Inside the Minimax Algorithm

In our previous article, we introduced the concept of adversarial search and met our two players: Max and Min. Now let's see how computers actually decide on the best move using the Minimax algorithm!

The Minimax Algorithm: Thinking Ahead

Imagine you're playing Tic-Tac-Toe and it's your turn. You want to make the best move possible. Here's how the Minimax algorithm would approach this:

1. For each possible move you could make, imagine making that move
2. Then, for each of your moves, consider all possible responses your opponent might make
3. Continue this pattern until reaching an end state (win, lose, or draw)
4. Assign values to these end states: +1 for a win, 0 for a draw, -1 for a loss
5. Work backward to determine which initial move leads to the best outcome

Let's Walk Through a Simple Example

Imagine we're near the end of a Tic-Tac-Toe game:

It's Max's turn (X). Let's see how Minimax would analyze this:

1. Max has 3 possible moves (top-middle-left, bottom-middle, top-right)
2. For each of Max's moves, Min (O) will have 2 remaining possible responses
3. We need to determine which of Max's initial moves will lead to the best outcome

Understanding Game Trees

This scenario creates what we call a game tree. The starting position is at the top, and each possible move creates a new branch.

Each position is assigned a value based on whether it leads to a win, loss, or draw.

Here is a partial game tree from the board above.

How Values Travel Up the Tree

Here's the clever part! The values from the end states travel upward:

• At a Max node, we take the highest value from the children (Max wants the best outcome)
• At a Min node, we take the lowest value from the children (Min tries to minimize Max's success)

So if Max has three possible moves leading to values of +1, 0, and -1, Max will choose the move that leads to +1.

Computers Don't Just Look at the Final Outcome

In more complex games like chess, looking at all possible moves until the end of the game would be impossible - there are more possible chess games than atoms in the universe!

Instead, computers use evaluation functions to estimate how good a position is without playing all the way to the end. For example, in chess, having more pieces than your opponent is generally good, so the computer might assign a higher value to those positions.

In the next part...

We'll discover how computers make these calculations faster using a technique called Alpha-Beta Pruning, and see how this applies to more complex games beyond Tic-Tac-Toe!