Support Contraction for Well-Supported Nash Equilibria

2026-08-01

AI Summary

This work addresses the efficient computation of a 1/2-approximate well-supported Nash equilibrium (WSNE) without additive slack in bimatrix games. It introduces a “support contraction” method that iteratively prunes the action space by alternately retaining the support sets of minimax strategies for each player, while preserving the value of the subgame. An exact Nash equilibrium is then constructed over a stable rectangle, with pruning errors controlled via zero-sum game values. The paper presents the first deterministic polynomial-time algorithm for computing a 1/2-WSNE, improving the query complexity’s dependence on ε from ε⁻⁴ to ε⁻². Additionally, it designs a two-player communication protocol using O(ε⁻² log²n) bits and an algorithm requiring O(ε⁻²n log n) payoff queries.

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 $ε^{-2}$.

Problem

• well-supported Nash equilibria
• bimatrix games
• approximation algorithms
• payoff queries
• computational complexity

Innovation

• support contraction
• well-supported Nash equilibrium
• bimatrix games
• minimax strategies
• payoff-query complexity