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Privacy and Truthful Equilibrium Selection for Aggregative 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 a very general class of games — multi-dimensional aggregative games — which in particular generalize both anonymous games and weighted congestion games. For any such game that is also large, we solve the equilibrium selection problem in a strong sense. In particular, we give an efficient weak mediator: a mechanism which has only the power to listen to reported types and provide non-binding suggested actions, such that (a) it is an asymptotic Nash equilibrium for every player to truthfully report their type to the mediator, and then follow its suggested action; and (b) that when players do so, they end up coordinating on a particular asymptotic pure strategy Nash equilibrium of the induced complete information game. In fact, truthful reporting is an ex-post Nash equilibrium of the mediated game, so our solution applies even in settings of incomplete information, and even when player types are arbitrary or worst-case (i.e. not drawn from a common prior). We achieve this by giving an efficient differentially private algorithm for computing a Nash equilibrium in such games. The rates of convergence to equilibrium in all of our results are inverse polynomial in the number of players n. We also apply our main results to a multi-dimensional market game.

Our results can be viewed as giving, for a rich class of games, a more robust version of the Revelation Principle, in that we work with weaker informational assumptions (no common prior), yet provide a stronger solution concept (ex-post Nash versus Bayes Nash equilibrium). In comparison to previous work, our main conceptual contribution is showing that weak mediators are a game theoretic object that exist in a wide variety of games – previously, they were only known to exist in traffic routing games. We also give the first weak mediator that can implement an equilibrium optimizing a linear objective function, rather than implementing a possibly worst-case Nash equilibrium.

The full version of this extended abstract can be found on arXiv [9].

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Notes

  1. 1.

    In the economics literature, aggregative games have more restricted aggregator function: \(S_k({\varvec{x}}) = \sum _{i=1}^n x_i\). The games we study are more general, and sometimes referred to as generalized aggregative games.

  2. 2.

    Note that the influence that any single player’s action has on the utility of others is also bounded by \(\gamma \). If \(\gamma =o(1/n)\), then any player’s utility is essentially independent of other players’ actions. Therefore, we further assume that \(\gamma = \varOmega (1/n)\) for the problem to be interesting. This will also simplify some statements.

  3. 3.

    Sometimes called best react [4], and apparent best response [19].

  4. 4.

    We show that \(\mathcal {E}(\zeta )\) is non-empty for \(\zeta \ge \gamma \sqrt{8n\log (2mn)}\) in the full version.

  5. 5.

    In the full version of this paper, we also present details of the non-private algorithm to compute equilibrium for aggregative games.

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Acknowledgments

Research supported in part by NSF grants 1253345, 1101389, CNS-1254169, US-Israel Binational Science Foundation grant 2012348, Simons Foundation grant 361105, a Google Faculty Research Award, and the Alfred P. Sloan Foundation. Research performed while the first author was visiting the University of Pennsylvania.

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Authors and Affiliations

  1. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, USA

    Rachel Cummings

  2. Department of Computer and Information Science, University of Pennsylvania, Philadelphia, USA

    Michael Kearns, Aaron Roth & Zhiwei Steven Wu

Authors
  1. Rachel Cummings
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  2. Michael Kearns
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  3. Aaron Roth
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  4. Zhiwei Steven Wu
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Correspondence to Rachel Cummings .

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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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Cummings, R., Kearns, M., Roth, A., Wu, Z.S. (2015). Privacy and Truthful Equilibrium Selection for Aggregative 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_21

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  • DOI: https://doi.org/10.1007/978-3-662-48995-6_21

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