Minimax Pathology and Real-Number Minimax Model

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Presentation transcript:

Minimax Pathology and Real-Number Minimax Model Mitja Luštrek Department of Intelligent Systems Jožef Stefan Institute Ljubljana, Slovenia

Minimax Basic mechanism in virtually all game-playing programs 2004-09-27 Minimax Basic mechanism in virtually all game-playing programs Game tree: nodes – positions arcs – moves Mitja Luštrek, JSI

2004-09-27 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

2004-09-27 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

2004-09-27 Experimental Results No pathology! Mitja Luštrek, JSI

Generality The model has a number of parameters: Branching factor (2) 2004-09-27 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

Verification in Chess Static values compared with chess program Crafty 2004-09-27 Verification in Chess Static values compared with chess program Crafty Mitja Luštrek, JSI

Mathematical Explanation (1) 2004-09-27 Mathematical Explanation (1) Mitja Luštrek, JSI

Mathematical Explanation (2) 2004-09-27 Mathematical Explanation (2) Mitja Luštrek, JSI

Conclusion Designed a non-pathological minimax model 2004-09-27 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

2004-09-27 Thank you. Questions? Mitja Luštrek, JSI