Game connectivity and adaptive dynamics in many-action games

TL;DR

The paper analyzes how frequently random, ordinal, generic $n$-player $k$-action games exhibit connectivity of their best-response graphs, a property linked to convergence of adaptive dynamics to pure Nash equilibria. In the many-actions regime ($k\to\infty$ with fixed $n\ge3$) the connected fraction tends to $1-\zeta_n$, where $\zeta_n>0$ is explicitly characterized via Poisson-Galton–Watson-type parameters; meanwhile, as $n\to\infty$ (with $k$ large or fixed) the connected fraction tends to 1, and thus connectivity is typical in both iterated limits and in the full limit. The core methodology recasts the problem in terms of random subgraphs of directed Hamming graphs $\vv{H}(n,k)$, defining a random process $\vv{L}(n,k)$ that yields Poisson limits for counts of sinks with distinct reachability properties, and then uses a multi-scale exploration to show the existence of a large strongly connected component formed by good cycles across slices. The results imply that a simple adaptive dynamic, such as best-response with inertia, converges to a pure Nash equilibrium in all but a vanishingly small fraction of generic games with a NE, thereby offering a probabilistic complement to Hart’s impossibility results and providing insight into equilibrium convergence in large-action spaces relevant to AI alignment and combinatorial settings.

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 $1-ζ_n$ as $k$ gets large, where $ζ_n>0$. In other words, a constant fraction of many-action games are not connected. However, $ζ_n$ 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.

Figures (1)

• Figure 1: A 3-player 2-action game (left) and its corresponding best-response graph (right). The $\bullet$ vertices are sinks. This figure is repeated from *johnston2023game.

Theorems & Definitions (65)

• Theorem 1
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• Definition 6
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• Lemma 7
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• Definition 9