Abstract
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.
Refer [9] for the full version. We thank Milind Sohoni for helpful comments and corrections.
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References
Adsul, B., Garg, J., Mehta, R., Sohoni, M.: Rank-1 bimatrix games: A homeomorphism and a polynomial time algorithm. In: STOC, pp. 195–204 (2011)
Avis, D., Rosenberg, G.D., Savani, R., von Stengel, B.: Enumeration of Nash equilibria for two-player games. Economic Theory 42, 9–37 (2010)
Charnes, A.: Constrained games and linear programming. Proceedings of the National Academy of Sciences of the USA 39, 639–641 (1953)
Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: FOCS, pp. 261–272 (2006)
Dantzig, G.B.: Linear Programming and Extensions. Princeton Univ. Press (1963)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. In: STOC, pp. 71–78 (2006)
Daskalakis, C., Papadimitriou, C.H.: On a Network Generalization of the Minmax Theorem. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 423–434. Springer, Heidelberg (2009)
Fudenberg, D., Tirole, J.: Game Theory. MIT Press (1991)
Garg, J., Jiang, A.X., Mehta, R.: Bilinear games: Polynomial time algorithms for rank based subclasses. arXiv:1109.6182 (2011)
Glicksberg, I.L.: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, vol. 3(1), pp. 170–174. AMS (1952)
Howson Jr., J.: Equilibria of polymatrix games. Management Science (1972)
Howson Jr, J., Rosenthal, R.: Bayesian equilibria of finite two-person games with incomplete information. Management Science, 313–315 (1974)
Immorlica, N., Kalai, A.T., Lucier, B., Moitra, A., Postlewaite, A., Tennenholtz, M.: Dueling algorithms. In: STOC (2011)
Kannan, R., Theobald, T.: Games of fixed rank: A hierarchy of bimatrix games. Economic Theory, 1–17 (2009)
Koller, D., Megiddo, N., von Stengel, B.: Fast algorithms for finding randomized strategies in game trees. In: STOC, pp. 750–759 (1994)
Koller, D., Megiddo, N., von Stengel, B.: Efficient computation of equilibria for extensive two-person games. Games and Economic Behavior 14, 247–259 (1996)
Kontogiannis, S., Spirakis, P.: Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 312–325. Springer, Heidelberg (2010)
Lipton, R., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: EC, pp. 36–41. ACM Press, New York (2003)
Ponssard, J.P., Sorin, S.: The LP formulation of finite zero-sum games with incomplete information. International Journal of Game Theory 9, 99–105 (1980)
von Stengel, B.: Equilibrium computation for two-player games in strategic and extensive form. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory, ch. 3, pp. 53–78 (2007)
Vavasis, S.: Approximation algorithms for indefinite quadratic programming. Mathematical Programming 57, 279–311 (1992)
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Garg, J., Jiang, A.X., Mehta, R. (2011). Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses. In: Chen, N., Elkind, E., Koutsoupias, E. (eds) Internet and Network Economics. WINE 2011. Lecture Notes in Computer Science, vol 7090. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25510-6_35
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