Combinatorial Game Theory

PSU CS410/510GAMES Lecture 9
May 30, 2002

• Review: Hidden Information
  • Failure of pure strategies
  • Mixed strategies
  • Opponent modeling: RSP
  • Iocaine Powder (Dan Egnor) and machine learning
    • Strategies
      • Naive prediction, adjust once, adjust twice
      • Three reverse strategies
    • Predictors
      • Random guess
      • Frequency analysis
      • History matching: find previous eg of current sequence
      • Variants of these
    • Meta-strategies: pick strategy and predictor based on performance over various history sizes
• Theory of Combinatorial Games (Berlekamp, Conway, Guy)
  • Our analysis: situational, may be approximate
  • CGT: side-independent, exact
    • Analyze games where first "stuck" player loses (e.g. Amazons)
    • First approximation: who wins in a state if
      • Left moves first
      • Right moves first
    • Moving first is
      • Disadvantage: run out of moves sooner
      • Advantage: can control what happens
  • Example: Blue-Red Hackenbush
  • Rules
    • Draw a rooted graph with blue and red edges
    • Left player deletes a blue edge, right deletes red
    • Any disconnected edges are discarded.
  • Analysis
    • Simplest positions are boring
    • Separable positions are more interesting
    • Mixed positions are complicated
      • Symmetry argument
      • Fractional moves
    • Value of a position is to Left
      • Computed by combining games
• Theory generalizes
  • Amazons endgames are analyzable using this theory
  • Endgame position values
  • Berlekamp analyzes all 2x10 positions
  • Seperated endgames are NP-complete! (Buro)