Minimax Pathology and RealNumber Minimax Model

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Minimax Pathology andReal-Number Minimax Model

• Mitja Lutrek
• Department of Intelligent Systems
• Joef Stefan Institute
• Ljubljana, Slovenia

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Minimax

• Basic mechanism in virtually all game-playing
  programs
• Game tree
• nodes positions
• arcs moves

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Minimax Pathology

• An accepted fact that the deeper a program
  searches a game tree, the better it plays
• Seemingly sensible mathematical model shows the
  opposite error in heuristic evaluation of the
  leaves is amplified through minimaxing Beal,
  1980
• Many attempts to explain, no definite conclusion
  Bratko Gams, 1982 Beal, 1982 Nau, 1982,
  1983 Pearl, 1983 Sadikov et al., 2003
• Analyses performed on two-value models

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Real-Number Model

• Static values of game-tree nodes assigned from
  the root downwards
• Values of nodes normally distributed around the
  value of their parent
• Rationale positions one move apart must have
  similar values
• Normally distributed error in the leaves
• Backed-up values computed from the leaves upwards

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Experimental Results

• No pathology!

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Generality

• The model has a number of parameters
• Branching factor (2)
• Types of distributions (normal)
• Standard deviation of error (0.2)
• ...
• Many combinations tried, absence of pathology
  persists

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Verification in Chess

• Static values compared with chess program Crafty

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Mathematical Explanation (1)
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Mathematical Explanation (2)
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Conclusion

• Designed a non-pathological minimax model
• Showed that it corresponds to chess as played by
  a high-quality program
• Explained the reason why increased depth of
  search reduces the error
• Pathology appears to be the product of
  limitations of two-value models
• In real-number model, minimax can be shown to
  work as expected

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Thank you.Questions?