimax relies on a scoring system to evaluate terminal nodes and propagate those values up the tree.

1. Maximizing Player (AI): The AI aims to maximize the score. At each max node, it chooses the move that gives the highest score among all children.

Formula:

Example: If the AI has 3 options with scores: child1 = 5, child2 = 3, child3 = 9.

So, the AI will choose the move that leads to a score of 9.

2. Minimizing Player (Opponent): The opponent aims to minimize the AI's score. At each min node, the opponent picks the move that leads to the lowest score for the AI.

Formula:

Example: If the opponent has 3 possible moves: child1 = 5, child2 = 2, child3 = 8
Then:

The opponent chooses the move that leads to score 2, reducing the AI’s chances of winning.

3. Evaluation Function (Terminal States): The evaluate(node) function scores the leaf nodes (endgame scenarios) based on the outcome:

This score is then used to guide decisions as the algorithm backtracks up the game tree.

Here’s a quick summary table to help you clearly differentiate between the roles of the maximizing and minimizing players in the Minimax algorithm:

Once the scoring mechanism and game logic are defined, implementing the MiniMax algorithm requires choosing the right technology stack. Python is widely used for prototyping the Minimax algorithm due to its simplicity, with libraries like Pygame for game development and NumPy for numerical operations.

For performance-intensive applications, languages such as Java, C++, and C# are often preferred. Tools like OpenAI Gym and advanced chess engines like Stockfish implement optimized versions of MiniMax, frequently enhanced with Alpha-Beta pruning for greater efficiency.

If you want to understand how GenAI can help you in software development, check out upGrad’s Generative AI Mastery Certificate for Software Development. The program offers expertise on QA engineering and AI automation to optimize development workflows. 

Also read: 5 Significant Benefits of Artificial Intelligence [Deep Analysis]

Let’s explore one of the best examples of the Minimax algorithm in action: the classic game of Tic-Tac-Toe.

Minimax Algorithm in AI Example

One of the best examples of a minimax algorithm is the tic-tac-toe game.In this implementation, the AI plays as 'X' (maximizing player), and the opponent plays as 'O' (minimizing player). The board is represented as a 3x3 list of lists, where each cell contains 'X', 'O', or None to indicate an empty spot.

Here’s a Python implementation of the Minimax algorithm for Tic-Tac-Toe:

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]}")

Output:

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

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. Returns None if the game is ongoing.
• minimax(board, depth, is_maximizing_player): Recursively explores all possible moves for both players. The AI tries to maximize the score while the opponent tries to minimize it. The depth parameter tracks recursion depth and can be used to prioritize quicker wins (not utilized here).
• find_best_move(board): This function runs the minimax algorithm for all possible moves and returns the AI's best move.

Elevate your skills with upGrad's Job-Linked Data Science Advanced Bootcamp. With 11 live projects and hands-on experience with 17+ industry tools, this program equips you with certifications from Microsoft, NSDC, and Uber, helping you build an impressive AI and machine learning portfolio.

Also Read: 5 Types of Classification Algorithms in Machine Learning

Let’s explore the different variations of the Minimax algorithm that have been developed to optimize performance and handle complex game scenarios more efficiently.

Key Variations of the Minimax 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 minimax algorithm.

Alpha-Beta Pruning (Optimization technique)

Alpha-beta pruning improves the efficiency of the minimax 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 algorithm prunes (stops exploring) a branch when it finds that it cannot possibly affect the final decision:

• 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.

This pruning significantly reduces the number of nodes evaluated and is widely used in complex games like chess.

Iterative Deepening 

Iterative Deepening Minimax (ID Minimax) combines the advantages of depth-first search (DFS) and breadth-first search (BFS) by progressively increasing the search depth. This approach allows the algorithm to quickly provide a reasonably good solution (like BFS) and continue improving it over time using the space-efficiency of DFS.

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.

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

This approach is useful in time-constrained scenarios such as timed games or real-time decision-making.

Negamax Algorithm

The negamax algorithm is a simplified version of minimax 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 minmax, the two players alternate between maximizing their score and minimizing the opponent’s score. However, in negamax, 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.

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

This simplification leads to cleaner, more maintainable code while preserving the logic of Minimax.

Negamax is a rule-based algorithm that evaluates moves using a static evaluation function, while machine learning models, like those in AlphaGo, learn and adapt strategies over time through experience. This makes ML approaches more dynamic and evolving in decision-making.

Enhance your ability to craft powerful, effective prompts with upGrad's Advanced Prompt Engineering with ChatGPT free course. Master how to interact with ChatGPT for complex use cases, driving impactful results in content generation and beyond.

Also Read: Generative AI in Practice: Real-World Use Cases and Success Stories

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

Comparing Minimax 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.

Minimax 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: MATLAB vs Python: Which Programming Language is Best for Your Needs?

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

Advantages and Challenges of the Minimax Algorithm in AI

The minimax 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 minimax algorithm.

What are the Advantages of Minimax Algorithms in AI?

You can check the following advantages of the minimax algorithm.

• Optimal decision-making

The minimax 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 minimax 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 minimax strategy can analyze the chess game tree at a reasonable depth and suggest the best possible move.

• Non-randomized approach

The minimax 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: How to Make a Chatbot in Python Step by Step [With Source Code] in 2025

What are the Disadvantages of Minimax Algorithm in AI?

While the minimax 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 minimax 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 minimax algorithm in AI takes high computation powers for games like chess, which has millions of possible game states.

• Exponential Growth of 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.

• Impracticality for Complex Games

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

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

• Limited to Perfect Information Scenarios

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

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

Also Read: How to Create a Perfect Decision Tree | Decision Tree Algorithm [With Examples]

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 Minimax Algorithms 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 the minimax 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 minimax 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 take decisions accordingly. 

For example, minimax 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 minimax algorithm, let's explore how you can leverage this knowledge to build a successful career in AI.

Transform AI Concepts Like Minimax into Career Skills with upGrad!

The Minimax Algorithm in AI​ is a decision-making method used primarily in two-player, zero-sum games to optimize moves by minimizing the possible loss for a worst-case scenario. It’s widely applied in game theory, robotics, and strategic decision systems for optimal outcomes.

To effectively utilize such powerful algorithms, upGrad’s courses provide comprehensive support at every step, helping you build the skills and knowledge needed 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
• ChatGPT for Developers

Not sure which course to choose for your career as an AI specialist? Reach out to upGrad for personalized counseling and expert insights. For more information, visit your nearest upGrad offline center.

Transform your passion for AI into expertise with our Best Machine Learning and AI Courses Online, designed for career success.

Best Machine Learning and AI Courses Online

Stay ahead in the tech race by mastering In-Demand Machine Learning Skills that drive innovation and career growth in AI-powered industries!

In-demand Machine Learning Skills

Learn, grow, and stay updated with our curated AI and ML Blogs & Free Courses, designed for tech enthusiasts and professionals alike.

Popular AI and ML Blogs & Free Courses

Reference:
https://www.tidio.com/blog/ai-statistics/