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Minimax Pathology and Real-Number Minimax Model

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Presentation on theme: "Minimax Pathology and Real-Number Minimax Model"— Presentation transcript:

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

2 Minimax Basic mechanism in virtually all game-playing programs
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)
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
Verification in Chess Static values compared with chess program Crafty Mitja Luštrek, JSI

8 Mathematical Explanation (1)
Mathematical Explanation (1) Mitja Luštrek, JSI

9 Mathematical Explanation (2)
Mathematical Explanation (2) Mitja Luštrek, JSI

10 Conclusion Designed a non-pathological minimax model
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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