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arXiv · 2309.10609

Game Connectivity and Adaptive Dynamics

Tom Johnston·Michael Savery·Alex Scott·Bassel Tarbush
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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.

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Tom Johnston, Michael Savery, Alex Scott, Bassel Tarbush. 2023-09-19. Game Connectivity and Adaptive Dynamics. https://arxiv.org/abs/2309.10609

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