Abstract
We study the fundamental problem of computing an arbitrary Nash equilibrium in bimatrix games. We start by proposing a novel characterization of the set of Nash equilibria, via a bijective map to the solution set of a (parameterized) quadratic program, whose feasible space is the (highly structured) set of correlated equilibria. We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible. In particular, we introduce the notion of mutual (quasi-) concavity of a bimatrix game, which assures (quasi-) convexity of our quadratic program, for at least one value of the parameter. For mutually concave bimatrix games, we provide a polynomial-time computation of a Nash equilibrium, based on the polynomial tractability of convex quadratic programming. For the mutually quasi-concave games, we provide (to our knowledge) the first FPTAS for the construction of a Nash equilibrium.
Of course, for these new polynomially tractable subclasses of bimatrix games to be useful, polynomial-time certificates are also necessary that will allow us to efficiently identify them. Towards this direction, we provide various characterizations of mutual concavity, which allow us to construct such a certificate. Interestingly, these characterizations also shed light to some structural properties of the bimatrix games satisfying mutual concavity. This subclass entirely contains the most popular subclass of polynomial-time solvable bimatrix games, namely, all the constant-sum games (rank− 0 games). It is though incomparable to the subclass of games with fixed rank [16]: Even rank− 1 games may not be mutually concave (eg, Prisoner’s dilemma), but on the other hand, there exist mutually concave games of arbitrary (even full) rank. Finally, we prove closeness of mutual concavity under (Nash equilibrium preserving) positive affine transformations of bimatrix games having the same scaling factor for both payoff matrices. For different scaling factors the property is not necessarily preserved.
Keywords
This work has been partially supported by the ICT Programme of the EU under contract number FP7-215270 (FRONTS), and the ERC/StG Programme of the EU under the contract number 210743 (RIMACO).
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References
Addario-Berry, L., Olver, N., Vetta, A.: A polynomial time algorithm for finding nash equilibria in planar win-lose games. Journal of Graph Algorithms and Applications 11(1), 309–319 (2007)
Althöfer, I.: On sparse approximations to randomized strategies and convex combinations. Linear Algebra and Applications 199, 339–355 (1994)
Bazaraa, M.S., Sherali, H.D., Shetty, C.: Nonlinear Programming: Theory and Algorithms, 2nd edn. John Wiley & Sons, Inc., Chichester (1993)
Bosse, H., Byrka, J., Markakis, E.: New algorithms for approximate nash equilibria in bimatrix games. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 17–29. Springer, Heidelberg (2007)
Boyd, S., Vandenberghe, L.: Convex Optimization, 7th edn. Cambridge University Press, Cambridge (2009)
Chen, X., Deng, X.: Settling the complexity of 2-player nash equilibrium. In: Proc. of 47th IEEE Symp. on Found. of Comp. Sci. (FOCS 2006), pp. 261–272. IEEE Comp. Soc. Press, Los Alamitos (2006)
Chen, X., Deng, X., Teng, S.-H.: Computing nash equilibria: Approximation and smoothed complexity. In: Proc. of 47th IEEE Symp. on Found. of Comp. Sci. (FOCS 2006), pp. 603–612. IEEE Comp. Soc. Press, Los Alamitos (2006)
Codenotti, B., Leoncini, M., Resta, G.: Efficient computation of nash equilibria for very sparse win-lose bimatrix games. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 232–243. Springer, Heidelberg (2006)
Conitzer, V., Sandholm, T.: Complexity results about nash equilibria. In: Proc. of 18th Int. Joint Conf. on Art. Intel. (IJCAI 2003), pp. 765–771. Morgan Kaufmann, San Francisco (2003)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a nash equilibrium. SIAM Journal on Computing 39(1), 195–259 (2009); Preliminary version in ACM STOC 2006
Daskalakis, C., Mehta, A., Papadimitriou, C.: A note on approximate equilibria. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 297–306. Springer, Heidelberg (2006)
Daskalakis, C., Mehta, A., Papadimitriou, C.: Progress in approximate nash equilibrium. In: Proc. of 8th ACM Conf. on El. Comm. (EC 2007), pp. 355–358 (2007)
Daskalakis, C., Papadimitriou, C.: Three player games are hard. Technical Report TR05-139, Electr. Coll. on Comp. Compl., ECCC (2005)
Even-Dar, E., Mansour, Y., Nadav, U.: On the convergence of regret minimization dynamics in concave games. In: Proc. of 41st ACM Symp. on Th. of Comp. (STOC 2009), pp. 523–532 (2009)
Gilboa, I., Zemel, E.: Nash and correlated equilibria: Some complexity considerations. Games & Econ. Behavior 1, 80–93 (1989)
Kannan, R., Theobald, T.: Games of fixed rank: A hierarchy of bimatrix games. Economic Theory 42, 157–173 (2010); Preliminary version appeared in ACM-SIAM SODA 2007
Kontogiannis, S., Panagopoulou, P., Spirakis, P.: Polynomial algorithms for approximating nash equilibria in bimatrix games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 286–296. Springer, Heidelberg (2006)
Kontogiannis, S., Spirakis, P.: Exploiting concavity in bimatrix games: New polynomially tractable subclasses. In: Proc. of 13th W. on Appr. Alg. for Comb. Opt., APPROX’10 (2010), http://www.cs.uoi.gr/~kontog/pubs/approx10paper-full.pdf
Kontogiannis, S., Spirakis, P.: Well supported approximate equilibria in bimatrix games. ALGORITHMICA 57, 653–667 (2010)
Lemke, C., Howson, J.: Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12, 413–423 (1964)
Lipton, R., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: Proc. of 4th ACM Conf. on El. Comm (EC 2003), pp. 36–41. Assoc. of Comp. Mach. (ACM), New York (2003)
Mangasarian, O.L., Stone, H.: Two-person nonzero-sum games and quadratic programming. Journal of Mathematical Analysis and Applications 9(3), 348–355 (1964)
Morgenstern, O., von Neumann, J.: The Theory of Games and Economic Behavior. Princeton University Press, Princeton (1947)
Robinson, J.: An iterative method of solving a game. Annals of Mathematics 54, 296–301 (1951)
Rosen, J.: Existence and uniqueness of equilibrium points for concave n −person games. Econometrica 33(3), 520–534 (1965)
Savani, R., von Stengel, B.: Exponentially many steps for finding a nash equilibrium in a bimatrix game. In: Proc. of 45th IEEE Symp. on Found. of Comp. Sci. (FOCS 2004), pp. 258–267 (2004)
Tsaknakis, H., Spirakis, P.G.: An optimization approach for approximate nash equilibria. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 42–56. Springer, Heidelberg (2007)
Tsaknakis, H., Spirakis, P.G.: A graph spectral approach for computing approximate nash equilibria. Technical report, Electronic Colloquium on Computational Complexity, Report No. 96 (2009)
Vavasis, S.: Approximation algorithms for indefinite quadratic programming. Mathematical Programming 57, 279–311 (1992)
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Kontogiannis, S., Spirakis, P. (2010). Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_24
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