Strategic planning for imperfect-information games

Although game-tree search works well in perfectinformation 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 ~ree 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 ill the worst case, the game tree contained only about 1300 nodes, as opposed to the approximately 6.01 x 1044 nodes that would have been produced by a brute-force game tree search in tile 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. This work supported in part by an ATT Levy and Newborn, 1982), checkers (Samuel, 1967), and othello (Lee and Mahajan, 1990)), it does not ways work so well in other games. One example is the game of bridge. Bridge is an imperfect-information game, in which no player has complete knowledge about the state of the world, the possible actions, and their effects. As a consequence, the branching factor of the game tree--and thus the size of the tree itself--is very large. Searching this game tree is completely infeasible, because the bridge deal must be played in just a few minutes (in contrast to a chess game, which can go on for several hours). Thus, a different approach is needed. In this paper, we describe an approach to this problem, based on the observation that bridge is a game of planning. The bridge literature describes a number of tactical and strategic schemes for dealing with various card-playing situations. It appears that there is a small number of such schemes for each bridge hand, and that each of them can be expressed relatively simply. To play bridge, many humans use these schemes to create plans. They then follow those plans for some number of tricks, replanning when appropriate. We have taken advantage of the planning nature of bridge, by adapting and extending some ideas from task-network planning. To represent the tactical and strategic schemes of card-playing in bridge, we use instances of multi-agent methods--structures similar to the task decompositions used in hierarchical singleagent planning systems such as Nonlin (Tate, 1976; Tate, 1977), NOAH (Sacerdoti, 1977), and MOLGEN (Stefik, 1981), but modified to represent multi-agency and uncertainty. To generate game trees, we use a procedure similar to task-network decomposition. This approach produces a game tree in which the number of branches from each state is determined not by the number of actions that an agent can perform, but instead by the number of different tactical and strategic schemes that the agent can employ. If at each node of the tree, the number of applicable schemes is From: AAAI Technical Report FS-93-02. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.