Finding pure Nash equilibria in large random games

TL;DR

This paper analyzes the behavior of Best Response Dynamics in large random games, demonstrating that such processes typically reach a pure Nash equilibrium before encountering traps, with high probability.

Contribution

It provides a probabilistic analysis of BRD in large random games, showing that PNE are reached efficiently despite the presence of obstacles.

Findings

BRD reaches PNE with high probability before traps

Analysis applies to a broad class of random walks on the hypercube

Results hold for games with ties in payoffs

Abstract

Best Response Dynamics (BRD) is a class of strategy updating rules to find Pure Nash Equilibria (PNE) in a game. At each step, a player is randomly picked, and the player switches to a "best response" strategy based on the strategies chosen by others, so that the new strategy profile maximises their payoff. If no such strategy exists, a different player will be chosen randomly. When no player wants to change their strategy anymore, the process reaches a PNE and will not deviate from it. On the other hand, either PNE may not exist, or BRD could be "trapped" within a subgame that has no PNE. We consider a random game with $N$ players, each with two actions available, and i.i.d. payoffs, in which the payoff distribution may have an atom, i.e. ties are allowed. We study a class of random walks in a random medium on the $N$-dimensional hypercube induced by the random game. The medium contains two types of obstacles corresponding to PNE and traps. The class of processes we analyze includes BRD, simple random walks on the hypercube, and many other nearest neighbour processes. We prove that, with high probability, these processes reach a PNE before hitting any trap.