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Minimax Pathology and Real-Number Minimax Model

This paper discusses the minimax algorithm in game-playing programs, the pathology of heuristic evaluation in the leaves of the game tree, and introduces a non-pathological real-number minimax model for chess.

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Minimax Pathology and Real-Number Minimax Model

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  1. Minimax Pathology andReal-Number Minimax Model Mitja Luštrek Department of Intelligent Systems Jožef Stefan Institute Ljubljana, Slovenia

  2. Minimax • Basic mechanism in virtually all game-playing programs • Game tree: • nodes – positions • arcs – moves Mitja Luštrek, JSI

  3. 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 Mitja Luštrek, JSI

  4. 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 Mitja Luštrek, JSI

  5. Experimental Results • No pathology! Mitja Luštrek, JSI

  6. 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 Mitja Luštrek, JSI

  7. Verification in Chess • Static values compared with chess program Crafty Mitja Luštrek, JSI

  8. Mathematical Explanation (1) Mitja Luštrek, JSI

  9. Mathematical Explanation (2) Mitja Luštrek, JSI

  10. 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 Mitja Luštrek, JSI

  11. Thank you.Questions? Mitja Luštrek, JSI

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