Game Connectivity and Adaptive Dynamics

TL;DR

This paper investigates the structure of games through their best-response graphs, revealing that most generic games with pure Nash equilibria are connected, which influences the convergence of adaptive dynamics.

Contribution

It demonstrates that nearly all generic games with pure Nash equilibria are connected, enabling simple adaptive dynamics to almost surely reach equilibrium in most cases.

Findings

Most generic games with pure Nash equilibria are connected.

Adaptive dynamics can almost surely reach equilibrium in these games.

Connectivity properties influence the convergence behavior of game dynamics.

Abstract

We analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. Our central result shows that, among games that are `generic' (without indifferences) and that have a pure Nash equilibrium, all but a small fraction are \emph{connected}, meaning that every action profile that is not a pure Nash equilibrium can reach every pure Nash equilibrium via best-response paths. This has important implications for dynamics in games. In particular, we show that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in all but a small fraction of generic games that have one (which contrasts with the known fact that there is no such dynamic that leads almost surely to a pure Nash equilibrium in \emph{every} generic game that has one). We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.