The Computational Complexity of Finding a Mixed Berge Equilibrium for a k-Person Noncooperative Game in Normal Form

Abstract

The mixed Berge equilibrium (MBE) is an extension of the Berge equilibrium (BE) to mixed strategies. The MBE models mutually support in a [Formula: see text]-person noncooperative game in normal form. An MBE is a mixed-strategy profile for the [Formula: see text] players in which every [Formula: see text] players have mixed strategies that maximize the expected payoff for the remaining player’s equilibrium strategy. In this paper, we study the computational complexity of determining whether an MBE exists in a [Formula: see text]-person normal-form game. For a two-person game, an MBE always exists and the problem of finding an MBE is PPAD-complete. In contrast to the mixed Nash equilibrium, the MBE is not guaranteed to exist in games with three or more players. Here we prove, when [Formula: see text], that the decision problem of asking whether an MBE exists for a [Formula: see text]-person normal-form game is NP-complete. Therefore, in the worst-case scenario there does not exist a polynomial algorithm that finds an MBE unless P=NP.