Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Monte Carlo *-Minimax Search Marc Lanctot Abdallah Saffidine Department of Knowledge Engineering LAMSADE, Maastricht University, Netherlands Universite´ Paris-Dauphine, France
[email protected] abdallah.saffi
[email protected] Joel Veness and Christopher Archibald, Mark H. M. Winands Department of Computing Science Department of Knowledge Engineering University of Alberta, Canada Maastricht University, Netherlands fveness@cs.,
[email protected] [email protected] Abstract the search enhancements from the classic αβ literature can- not be easily adapted to MCTS. The classic algorithms for This paper introduces Monte Carlo *-Minimax stochastic games, EXPECTIMAX and *-Minimax (Star1 and Search (MCMS), a Monte Carlo search algorithm Star2), perform look-ahead searches to a limited depth. How- for turned-based, stochastic, two-player, zero-sum ever, the running time of these algorithms scales exponen- games of perfect information. The algorithm is de- tially in the branching factor at chance nodes as the search signed for the class of densely stochastic games; horizon is increased. Hence, their performance in large games that is, games where one would rarely expect to often depends heavily on the quality of the heuristic evalua- sample the same successor state multiple times at tion function, as only shallow searches are possible. any particular chance node. Our approach com- bines sparse sampling techniques from MDP plan- One way to handle the uncertainty at chance nodes would ning with classic pruning techniques developed for be forward pruning [Smith and Nau, 1993], but the perfor- adversarial expectimax planning. We compare and mance gain until now has been small [Schadd et al., 2009].