Equilibria of Graphical Games with Symmetries

  • Authors:

  • Felix Brandt;Felix Fischer;Markus Holzer

  • Affiliations:

  • Institut für Informatik, Universität München, München, Germany 80538;Institut für Informatik, Universität München, München, Germany 80538;Institut für Informatik, Technische Universität München, Garching, Germany 85748

  • Venue:
  • WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
  • Year:
  • 2008

  • Pairwise-interaction games

    ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I

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Abstract

We study graphical games where the payoff function of each player satisfies one of four types of symmetry in the actions of his neighbors. We establish that deciding the existence of a pure Nash equilibrium is NP-hard in general for all four types. Using a characterization of games with pure equilibria in terms of even cycles in the neighborhood graph, as well as a connection to a generalized satisfiability problem, we identify tractable subclasses of the games satisfying the most restrictive type of symmetry. Hardness for a different subclass is obtained via a satisfiability problem that remains NP-hard in the presence of a matching, a result that may be of independent interest. Finally, games with symmetries of two of the four types are shown to possess a symmetric mixed equilibrium which can be computed in polynomial time. We thus obtain a class of games where the pure equilibrium problem is computationally harder than the mixed equilibrium problem, unless P=NP.