[Paper Review] Approximate well-supported Nash equilibria in symmetric bimatrix games

Artur Czumaj, Michail Fasoulakis+1 more|arXiv (Cornell University)|Jul 11, 2014

Game Theory and Applications11 references4 citations

TL;DR

This paper presents a polynomial-time algorithm that computes a (1/2 + δ)-well-supported Nash equilibrium in symmetric bimatrix games for any arbitrarily small δ > 0. The approach combines two relaxations of symmetric Nash equilibria: one based on payoff prevention using linear programming for the 1/2 case, and a probabilistic analysis using Hoeffding's inequality to handle the remaining case, ensuring high-probability convergence to an approximate equilibrium.

ABSTRACT

The $\varepsilon$-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than $\varepsilon$ to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant $\varepsilon$ currently known for which there is a polynomial-time algorithm that computes an $\varepsilon$-well-supported Nash equilibrium in bimatrix games is slightly below $2/3$. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a $(1/2+δ)$-well-supported Nash equilibrium, for an arbitrarily small positive constant $δ$.

Motivation & Objective

• To improve the approximation guarantee for well-supported Nash equilibria in symmetric bimatrix games.
• To address the computational hardness of exact Nash equilibrium computation in symmetric games.
• To develop a polynomial-time algorithm that achieves a better approximation than the best-known bounds for general bimatrix games.
• To leverage structural properties of symmetric games to design a more effective approximation method.

Proposed method

• The algorithm first checks for a symmetric strategy profile that prevents payoff exceeding 1/2 using a single linear program.
• If no such profile exists, the method employs a probabilistic construction based on sampling mixed strategies with bounded support.
• It uses Hoeffding’s inequality to bound the probability of large deviations in expected payoffs across sampled strategies.
• The method defines a random variable for each pure strategy’s payoff under random sampling, ensuring concentration around the true expected value.
• It applies a union bound over all strategies in the support to show that with positive probability, all strategies in the support yield payoffs within δ of the optimal value.
• The algorithm constructs a pair of mixed strategies (x, y) that well-support a value close to the optimal, ensuring (1/2 + δ)-well-supported equilibrium.

Experimental results

Research questions

• RQ1 Can a better approximation guarantee than 2/3 be achieved for well-supported Nash equilibria in symmetric bimatrix games?
• RQ2 Is it possible to compute a (1/2 + δ)-well-supported Nash equilibrium in polynomial time for symmetric games?
• RQ3 Can the relaxation of symmetric Nash equilibrium via payoff prevention be used to design efficient approximation algorithms?
• RQ4 How can probabilistic sampling and concentration inequalities be used to construct approximate equilibria in symmetric games?

Key findings

• The paper presents a polynomial-time algorithm that computes a (1/2 + δ)-well-supported Nash equilibrium for any δ > 0 in symmetric bimatrix games.
• When a symmetric Nash equilibrium with value at most 1/2 exists, the algorithm finds it via a single linear program.
• For games without such an equilibrium, the algorithm uses random sampling of strategies with support size bounded by κ(δ) = 2δ² ln(1/δ).
• With positive probability, the sampled strategies yield payoffs within δ of the optimal for all strategies in the support, ensuring (1/2 + δ)-well-supported equilibrium.
• The analysis uses Hoeffding’s inequality to bound the tail probability of payoff deviations, and the union bound ensures high-probability success across all support strategies.
• The method guarantees that the resulting strategy profile is a (1/2 + δ)-well-supported Nash equilibrium, improving upon the best-known bound of ~0.6619 for general bimatrix games.