Minimax Pathology and Real-Number Minimax Model

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

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

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

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

Experimental Results • No pathology! Mitja Luštrek, JSI

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 Mitja Luštrek, JSI

Mathematical Explanation (1) Mitja Luštrek, JSI

Mathematical Explanation (2) Mitja Luštrek, JSI

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

Thank you.Questions? Mitja Luštrek, JSI

Minimax Pathology and Real-Number Minimax Model