An Empirical Study on Computing Equilibria in Polymatrix Games

Abstract

The Nash equilibrium is an important benchmark for behaviour in systems of strategic autonomous agents. Polymatrix games are a succinct and expressive representation of multiplayer games that model pairwise interactions between players. The empirical performance of algorithms to solve these games has received little attention, despite their wide-ranging applications. In this paper we carry out a comprehensive empirical study of two prominent algorithms for computing a sample equilibrium in these games, Lemke's algorithm that computes an exact equilibrium, and a gradient descent method that computes an approximate equilibrium. Our study covers games arising from a number of interesting applications. We find that Lemke's algorithm can compute exact equilibria in relatively large games in a reasonable amount of time. If we are willing to accept (high-quality) approximate equilibria, then we can deal with much larger games using the descent method. We also report on which games are most challenging for each of the algorithms.

Turbocharging Solution Concepts: Solving NEs, CEs and CCEs with Neural Equilibrium Solvers

“…The main limitation of the approach is that the activation space of the network is large, particularly with a large number of players and strategies which limits the size of games that can be tackled. Future work could look at restricted classes of games, such as polymatrix games [10,11], or graphical games [31], which consider only local payoff structure and have much smaller payoff representations. This is a promising direction because NES otherwise has good scaling properties: (i) the dual variables are space efficient, (ii) there are relatively few parameters, (iii) the number of parameters is independent of the number of strategies in the game, (iv) equivariance means each training sample is equivalent to training under all payoff permutations, and (v) there are promising zero-shot generalization results to larger games.…”

“…The duals are a significantly more space efficient objective target ( p |A p | 2 for CEs and p |A p | for CCEs) than the full joint ( p |A p |), particularly when scaling the number of strategies and players. The joint, σ(a), and approximation, p , can be computed analytically from the dual deviation gains and the inputs using Equations ( 9) and (10). The network is trained by minimizing the loss, L (C)CE (Equation ( 7)).…”

Sample-based Approximation of Nash in Large Many-Player Games via Gradient Descent

“…Lyapunov approaches minimize non-convex energy functions with the property that zero energy implies Nash (Shoham & Leyton-Brown, 2009), however these approaches may suffer from convergence to local minima with positive energy. In some settings, such as polymatrix games with payoffs in [0, 1], gradient descent on appropriate energy functions 4 guarantees a ( 1 2 +δ)-Nash in time polynomial in 1 δ (Deligkas et al, 2017) and performs well in practice (Deligkas et al, 2016).…”