Lecture Notes: Game Playing Algorithms and Minimax Algorithm

Introduction to Game Playing Algorithms

• Computers excel at playing games (e.g., Tic-Tac-Toe, Chess)
• Ability to evaluate game states and determine optimal moves

Minimax Algorithm

• Purpose: Determine optimal decision in adversarial games
• Adversarial Games: Two players with opposite goals (win vs. lose)
• Optimal Play: Best outcome for oneself even against an optimally playing opponent
• Key Idea: Not only play to win but also prevent opponent from winning

Understanding Game States

• Terminal States: Game over (e.g., X wins, O wins, tie)
• Assign numerical values to terminal states:
  • X wins = 1
  • O wins = -1
  • Tie = 0

Player Goals

• MAX Player (X): Goal is to maximize score (win or tie)
• MIN Player (O): Goal is to minimize score (win or tie)

Evaluating Non-Terminal States

• For O’s turn (MIN player):
  • Consider possible actions leading to new game states
  • Calculate outcomes for possible actions
• Examples of outcomes:
  • One move leads to X winning (value = 1)
  • Another leads to tie (value = 0)

Minimax Algorithm Pseudocode

• Functions needed:
  • Terminal: Determines if the game is over
  • Value: Get value of terminal state
  • Player: Determines whose turn it is
  • Actions: Possible actions in a game state
  • Result: New game state after an action

Algorithm Procedure

1. Check if state is terminal
2. If terminal, return its value
3. If not terminal:
   • If MAX’s turn:
     • Loop through actions, calculate value of resulting states, return max value
   • If MIN’s turn:
     • Loop through actions, calculate value of resulting states, return min value

Recursion and Optimality

• The algorithm recursively calls itself until reaching terminal states
• If implemented correctly, it ensures optimal moves
• Works well for small games (e.g., Tic-Tac-Toe) but becomes complex for larger games (e.g., Chess)

Challenges with More Complex Games

• Larger game spaces lead to intractability (e.g., Chess)
• Many possible moves and potential game states increase exponentially

Optimizations for Minimax Algorithm

Alpha-Beta Pruning

• Skip irrelevant game states when outcomes are worse than already explored options
• Improves efficiency without affecting correctness of the algorithm

Depth Limitation and Evaluation Functions

• Limit the depth of exploration (e.g., 3, 5, or 10 moves ahead)
• Use of evaluation functions to estimate game state value
  • A simple function could sum piece values in Chess, but not always accurate
• Importance of robust evaluation functions for better performance

Machine Learning Approaches

• Recent techniques for computers to learn how to evaluate positions
• Enhanced methods to play more efficiently against opponents

Conclusion

• Minimax remains a foundational algorithm for game-playing agents
• Continues to evolve with optimizations and new techniques in AI development.