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Fast Algorithms for Rank-1 Bimatrix Games

Author

Listed:
  • Bharat Adsul

    (Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India;)

  • Jugal Garg

    (Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801;)

  • Ruta Mehta

    (Department of Computer Science, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801;)

  • Milind Sohoni

    (Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India;)

  • Bernhard von Stengel

    (Department of Mathematics, London School of Economics, London WC2A 2AE, United Kingdom)

Abstract

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one and shows the following: (1) For a game of rank r , the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r − 1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component.

Suggested Citation

  • Bharat Adsul & Jugal Garg & Ruta Mehta & Milind Sohoni & Bernhard von Stengel, 2021. "Fast Algorithms for Rank-1 Bimatrix Games," Operations Research, INFORMS, vol. 69(2), pages 613-631, March.
    • Handle: RePEc:inm:oropre:v:69:y:2021:i:2:p:613-631
    DOI: 10.1287/opre.2020.1981
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    References listed on IDEAS

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