2004 09 27 Minimax Pathology and RealNumber Minimax

2004 -09 -27 Minimax Pathology and Real-Number Minimax Model Mitja Luštrek Department of Intelligent Systems Jožef Stefan Institute Ljubljana, Slovenia Mitja Luštrek, JSI

2004 -09 -27 Minimax n Basic mechanism in virtually all game-playing programs n Game tree: n nodes – positions n arcs – moves Mitja Luštrek, JSI 2

2004 -09 -27 Minimax Pathology n An accepted fact that the deeper a program searches a game tree, the better it plays n Seemingly sensible mathematical model shows the opposite: error in heuristic evaluation of the leaves is amplified through minimaxing [Beal, 1980] n Many attempts to explain, no definite conclusion [Bratko & Gams, 1982; Beal, 1982; Nau, 1982, 1983; Pearl, 1983; Sadikov et al. , 2003] n Analyses performed on two-value models Mitja Luštrek, JSI 3

2004 -09 -27 Real-Number Model n Static values of game-tree nodes assigned from the root downwards: n Values of nodes normally distributed around the value of their parent n Rationale: positions one move apart must have similar values n Normally distributed error in the leaves n Backed-up values computed from the leaves upwards Mitja Luštrek, JSI 4

2004 -09 -27 Experimental Results n No pathology! Mitja Luštrek, JSI 5

2004 -09 -27 Generality n The model has a number of parameters: n Branching factor (2) n Types of distributions (normal) n Standard deviation of error (0. 2) n. . . n Many combinations tried, absence of pathology persists Mitja Luštrek, JSI 6

2004 -09 -27 Verification in Chess n Static values compared with chess program Crafty Mitja Luštrek, JSI 7

2004 -09 -27 Mathematical Explanation (1) Mitja Luštrek, JSI 8

2004 -09 -27 Mathematical Explanation (2) Mitja Luštrek, JSI 9

2004 -09 -27 Conclusion n Designed a non-pathological minimax model n Showed that it corresponds to chess as played by a high-quality program n Explained the reason why increased depth of search reduces the error n Pathology appears to be the product of limitations of two-value models n In real-number model, minimax can be shown to work as expected Mitja Luštrek, JSI 10

2004 -09 -27 Thank you. Questions? Mitja Luštrek, JSI 11