Finding Mixed Strategies with Small Supports in Extensive Form Games
نویسندگان
چکیده
The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved One measure for the size of a mixed strategy is the number of strategies in its support the set of pure strategies to which it gives positive probability This paper investigates the existence of small mixed strategies in exten sive form games and how such strategies can be used to create more e cient algorithms The basic idea is that in an extensive form game a mixed strategy induces a small set of realization weights that completely describe its observable behavior This fact can be used to show that for any mixed strategy there exists a realization equivalent mixed strategy whose size is at most the size of the game tree For a player with imperfect recall the problem of nding such a strategy given the realization weights is NP hard On the other hand if is a behavior strategy can be constructed from in time polynomial in the size of the game tree In either case we can use the fact that mixed strategies need never be too large for constructing e cient algorithms that search for equilibria In particular we construct the rst exponential time algorithm for nding all equilibria of an arbitrary two person game in extensive form Research supported in part by ONR Contract N C by the Air Force O ce of Scienti c Research AFSC under Contract F C and by a University of California President s Postdoctoral Fellowship Some of this work was done while Daphne Koller was at Stanford University The United States Government is authorized to reproduce and distribute reprints for governmental purposes Computer Science Division University of California Berkeley California and IBM Almaden Research Center Harry Road San Jose California IBM Almaden Research Center Harry Road San Jose California and School of Mathematical Sciences Tel Aviv University Tel Aviv Israel
منابع مشابه
Finding Mixed Strategies with Small Supports in Extensive Form Games1
The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its support the set of pure strategies to which it gives positive probability. This paper investigates the existence of "small" mixed strategies in extensive form games, and ...
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