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Unformatted text preview: An Analysis of UCT in Multi-Player Games 195 AN ANALYSIS OF UCT IN MULTI-PLAYER GAMES Nathan Sturtevant 1 Department of Computing Science University of Alberta ABSTRACT The UCT algorithm has been exceedingly popular for Go, a two-player game, significantly increasing the playing strength of Go programs in a very short time. This paper provides an analysis of the UCT algorithm in multi-player games, showing that UCT is computing a mixed-strategy equilibrium, as opposed to max n , which computes a pure-strategy equilibrium. We analyze the performance of UCT in several known domains and show that it performs as well or better than existing algorithms. We also test several well-known UCT learning and playout rules. We suggest two criteria, branching factor and n-ply state variance. The experiments show that they are correlated with the success of the techniques. 1. INTRODUCTION Monte-Carlo methods have become popular in the game of Go over the last few years, and even more so with the introduction of the UCT algorithm (Kocsis and Szepesv´ari, 2006). Go is probably the best-known two-player game in which computer players are still significantly weaker than humans. UCT works particularly well in Go for several reasons. We mention two of them. First, in Go it is difficult to evaluate states in the middle of a game, but UCT only evaluates endgames states, which is relatively easy. Second, the game of Go converges for random play, meaning that it is not very difficult to arrive at an end-game state. Multi-player games are also difficult for computers to play well. First, it is more difficult to prune in multi-player games, meaning that normal search algorithms are less effective at obtaining deep lookahead. While alpha-beta pruning reduces the size of a game tree from O ( b d ) to O ( b d/ 2 ) , the best techniques in multi-player games only reduce the size of the game tree to O ( b n- 1 n d ) , where n is the number of players in the game (Sturtevant and Korf, 2000; Sturtevant, 2003a). A second reason why multi-player games are difficult is because of opponent modelling. In two-player zero-sum games opponent modelling has never been shown to be necessary for high- quality play, while in multi-player games, opponent modelling is a necessity for robust play versus unknown opponents in some domains (Sturtevant and Bowling, 2006). As a result, it is worth investigating UCT to see how it performs in multi-player games. We first present a theoretical analysis, where we show that UCT may compute a mixed-strategy equilibrium in multi-player games and discuss the implications. Then, we analyze UCT’s performance in a variety of domains, showing that it performs as well or better as the best previous approaches. This is an extended version of Sturtevant (2008) with additional results and analysis. An preliminary analysis of UCT for multi-player Go can be found in Cazenave (2008) ....
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