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Motivated by the sequence form formulation of Koller et al. [16], this paper considers bilinear games , represented by two payoff matrices (A ,B ) and compact polytopal strategy sets. Bilinear games are very general and capture many interesting classes of games including bimatrix games, two player Bayesian games, polymatrix games, and two-player extensive form games with perfect recall as special cases, and hence are hard to solve in general. For a bilinear game, we define its best response polytopes (BRPs) and characterize its Nash equilibria as the fully-labeled pairs of the BRPs. Rank of a game (A ,B ) is defined as rank (A +B ). In this paper, we give polynomial-time algorithms for computing Nash equilibria of (i ) rank-1 games, (ii ) FPTAS for constant-rank games, and (iii ) when rank (A ) or rank (B ) is constant.