Perfect equilibria in games of incomplete information
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Perfect equilibria in games of incomplete information Oriol Carbonell-Nicolau1 Received: 2 November 2018 / Accepted: 11 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper extends Selten’s (Int J Game Theory 4:25–55, 1975) notion of perfection to normal-form games of incomplete information and provides conditions on the primitives of a game that ensure the existence of a perfect Bayes–Nash equilibrium. The existence results, which allow for arbitrary (compact, metric) type and/or action spaces and payoff discontinuities, are illustrated in the context of all-pay auctions and Cournot games with incomplete information and cost discontinuities. Keywords Infinite game of incomplete information · Perfect Bayes–Nash equilibrium · Payoff security JEL Classification C72
1 Introduction The notion of perfect equilibrium was introduced by Selten (1975). For normal-form games with complete information, Selten’s (1975) perfect equilibrium refines the Nash equilibrium concept by requiring that equilibrium strategies be immune to slight trembles in the execution of the players’ actions. The standard definition of perfect equilibrium for normal-form games with finite action spaces (see, e.g., van Damme 2002) can be extended to normal-form games with infinitely many actions, and these extensions have been studied by several authors (see, e.g., Al-Najjar 1995; Simon and Stinchcombe 1995; Carbonell-Nicolau 2011a, b, c, 2014b; Carbonell-Nicolau and McLean 2013, 2014, 2015; Scalzo 2014; Bajoori et al. 2013). For applications of the notion of perfection as an equilibrium selection criterion in complete-information games, see, e.g., Bagnoli and Lipman (1989), Broecker (1990), Pitchik and Schotter (1988), and Allen (1988).
Thanks to the anonymous referees and the associate editor for their valuable comments.
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Oriol Carbonell-Nicolau [email protected] Department of Economics, Rutgers University, 75 Hamilton Street, New Brunswick, NJ 08901, USA
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O. Carbonell-Nicolau
This paper considers an extension of the notion of perfection to normal-form games of incomplete information, also called Bayesian games, that refines the standard Bayes–Nash equilibrium concept. Roughly, a Bayes–Nash equilibrium is perfect if there are nearby Bayes–Nash equilibria in slightly perturbed Bayesian games in which each type of each player makes slight mistakes in the execution of her strategies. Conditions on the primitives of a Bayesian game are furnished under which a Bayesian game with infinitely many types and/or actions (henceforth infinite Bayesian games), and possibly with payoff discontinuities in type and action profiles, possesses a perfect Bayes–Nash equilibrium.1 While there is a substantial literature on the existence of Bayes–Nash equilibria in infinite Bayesian games,2 there is very little work dealing with refinements. Jackson et al. (2002) employ the notion of perfection to eliminate “undesirable” Nash equilibria in one of their applications to second-price auction games. A stron
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