The Frequency of Convergent Games under Best-Response Dynamics

Samuel C. Wiese () and Torsten Heinrich ()
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Samuel C. Wiese: University of Oxford
Torsten Heinrich: University of Oxford

Dynamic Games and Applications, 2022, vol. 12, issue 2, No 15, 689-700

Abstract: Abstract We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

Keywords: Pure Nash equilibrium; Best-response dynamics; Random games; 91A10; 91A06 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13235-021-00401-3

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