Polylogarithmic Supports Are Required for Approximate Well-Supported Nash Equilibria below 2/3

Abstract

In an ε-approximate Nash equilibrium, a player can gain at most ε in expectation by unilateral deviation. An ε-well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within ε of the best response payoff. Daskalakis, Mehta and Papadimitriou [8] conjectured that every win-lose bimatrix game has a $\frac{2}{3}$-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a $\frac23$ payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least $\Omega(\sqrt[3]{\log n})$ in any ε-well supported equilibrium with $\epsilon < \frac23$. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Häggkvist conjecture [4]. A probabilistic argument [13] shows that there exist ε-well-supported equilibria with supports of cardinality $O(\frac{1}{\epsilon^2}\cdot \log n)$, for any ε > 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any δ > 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1 − δ)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a $\frac{1}{2}$-approximate Nash equilibrium where the supports of the players have cardinality at most two [8].