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Variational convergence: Approximation and existence of equilibria in discontinuous games

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  • Bagh, Adib
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    Abstract

    We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Economic Theory.

    Volume (Year): 145 (2010)
    Issue (Month): 3 (May)
    Pages: 1244-1268

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    Handle: RePEc:eee:jetheo:v:145:y:2010:i:3:p:1244-1268

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    Web page: http://www.elsevier.com/locate/inca/622869

    Related research

    Keywords: Convergence of games Discontinuous games Equilibrium map Bertrand-Edgeworth games;

    References

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    1. Carmona, Guilherme, 2008. "An Existence Result for Discontinuous Games," FEUNL Working Paper Series wp530, Universidade Nova de Lisboa, Faculdade de Economia.
    2. Green, Edward J, 1984. "Continuum and Finite-Player Noncooperative Models of Competition," Econometrica, Econometric Society, vol. 52(4), pages 975-93, July.
    3. Simon, Leo K, 1987. "Games with Discontinuous Payoffs," Review of Economic Studies, Wiley Blackwell, vol. 54(4), pages 569-97, October.
    4. Maskin, Eric, 1986. "The Existence of Equilibrium with Price-Setting Firms," American Economic Review, American Economic Association, vol. 76(2), pages 382-86, May.
    5. Drew Fudenberg & David Levine, 1983. "Limit Games and Limit Equilibria," UCLA Economics Working Papers 289, UCLA Department of Economics.
    6. Carlos Alós-Ferrer, 2006. "The Discretization Of Continuum Strategy Spaces," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 499-514.
    7. HILDENBRAND, Werner & MERTENS, Jean-François, . "Upper hemi-continuity of the equilibrium set correspondence for pure exchange economies," CORE Discussion Papers RP -109, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Wiley Blackwell, vol. 53(1), pages 1-26, January.
    9. Walker, Mark, 1979. "A Generalization of the Maximum Theorem," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 20(1), pages 267-72, February.
    10. Dasgupta, Partha & Maskin, Eric, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, II: Applications," Review of Economic Studies, Wiley Blackwell, vol. 53(1), pages 27-41, January.
    11. Osborne, Martin J. & Pitchik, Carolyn, 1986. "Price competition in a capacity-constrained duopoly," Journal of Economic Theory, Elsevier, vol. 38(2), pages 238-260, April.
    12. Dan Kovenock & Raymond J. Deneckere, 1996. "Bertrand-Edgeworth duopoly with unit cost asymmetry (*)," Economic Theory, Springer, vol. 8(1), pages 1-25.
    13. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
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    Cited by:
    1. Oriol Carbonell-Nicolau & Richard McLean, 2012. "Approximation Results for Discontinuous Games with an Application to Equilibrium Refinement," Departmental Working Papers 201206, Rutgers University, Department of Economics.
    2. Holmberg, Pär & Newbery, David & Ralph, Daniel, 2009. "Supply Function Equilibria: Step Functions and Continuous Representations," Working Paper Series 788, Research Institute of Industrial Economics.
    3. Pavlo Prokopovych & Nicholas C. Yannelis, 2012. "On Uniform Conditions for the Existence of Mixed Strategy Equilibria," Discussion Papers 48, Kyiv School of Economics.
    4. Tasnádi, Attila, 2012. "Endogenous Timing of Moves in Bertrand-Edgeworth Triopolies," MPRA Paper 47610, University Library of Munich, Germany.

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