Riascos Villegas, Alvaro and Torres-Martínez, Juan Pablo
(2012):
*On the existence of pure strategy equilibria in large generalized games with atomic players.*

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## Abstract

We consider a game with a continuum of players where only a finite number of them are atomic. Objective functions and admissible strategies may depend on the actions chosen by atomic players and on aggregate information about the actions chosen by non-atomic players. Only atomic players are required to have convex sets of admissible strategies and quasi-concave objective functions.

We prove the existence of a pure strategy Nash equilibria. Thus, we extend to large generalized games with atomic players the results of equilibrium existence for non-atomic games of Schemeidler (1973) and Rath (1992). We do not obtain a pure strategy equilibrium by purification of mixed strategy equilibria. Thus, we have a direct proof of both Balder (1999, Theorem 2.1) and Balder (2002, Theorem 2.2.1), for the case where non-atomic players have a common non-empty set of strategies and integrable bounded codification of action profiles.

Our main result is readily applicable to many interesting problems in general equilibrium. As an application, we extend Aumann (1966) result on the existence of equilibrium with a continuum of traders to a standard general equilibrium model with incomplete asset markets.

Item Type: | MPRA Paper |
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Original Title: | On the existence of pure strategy equilibria in large generalized games with atomic players |

Language: | English |

Keywords: | Generalized games; Non-convexities; Pure-strategy Nash equilibrium |

Subjects: | C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium |

Item ID: | 36626 |

Depositing User: | Juan Pablo Torres-Martínez |

Date Deposited: | 13 Feb 2012 16:59 |

Last Modified: | 05 Oct 2019 06:13 |

References: | Aumann, R.J. (1965): "Integrals of set-valued functions," Journal of Mathematical Analysis and Applications, volume 12, pages 1-12. Aumann, R.J. (1966): "Existence of competitive equilibria in markets with a continuum of traders," Econometrica, volume 34, pages 1-11. Aumann, R.J. (1976): "An elementary proof that integration preserves uppersemicontinuity," Journal of Mathematical Economics, volume 3, pages 15-18. Balder, E.J. (1999): "On the existence of Cournot-Nash equilibria in continuum games," Journal of Mathematical Economics, volume 32, pages 207-223. Balder, E.J. (2002): "A Unifying Pair of Cournot-Nash Equilibrium Existence Results," Journal of Economic Theory, volume 102, pages 437-470. Debreu, G. (1952): "A social equilibrium existence theorem", Proceedings of the National Academy of Science, volume 38, pages 886-893. Geanakoplos, J and H. Polemarchakis (1986): "Existence Regularity and Constrained Suboptimality of Competitive Allocations when Asset Market is Incomplete." Uncertainty, Information and Communication. Essays in honor of Kenneth J. Arrow, Volume III. Edited by Walter P Heller, Ross M Starr and David Starrett. Cambridge University Press. Hildenbrand, W. (1974): "Core and equilibria of a large economy," Princeton University Press, Princeton, New Jersey. Poblete-Cazenave, R., and J.P. Torres-Martnez (2012): "Equilibrium with limited-recourse collateralized loans," Economic Theory, doi: 10.1007/s00199-011-0685-8. Rath, K.P. (1992): "A direct proof of the existence of pure strategy equilibria in games with a continuum of players," Economic Theory, volume 2, pages 427-433. Schmeidler, D.(1973): "Equilibrium point of non-atomic games," Journal of Statistical Physics, volume 17, pages 295-300. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/36626 |