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International Workshop on Randomization and Approximation Techniques in Computer Science

International Workshop on Approximation Algorithms for Combinatorial Optimization

RANDOM 2010, APPROX 2010: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques pp 312–325Cite as

Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6302))

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.

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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Author information

Authors and Affiliations

  1. Dept. of Computer Science, University of Ioannina, 45110, Ioannina, Greece

    Spyros Kontogiannis

  2. R.A. Computer Technology Institute, Patras Univ. Campus, 26504, Rio-Patra, Greece

    Spyros Kontogiannis & Paul Spirakis

Authors
  1. Spyros Kontogiannis
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  2. Paul Spirakis
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Editor information

Editors and Affiliations

  1. Universitat Politecnica de Catalunya, Departament de Llenguatges i Sistemes Informatics, Campus Nord - Ed Omega, Desp. 235, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Maria Serna

  2. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

  3. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24098, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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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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  • DOI: https://doi.org/10.1007/978-3-642-15369-3_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15368-6

  • Online ISBN: 978-3-642-15369-3

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