Empirical distribution of equilibrium play and its testing application

Yakov Babichenko, Siddharth Barman, Ron Peretz

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We show that in any n-player, m-action normal-form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three types of equilibria: Nash, correlated, and coarse-correlated. For each one we obtain upper and lower bounds on the number of samples required for the empirical distribution over the sampled action profiles to form an approximate equilibrium with probability close to 1. These bounds imply that using a small number of samples we can test whether or not players are playing according to an approximate equilibrium, even in games where n and m are large. In addition, our results substantially improve previously known upper bounds on the support size of approximate equilibria in games with many players. In particular, for the three types of equilibria we show the existence of approximate equilibrium with support-size polylogarithmic in n and m, whereas the previously bestknown upper bounds were polynomial in n (Hémon et al. [Hémon S, de Rougemont M, Santha M (2008) Approximate nash equilibria for multi-player games. Algorithmic Game Theory (Springer), 267-278], Germano and Lugosi [Germano F, Lugosi G (2007) Existence of sparsely supported correlated equilibria. Econom. Theory 32(3):575-578], Hart et al. [Hart S, Mas-Colell A, Babichenko Y (2013) Simple Adaptive Strategies: From Regret-Matching to Uncoupled Dynamics, Vol. 4 (World Scientific Publishing Company)]).

Original languageEnglish
Pages (from-to)15-29
Number of pages15
JournalMathematics of Operations Research
Issue number1
Early online date17 Oct 2016
StatePublished - Feb 2017

Bibliographical note

Publisher Copyright:
© 2016 INFORMS.


Yakov Babichenko would like to acknowledge the support of Israel Science Foundation [Grant 2021296], the Air Force Office of Scientific Research [Grant FA9550-09-1-0538], and the Office of Naval Research [Grant N00014-09-1-0751].

FundersFunder number
Office of Naval ResearchN00014-09-1-0751
Air Force Office of Scientific ResearchFA9550-09-1-0538
Israel Science Foundation2021296


    • Approximate Nash equilibrium
    • Hypothesis testing
    • Sampling method


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