Approximation and characterization of Nash equilibria of large games
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Approximation and characterization of Nash equilibria of large games Guilherme Carmona1
· Konrad Podczeck2
Received: 21 November 2019 / Accepted: 18 September 2020 © The Author(s) 2020
Abstract We characterize Nash equilibria of games with a continuum of players in terms of approximate equilibria of large finite games. This characterization precisely describes the relationship between the equilibrium sets of the two classes of games. In particular, it yields several approximation results for Nash equilibria of games with a continuum of players, which roughly state that all finite-player games that are sufficiently close to a given game with a continuum of players have approximate equilibria that are close to a given Nash equilibrium of the non-atomic game. Keywords Nash equilibrium · Non-atomic games · Large games · Approximation JEL Classification C72
1 Introduction Models with a continuum of agents are viewed as a tractable idealization of situations involving a large but finite number of negligible agents. This view requires considerations of the relationship between results in models with a continuum of agents and models with a large but finite number of them. In the context of general equilibrium theory, this issue was taken up e.g. by Hildenbrand (1970), Hildenbrand and Mertens (1972), Hildenbrand (1974), Emmons and Yannelis (1985) and Yannelis (1985), amongst others. Analogous results have been obtained in the context of game
We wish to thank three anonymous referees for helpful comments. Financial support from Fundação para a Ciência e a Tecnologia is gratefully acknowledged.
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Guilherme Carmona [email protected] Konrad Podczeck [email protected]
1
School of Economics, University of Surrey, Guildford GU2 7XH, UK
2
Institut für Volkswirtschaftslehre, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
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G. Carmona, K. Podczeck
theory by Green (1984), Housman (1988), Khan and Sun (1999), Khan et al. (2013) and He et al. (2017). In this paper, we revisit these latter results. In Carmona and Podczeck (2020a, b), we have recently shown that, under an equicontinuity assumption on players’ payoff functions, equilibria of games with a continuum of players are the limit points of equilibria of games with a large but finite number of players. These results bear some surprise: It is well known that there is often an “explosion” of Nash equilibria in the limit, which suggested that limits of pure strategy Nash equilibria in sequences of finite-player games converging to a given non-atomic game could form a proper subset of the equilibrium set of the limit non-atomic game. As our results in those previous papers show, this intuition is incorrect. However, this discussion also means that, while there is at least one sequence of finite-player games converging to a given game with a continuum of players and having equilibria converging to a given equilibrium of the game with a continuum of players, not all sequences of finite-player games converging to a given game with a contin
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