Game connectivity and adaptive dynamics in many-action games

2026-01-09

AI Summary

This study investigates the asymptotic connectivity structure in multi-player, multi-action games—specifically, whether all non-equilibrium states can reach a pure-strategy Nash equilibrium via best-response paths. Addressing the limitation of prior techniques that only applied to small numbers of actions \(k\), this work introduces novel probabilistic and combinatorial methods to handle large \(k\). It establishes, for the first time, that as \(k\) grows, the fraction of disconnected games converges to a positive constant \(\zeta_n\), which decays rapidly to zero with the number of players \(n\). Consequently, for sufficiently large \(k\) and moderate \(n\), almost all such games are connected, ensuring that simple adaptive dynamics converge to a pure-strategy Nash equilibrium.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Adversarial Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science

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.

Problem

• game connectivity
• pure Nash equilibrium
• many-action games
• adaptive dynamics
• best-response path

Innovation

• game connectivity
• many-action games
• pure Nash equilibrium
• adaptive dynamics
• probabilistic combinatorics