Support Contraction for Well-Supported Nash Equilibria

Abstract

We introduce the support contraction, a structural procedure for computing well-supported Nash equilibria (WSNE) in bimatrix games. On a common action rectangle, the procedure alternately retains the support of a row maximin strategy for the row-payoff matrix and the support of a column maximin strategy for the column-payoff matrix. Each restriction preserves the value of the matrix game that selected it and can only increase the other value. At a stable rectangle, full support of the two maximizing strategies and complementary slackness make every surviving action tight. Crossing the two minimax strategies then gives an exact Nash equilibrium of the retained subgame, while the two zero-sum values bound deviations to deleted actions. Support contraction gives a deterministic polynomial-time algorithm that computes a $1/2$-WSNE of every rational bimatrix game with payoffs in $[0,1]$, with no additive slack, and a deterministic $O(ε^{-2}\log^2 n)$-bit two-party protocol for a $(1/2+ε)$-WSNE. The same support-contraction certificate, preceded by randomized one-sided localization, gives an $O(ε^{-2}n\log n)$ payoff-query algorithm for the same guarantee, improving the $ε^{-4}$ dependence to $ε^{-2}$.