Strategic Planning for Imperfect-Information Games

Stephen J. Smith and Dana S. Nau

Although game-tree search works well in perfect-information games, there are problems in trying to use it for imperfect-information games such as bridge. The lack of knowledge about the opponents’ possible moves gives the game tree a very large branching factor, making the tree so immense that game-tree searching is infeasible. In this paper, we describe our approach for overcoming this problem. We develop a model of imperfect-information games, and describe how to represent information about the game using a modified version of a task network that is extended to represent multi-agency and uncertainty. We present a game-playing procedure that uses this approach to generate game trees in which the set of alternative choices is determined not by the set of possible actions, but by the set of available tactical and strategic schemes. In our tests of this approach on the game of bridge, we found that it generated trees having a much smaller branching factor than would have been generated by conventional game-tree search techniques. Thus, even in the worst case, the game tree contained only about 1300 nodes, as opposed to the approximately 6.01 times 10^44 nodes that would have been produced by a brute-force game tree search in the worst case. Furthermore, our approach successfully solved typical bridge problems that matched situations in its knowledge base. These preliminary tests suggest that our approach has the potential to yield bridge-playing programs much better than existing ones---and thus we have begun to build a full implementation.


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