Game connectivity and adaptive dynamics in many-action games

TL;DR

This paper investigates the connectivity properties of many-action games, showing that as the number of players increases, most such games remain connected, facilitating equilibrium convergence through adaptive dynamics.

Contribution

The study extends previous work by analyzing the large-action regime, revealing that most many-player games are connected despite the increased number of actions, using new probabilistic and combinatorial methods.

Findings

Most large-action games are connected as the number of players grows.

A small but positive fraction of many-action games are not connected.

Almost all games with many players are connected, ensuring equilibrium convergence.

Abstract

We study the typical structure of games in terms of their connectivity properties. A game is said to be `connected' if it has a pure Nash equilibrium and the property that there is a best-response path from every action profile which is not a pure Nash equilibrium to every pure Nash equilibrium, and it is generic if it has no indifferences. In previous work we showed that, among all $n$-player $k$-action generic games that admit a pure Nash equilibrium, the fraction that are connected tends to $1$ as $n$ gets sufficiently large relative to $k$. The present paper considers the large-$k$ regime, which behaves differently: we show that the connected fraction tends to $1-\zeta_n$ as $k$ gets large, where $\zeta_n>0$. In other words, a constant fraction of many-action games are not connected. However, $\zeta_n$ is small and tends to $0$ rapidly with $n$, so as $n$ increases all but a vanishingly small fraction of many-player-many-action games are connected. Since connectedness is conducive to equilibrium convergence we obtain, by implication, that there is a simple adaptive dynamic that is guaranteed to lead to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one. Our results are based on new probabilistic and combinatorial arguments which allow us to address the large-$k$ regime that the approach used in our previous work could not tackle. We thus complement our previous work to provide a more complete picture of game connectivity across different regimes.