# Computing Approximate Nash Equilibria in Polymatrix Games

arXiv: https://arxiv.org/abs/1409.3741 · HTML: https://mlsanity.com/paper/1409.3741/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2014.

## Built on by (most cited first, 5 of 5)

- [Provably Efficient Policy Optimization for Two-Player Zero-Sum Markov Games](https://mlsanity.com/paper/2102.08903/) (2021)
- [Turbocharging Solution Concepts: Solving NEs, CEs and CCEs with Neural Equilibrium Solvers](https://mlsanity.com/paper/2210.09257/) (2022)
- [Discovering Expert-Level Nash Equilibrium Algorithms with Large Language Models](https://mlsanity.com/paper/2508.11874/) (2025)
- [Approximating Nash Equilibrium in Random Graphical Games](https://mlsanity.com/paper/2112.03442/) (2021)
- [Differentially Private Equilibrium Finding in Polymatrix Games](https://mlsanity.com/paper/2503.09538/) (2025)

## Abstract (arXiv)

In an ε-Nash equilibrium, a player can gain at most ε by unilaterally changing his behaviour. For two-player (bimatrix) games with payoffs in [0,1], the best-knownε achievable in polynomial time is 0.3393. In general, for n-player games an ε-Nash equilibrium can be computed in polynomial time for an ε that is an increasing function of n but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of ε are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general n-player games where a player's payoff is the sum of payoffs from a number of bimatrix games. There exists a very small but constant ε such that computing an ε-Nash equilibrium of a polymatrix game is PPAD-hard. Our main result is that a (0.5+δ)-Nash equilibrium of an n-player polymatrix game can be computed in time polynomial in the input size and frac1δ. Inspired by the algorithm of Tsaknakis and Spirakis, our algorithm uses gradient descent on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a (0.5+δ)-Nash equilibrium in a two-player Bayesian game.