Swim till You Sink: Computing the Limit of a Game

TL;DR

This paper investigates the asymptotic behavior of natural game dynamics, particularly noisy replicator dynamics, and demonstrates efficient computation and sampling methods to understand the limit distributions over sink equilibria in large games.

Contribution

It introduces a computational framework for analyzing the limit behavior of game dynamics, including efficient algorithms and sampling techniques for large games.

Findings

Limit distribution can be computed efficiently for pure strategy priors.

Sampling methods accurately estimate limit distributions in large games.

Sink equilibria effectively capture the long-term behavior of game dynamics.

Abstract

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.