My Blog.

Illustrate Mini-Max search for the tic-tac-toe game.

Mini-Max Search for Tic-Tac-Toe

Definition

The Mini-Max search algorithm is a recursive method used in decision-making and game theory to determine the optimal move for a player, assuming that the opponent is also playing optimally. It is commonly used in zero-sum games such as tic-tac-toe, chess, and checkers.

Key Concepts

  • Maximizer and Minimizer: In tic-tac-toe, the algorithm considers two players: the maximizer (usually the AI or the player making the move) and the minimizer (the opponent). The maximizer aims to maximize their score, while the minimizer aims to minimize the maximizer's score.
  • Game Tree: A tree structure representing all possible moves from the current position, with nodes representing game states and edges representing moves.
  • Terminal States: The end points of the game tree, which represent win, lose, or draw scenarios.
  • Utility Function: A function that assigns a value to terminal states (e.g., +1 for a win, -1 for a loss, and 0 for a draw).

Detailed Explanation

  1. Initialization:

    • The algorithm starts with the current game state (board configuration).
    • It constructs a game tree with the current state as the root.

  2. Building the Game Tree:

    • The game tree is constructed by recursively generating all possible moves for both players.
    • Each node in the tree represents a possible state of the board after a move by either player.
    • The tree alternates between levels representing the maximizer's move and the minimizer's move.

  3. Evaluating Terminal States:

    • When a terminal state (win, lose, draw) is reached, the utility function assigns a value to that state.
    • For example:
      • +1 if the maximizer wins
      • -1 if the minimizer wins
      • 0 for a draw

  4. Backpropagation:

    • The algorithm backpropagates the utility values from the terminal states up the tree.
    • At each node, if it is the maximizer's turn, the node's value is the maximum value of its children (maximizing player chooses the best move for itself).
    • If it is the minimizer's turn, the node's value is the minimum value of its children (minimizing player chooses the move that is least favorable to the maximizer).

  5. Choosing the Optimal Move:

    • The root node's value is determined by the best possible move the maximizer can make.
    • The move leading to the child node with this value is selected as the optimal move.

Example: Mini-Max in Tic-Tac-Toe

Let's illustrate the Mini-Max algorithm with a simplified example of tic-tac-toe.

Initial Board State

 X | O | X
-----------
 O | X |  
-----------
   |   | O

It's X's turn to move.

Step-by-Step Illustration

  1. Current Game State (Root Node):

    • X has to decide between the following possible moves:
      • (2, 1) or (2, 2)

  2. Generate Game Tree:

    (2, 1)      (2, 2)
     /           \

  1. Evaluating Terminal States:

    • After X moves at (2, 1):

       X | O | X
 -----------
 O | X |  
 -----------
 X |   | O
      • X wins, utility = +1

    • After X moves at (2, 2):

       X | O | X
 -----------
 O | X |  
 -----------
   | X | O
      • No win or draw, O's turn.

  2. Generate Next Level of the Game Tree (O's Move):

    (2, 1)      (2, 2)
     /           \
    +1            (2, 1)
                  /
                 -1

  • After O moves at (2, 1):

     X | O | X
 -----------
 O | X |  
 -----------
 X | O | O
    • O wins, utility = -1

  1. Backpropagation:

    • The node (2, 1) has a utility of +1.
    • The node (2, 2) has children with utilities of -1.
    • X (maximizer) will choose the move with the highest utility:
      • Move (2, 1) results in +1.
      • Move (2, 2) results in -1 (worst-case scenario for X).

  2. Choosing the Optimal Move:

    • X will choose to move at (2, 1) as it leads to a win (utility = +1).

Diagram of the Mini-Max Tree

                     [Root Node (Current State)]
                        /                   \
              [Move (2, 1), +1]         [Move (2, 2)]
                                          /     
                               [Move (2, 1), -1]

Links to Resources

Notes and Annotations

  • Summary of Key Points:

    • Mini-Max algorithm helps in making optimal decisions in zero-sum games.
    • It evaluates all possible moves and chooses the one that maximizes the player's minimum gain.
    • The game tree represents all possible moves and outcomes, with utility values assigned to terminal states.
    • Backpropagation ensures that the best move is selected based on the maximizer's and minimizer's optimal plays.

  • Personal Annotations and Insights:

    • Implementing Mini-Max for tic-tac-toe is straightforward due to the game's limited complexity.
    • The efficiency of the algorithm can be improved using alpha-beta pruning to eliminate unnecessary branches.
    • Understanding Mini-Max is fundamental for developing more complex game AI, such as for chess or Go.

Backlinks

  • Adversarial Search in AI
  • Optimal Decision in Game Theory
  • Heuristic Search Techniques