2011/09/28 by Jugal Garg, Garg, Jugal, Albert Xin Jiang +3
Computer Science · Decision Sciences · Economics, Econometrics and Finance · #Auction Theory and Applications #Computer Science and Game Theory (cs.GT) #FOS: Computer and information sciences #Game Theory and Applications #Game Theory and Voting Systems #cs.GT
paper · pdf · doi:10.48550/arxiv.1109.6182
arxiv created 2011/09/28 · openalex publication_date 2011/09/28 · arxiv updated 2011/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the sequence form formulation of Koller et al. (GEB'96), this paper defines \em 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 xAy and xBy 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 \em Best Response Polytopes (BRPs) and characterize the set of Nash equilibria as \em 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: (i) Rank-1 games (i.e., rank(A+B)=1). (ii) FPTAS for constant rank games (i.e., rank(A+B) is constant). (iii) When rank(A) or rank(B) is constant. This improves the results by Lipton et al. (EC'03) and Kannan et al. (ET'09), for bimatrix games with low rank matrices.