Implementing Alpha-Beta Pruning for Board Game AI

Building a smart AI opponent for board games is one of the most rewarding challenges in game development. In this tutorial, we'll explore how to implement the Minimax algorithm with Alpha-Beta pruning - the same foundational technique used in chess engines and countless other games.

Who is this for? Developers with basic programming knowledge who want to understand and implement game AI. We'll keep the code examples language-agnostic where possible.

Understanding the Problem

When building AI for turn-based games, we need to answer one question: "What's the best move?"

To answer this, the AI must:

1. Consider all possible moves
2. Simulate what happens after each move
3. Assume the opponent will also play optimally
4. Choose the move leading to the best outcome

This is exactly what the Minimax algorithm does.

The Minimax Algorithm

Core Concept

Minimax operates on a simple principle:

• The AI (maximizing player) wants to maximize their score
• The opponent (minimizing player) wants to minimize the AI's score

The algorithm builds a game tree - a branching structure of all possible game states - and traverses it to find the optimal move.

How It Works

function minimax(state, depth, isMaximizing):
    if depth == 0 or game_over(state):
        return evaluate(state)
    
    if isMaximizing:
        bestValue = -INFINITY
        for each possible move:
            value = minimax(result(state, move), depth - 1, false)
            bestValue = max(bestValue, value)
        return bestValue
    else:
        bestValue = +INFINITY
        for each possible move:
            value = minimax(result(state, move), depth - 1, true)
            bestValue = min(bestValue, value)
        return bestValue

The Evaluation Function

The evaluate() function is crucial - it assigns a numerical score to any game state. For example:

• Win state: +1000
• Lose state: -1000
• Draw: 0
• Advantageous position: Positive value
• Disadvantageous position: Negative value

Designing a good evaluation function is an art. It encodes your understanding of what makes a position "good" in your specific game.

The Problem: Exponential Complexity

Here's the issue: the game tree grows exponentially.

If each position has 10 possible moves and we look 6 moves ahead:

• Level 1: 10 positions
• Level 2: 100 positions
• Level 3: 1,000 positions
• Level 6: 1,000,000 positions

For complex games, this becomes computationally impossible. Enter Alpha-Beta Pruning.

Alpha-Beta Pruning: The Optimization

The Key Insight

Many branches of the game tree don't need to be explored. If we've already found a good move to ensure a certain score at a certain depth, we can cut off the exploration of nodes that cannot possibly affect the final decision - this is 'pruning'.

The Algorithm

Alpha-Beta pruning introduces two values:

• Alpha (α): Best score the maximizer can guarantee (starts at -∞)
• Beta (β): Best score the minimizer can guarantee (starts at +∞)

function alphabeta(state, depth, alpha, beta, isMaximizing):
    if depth == 0 or game_over(state):
        return evaluate(state)
    
    if isMaximizing:
        value = -INFINITY
        for each possible move:
            value = max(value, alphabeta(result(state, move), 
                                         depth - 1, alpha, beta, false))
            alpha = max(alpha, value)
            if alpha >= beta:
                break  // Beta cutoff (prune!)
        return value
    else:
        value = +INFINITY
        for each possible move:
            value = min(value, alphabeta(result(state, move), 
                                         depth - 1, alpha, beta, true))
            beta = min(beta, value)
            if beta <= alpha:
                break  // Alpha cutoff (prune!)
        return value

Why It Works

Imagine you're the maximizer at the root, and you've already found a move giving you +5. Now you're exploring another branch:

1. You go down one level (minimizer's turn)
2. The minimizer finds an option giving them -10 (which is +10 from your perspective... wait, no)
3. Actually, if the minimizer can guarantee you get at most +3 in this branch...
4. ...and you already have +5 guaranteed elsewhere...
5. Why explore further? This branch can't help you!

This is the pruning in action. We skip entire subtrees that can't affect our decision.

Performance Gains

With optimal move ordering (checking likely good moves first), Alpha-Beta can reduce the effective branching factor from b to approximately √b.

That's potentially a 1000x speedup!

Implementation Tips

1. Move Ordering Matters

The effectiveness of pruning depends heavily on move ordering. Try good moves first:

• Captures before quiet moves
• Center squares before edges (in many games)
• Historically successful moves
• Killer heuristic (moves that caused cutoffs before)

2. Iterative Deepening

Instead of searching directly to depth 6:

1. Search to depth 1
2. Search to depth 2
3. ...continue until time runs out

Benefits:

• Any-time algorithm (always has some answer)
• Results from shallower searches inform move ordering
• Natural time management

3. Transposition Table

Many game positions can be reached via different move sequences. Store already-evaluated positions in a hash table to avoid recalculating them.

transposition_table = {}

function alphabeta_with_memory(state, depth, alpha, beta, isMax):
    key = hash(state)
    if key in transposition_table:
        return transposition_table[key]
    
    // ... normal alphabeta logic ...
    
    transposition_table[key] = result
    return result

Practical Considerations for Board Games

Handling Randomness (e.g., Dice Games)

For games with random elements, you'll need Expectiminimax - an extension that adds "chance nodes" to the tree, averaging over possible random outcomes.

Adjustable Difficulty

Make your AI accessible:

• Easy: Shallow search depth (1-2)
• Medium: Moderate depth (3-4) + occasional random moves
• Hard: Full depth + optimal play

Time Management

In real-time games, use iterative deepening with a time limit rather than a fixed depth.

Further Reading

This tutorial covers the fundamentals. To go deeper:

• Negamax: A simplified Minimax formulation
• Principal Variation Search (PVS): Advanced Alpha-Beta variant
• Monte Carlo Tree Search (MCTS): Alternative approach used in Go AI
• Neural Network Evaluation: Modern AI like AlphaZero

Conclusion

Alpha-Beta pruning transforms an exponentially complex problem into something tractable. With this foundation, you can build AI opponents that provide genuine challenge while remaining computationally efficient.

The beauty of this algorithm lies in its elegance: we're not making the AI "smarter" - we're making it faster at being smart, by ignoring obviously bad options.

Want to see these algorithms in action? Play against our AI at play-ludo.com and experience smart board game AI firsthand!

Have questions about implementing game AI? Join the discussion in our community!