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Query Complexity of Approximate Equilibria in Anonymous Games

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Web and Internet Economics (WINE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 9470))

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Abstract

We study the computation of equilibria of two-strategy anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game’s payoff function, assumed to be unknown initially. The general topic we consider is query complexity, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium.

We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. Our main result is a new randomized query-efficient algorithm that finds a \(O(n^{-1/4})\)-approximate Nash equilibrium querying \(\tilde{O}(n^{3/2})\) payoffs and runs in time \(\tilde{O}(n^{3/2})\). This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be used to get an efficient PTAS. Furthermore, we prove that \(\varOmega (n \log {n})\) payoffs must be queried in order to find any \(\epsilon \)-well-supported Nash equilibrium, even by randomized algorithms.

P.W.Goldberg – Supported by EPSRC project EP/K01000X/1.

S. Turchetta – Work based on MSc. thesis at the Technische Universität München.

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Notes

  1. 1.

    Due to space constraints, we defer this discussion to the full version of the paper (http://arxiv.org/abs/1412.6455).

  2. 2.

    To make Theorem 4 easier to read, we state it only for the best attainable approximation (i.e., \(n^{-1/4}\)); however, it is possible to set parameters to get any approximation \(\epsilon \ge n^{-1/4}\). For details, see the proof of Theorem 4.

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Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Paul W. Goldberg & Stefano Turchetta

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  1. Paul W. Goldberg
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  2. Stefano Turchetta
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Correspondence to Stefano Turchetta .

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Editors and Affiliations

  1. Athens University of Economics and Business, Athens, Greece

    Evangelos Markakis

  2. CWI and VU Amsterdam, Amsterdam, The Netherlands

    Guido Schäfer

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Goldberg, P.W., Turchetta, S. (2015). Query Complexity of Approximate Equilibria in Anonymous Games. In: Markakis, E., Schäfer, G. (eds) Web and Internet Economics. WINE 2015. Lecture Notes in Computer Science(), vol 9470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48995-6_26

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  • DOI: https://doi.org/10.1007/978-3-662-48995-6_26

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