The Cut-and-Play Algorithm: Computing Nash Equilibria via Outer Approximations

TL;DR

The paper presents Cut-and-Play, an efficient algorithm for computing Nash equilibria in complex nonconvex games by iteratively refining convex approximations, outperforming existing methods.

Contribution

It introduces a novel algorithm that handles nonconvex, unbounded, and separable payoff games without requiring convexity or continuity.

Findings

Efficiently computes Nash equilibria in challenging nonconvex games.

Outperforms existing game-specific algorithms in experiments.

Handles discrete decisions and bilevel problems effectively.

Abstract

We introduce Cut-and-Play, a practically-efficient algorithm for computing Nash equilibria in simultaneous non-cooperative games where players decide via nonconvex and possibly unbounded optimization problems with separable payoff functions. Our algorithm exploits an intrinsic relationship between the equilibria of the original nonconvex game and the ones of a convexified counterpart. In practice, Cut-and-Play formulates a series of convex approximations of the game and iteratively refines them with cutting planes and branching operations. Our algorithm does not require convexity or continuity of the player's optimization problems and can be integrated with existing optimization software. We test Cut-and-Play on two families of challenging nonconvex games involving discrete decisions and bilevel problems, and we empirically demonstrate that it efficiently computes equilibria while outperforming existing game-specific algorithms.