Decision algorithms for multiplayer noncooperative games of incomplete information

Abstract

Extending the complexity results of Reif [1,2] for two player games of incomplete information, this paper (see also [3]) presents algorithms for deciding the outcome for various classes of multiplayer games of incomplete information, i.e., deciding whether or not a team has a winning strategy for a particular game. Our companion paper, [4] shows that these algorithms are indeed asymptotically optimal by providing matching lower bounds. The classes of games to which our algorithms are applicable include games which were not previously known to be decidable. We apply our algorithms to provide alternative upper bounds, and new time-space trade-offs on the complexity of multiperson alternating Turing machines [3]. We analyze the algorithms to characterize the space complexity of multiplayer games in terms of the complexity of deterministic computation on Turing machines. In hierarchical multiplayer games, each additional clique (subset of players with the same information) increases the complexity of the outcome problem by a further exponential. We show that an S( n) space bounded k-player game of incomplete information has a deterministic time upper bound of k + 1 repeated exponentials of S( n). Furthermore, S( n) space bounded k-player blindfold games have a deterministic space upper bound of k repeated exponentials of S( n). This paper proves that this exponential blow-up can occur. We also show that time bounded games do not exhibit such hierarchy. A T( n) time bounded blindfold multiplayer game, as well as a T( n) time bounded multiplayer game of incomplete information, has a deterministic space bound of T( n).