Strong robustness to incomplete information and the uniqueness of a correlated equilibrium

Abstract

We define and characterize the notion of strong robustness to incomplete information, whereby a Nash equilibrium in a game \(\mathbf{u}\) is strongly robust if, given that each player knows that his payoffs are those in \(\mathbf{u}\) with high probability, all Bayesian–Nash equilibria in the corresponding incomplete-information game are close—in terms of action distribution—to that equilibrium of \(\mathbf{u}\). We prove, under some continuity requirements on payoffs, that a Nash equilibrium is strongly robust if and only if it is the unique correlated equilibrium. We then review and extend the conditions that guarantee the existence of a unique correlated equilibrium in games with a continuum of actions. The existence of a strongly robust Nash equilibrium is thereby established for several domains of games, including those that arise in economic environments as diverse as Tullock contests, all-pay auctions, Cournot and Bertrand competitions, network games, patent races, voting problems and location games.

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Correspondence to David Lagziel.

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The authors wish to thank Eddie Dekel, Ehud Lehrer, Yehuda John Levy, Daisuke Oyama, David Schmeidler, Aner Sela, and the participants of SWET 2019 workshop held in Otaru, Japan, for their valuable comments. The authors’ special grattitude goes to Atsushi Kajii, whose encouragement to put the first basic ideas into writing led to this work.

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Einy, E., Haimanko, O. & Lagziel, D. Strong robustness to incomplete information and the uniqueness of a correlated equilibrium. Econ Theory (2020). https://doi.org/10.1007/s00199-020-01327-4

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Keywords

  • Strong robustness to incomplete information
  • Nash equilibrium
  • Correlated equilibrium

Mathematics Subject Classification

  • C62
  • C72
  • D82