🤖 AI Summary
This paper investigates the connectivity of best-response graphs in generic games: Why, in large random games admitting pure Nash equilibria (PNE), do almost all non-equilibrium strategy profiles admit a best-response path to some PNE?
Method: Leveraging an interdisciplinary approach combining probabilistic combinatorics, game theory, and graph theory, the authors establish the first universal probabilistic relationship among game size, existence of PNE, and best-response graph connectivity.
Contribution/Results: They prove that for almost all large generic games possessing at least one PNE, the associated best-response graph is asymptotically almost surely connected. This graph-theoretic connectivity serves as a sufficient structural condition ensuring almost-sure convergence of adaptive dynamics—such as inertia-augmented best-response processes—to a PNE. The result uncovers an intrinsic robustness in equilibrium accessibility within high-dimensional games and provides a foundational topological explanation for the convergence of uncoupled learning mechanisms.
📝 Abstract
We analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. In particular, we show that almost every 'large' generic game that has a pure Nash equilibrium is connected, meaning that every non-equilibrium action profile can reach every pure Nash equilibrium via best-response paths. This has implications for dynamics in games: many adaptive dynamics, such as the best-response dynamic with inertia, lead to equilibrium in connected games. It follows that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in almost every 'large' generic game that has one. We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.
