Abstract. Motivated by the sequence form formulation of Koller et al. [19], this paper defines bilinear games, and proposes efficient algorithms for its rank based subclasses. Bilinear games are two-player non-cooperative single-shot games with compact polytopal strategy sets and two payoff matrices (A, B) such that when (x, y) is the played strategy profile, the payoffs of the players are x T Ay and x T By respectively. We show that bilinear games are very general and capture many interesting classes of games like bimatrix games, two player Bayesian games, polymatrix games, two-player extensive form games with perfect recall etc. as special cases, and hence are hard to solve in general. Existence of a (symmetric) Nash equilibrium for (symmetric) bilinear games follow directly from the known results. For a given bilinear game, we define its Best Response Polytopes (BRPs) and characterize the set of Nash equilibria as fully-labeled pairs in the BRPs. We consider a rank based hierarchy of bilinear games, where rank of a game (A, B) is defined as rank(A + B). In this paper, we give polynomial time algorithms to compute Nash equilibrium for special classes of bilinear games: -Rank-1 games (i.e., rank(A + B) = 1). -FPTAS for constant rank games (i.e., rank(A + B) is constant). -When rank(A) or rank(B) is constant. This improves the results by Lipton et al. [22] and Kannan et al. [17], for bimatrix games with low rank matrices.
REF ] consider bilinear general-sum games, and show that such games are general enough to capture many interesting classes of games, hence are hard to solve in general.
Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses
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Mathematics Computer Science