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Nash equilibrium uniqueness in nice games with isotone best replies

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  • Ceparano, Maria Carmela
  • Quartieri, Federico

Abstract

We prove the existence of a unique pure-strategy Nash equilibrium in nice games with isotone chain-concave best replies and compact strategy sets. We establish a preliminary fixpoint uniqueness argument showing sufficient assumptions on the best replies of a nice game that guarantee the existence of exactly one Nash equilibrium. Then, by means of a comparative statics analysis, we examine the necessity and sufficiency of the conditions on (marginal) utility functions for such assumptions to be satisfied; in particular, we find necessary and sufficient conditions for the isotonicity and chain-concavity of best replies. We extend the results on Nash equilibrium uniqueness to nice games with upper unbounded strategy sets and we present "dual" results for games with isotone chain-convex best replies. A final application to Bayesian games is exhibited.

Suggested Citation

  • Ceparano, Maria Carmela & Quartieri, Federico, 2015. "Nash equilibrium uniqueness in nice games with isotone best replies," MPRA Paper 67765, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:67765
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    1. Van Zandt, Timothy, 2010. "Interim Bayesian Nash equilibrium on universal type spaces for supermodular games," Journal of Economic Theory, Elsevier, vol. 145(1), pages 249-263, January.
    2. Moshe Shaked & J. Shanthikumar, 1990. "Parametric stochastic convexity and concavity of stochastic processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(3), pages 509-531, September.
    3. G. Giorgi & S. Komlósi, 1992. "Dini derivatives in optimization — Part I," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 15(1), pages 3-30, March.
    4. Dubey, Pradeep & Haimanko, Ori & Zapechelnyuk, Andriy, 2006. "Strategic complements and substitutes, and potential games," Games and Economic Behavior, Elsevier, vol. 54(1), pages 77-94, January.
    5. Pierre von Mouche & Federico Quartieri (ed.), 2016. "Equilibrium Theory for Cournot Oligopolies and Related Games," Springer Series in Game Theory, Springer, number 978-3-319-29254-0, June.
    6. Massimo Marinacci & Luigi Montrucchio, 2005. "Ultramodular Functions," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 311-332, May.
    7. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-180, January.
    8. Moulin, Herve, 1984. "Dominance solvability and cournot stability," Mathematical Social Sciences, Elsevier, vol. 7(1), pages 83-102, February.
    9. Milgrom, Paul & Roberts, John, 1996. "Coalition-Proofness and Correlation with Arbitrary Communication Possibilities," Games and Economic Behavior, Elsevier, vol. 17(1), pages 113-128, November.
    10. John Kennan, 2001. "Uniqueness of Positive Fixed Points for Increasing Concave Functions on Rn: An Elementary Result," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 4(4), pages 893-899, October.
    11. Agliardi, Elettra, 2000. "A generalization of supermodularity," Economics Letters, Elsevier, vol. 68(3), pages 251-254, September.
    12. Vives, Xavier, 1990. "Nash equilibrium with strategic complementarities," Journal of Mathematical Economics, Elsevier, vol. 19(3), pages 305-321.
    13. G. Giorgi & S. Komlósi, 1992. "Dini derivatives in optimization — Part II," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 15(2), pages 3-24, September.
    14. Belhaj, Mohamed & Bramoullé, Yann & Deroïan, Frédéric, 2014. "Network games under strategic complementarities," Games and Economic Behavior, Elsevier, vol. 88(C), pages 310-319.
    15. Ui, Takashi, 2016. "Bayesian Nash equilibrium and variational inequalities," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 139-146.
    16. Nikolai S. Kukushkin, 2016. "Cournot Tatonnement in Aggregative Games with Monotone Best Responses," Springer Series in Game Theory, in: Pierre von Mouche & Federico Quartieri (ed.), Equilibrium Theory for Cournot Oligopolies and Related Games, pages 31-45, Springer.
    17. ,, 2015. "Social activity and network formation," Theoretical Economics, Econometric Society, vol. 10(2), May.
    18. Müller, Alfred & Scarsini, Marco, 2012. "Fear of loss, inframodularity, and transfers," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1490-1500.
    19. Maschler,Michael & Solan,Eilon & Zamir,Shmuel, 2013. "Game Theory," Cambridge Books, Cambridge University Press, number 9781107005488.
    20. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-1277, November.
    21. Donald M. Topkis, 1978. "Minimizing a Submodular Function on a Lattice," Operations Research, INFORMS, vol. 26(2), pages 305-321, April.
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    Cited by:

    1. Maria Carmela Ceparano & Federico Quartieri, 2020. "On Pareto Dominance in Decomposably Antichain-Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 68-85, July.
    2. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2019. "Subgame Perfect Nash Equilibrium: A Learning Approach via Costs to Move," Dynamic Games and Applications, Springer, vol. 9(2), pages 416-432, June.
    3. Ceparano, Maria Carmela & Quartieri, Federico, 2019. "A second welfare theorem in a non-convex economy: The case of antichain-convexity," Journal of Mathematical Economics, Elsevier, vol. 81(C), pages 31-47.
    4. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2017. "Uniqueness of Nash Equilibrium in Continuous Weighted Potential Games," CSEF Working Papers 471, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, revised 18 Jun 2017.
    5. Vincenzo Scalzo, 2020. "On the uniqueness of Nash equilibrium in discontinuous ordinal and normal form games," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 163-168, April.
    6. Maria Carmela Ceparano & Jacqueline Morgan, 2015. "Equilibria Under Passive Beliefs for Multi-leader-follower Games with Vertical Information: Existence Results," CSEF Working Papers 417, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, revised 09 Jan 2017.
    7. Maria Carmela Ceparano & Jacqueline Morgan, 2017. "Equilibrium selection in multi-leader-follower games with vertical information," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 526-543, October.

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    More about this item

    Keywords

    Nash equilibrium uniqueness; Chain-concave best replies; Nice games; Comparative statics; Strategic complementarity.;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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