Strong robustness to incomplete information and the uniqueness of a correlated equilibrium
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Strong robustness to incomplete information and the uniqueness of a correlated equilibrium Ezra Einy1 · Ori Haimanko1
· David Lagziel1
Received: 5 June 2020 / Accepted: 5 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We define and characterize the notion of strong robustness to incomplete information, whereby a Nash equilibrium in a game u is strongly robust if, given that each player knows that his payoffs are those in 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 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. Keywords Strong robustness to incomplete information · Nash equilibrium · Correlated equilibrium Mathematics Subject Classification C62 · C72 · D82
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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David Lagziel [email protected] Ezra Einy [email protected] Ori Haimanko [email protected]
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Department of Economics, Ben-Gurion University of the Negev, Beer Sheva, Israel
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E. Einy et al.
1 Introduction Nash equilibrium is an immensely popular and long-established solution concept in economics. By comparison, its generalized version, the correlated equilibrium of Aumann (1974), is a far less frequent choice in economic modelling. Clearly, the two solution concepts have their merits and drawbacks: Nash equilibrium is believed to have a high predictive power and does not require a mediation or a correlation device, while the correlated equilibrium is superior in terms of computational complexity and arises naturally in a range of simple learning processes.1 In this paper we bring to light the effect produced when the two concepts happen to coincide on the robustness of equilibrium outcome to the presence of incomplete information. To motivate our notion of robustness, we first take a step back to discuss an important issue in the field of economics—the need to predict. 1.1 The need to predict, and strong robustness to incomplete information A major difficulty in the profession of economics is the perpetual requirement to provide accurate predictions in a realm affected by uncertainty and randomness. Similarl
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