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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 reply functions and compact strategy sets. We show a preliminary fixpoint uniqueness argument which provides sufficient assumptions on the best replies of a nice game for the existence of exactly one Nash equilibrium. Then we examine the necessity and sufficiency of the conditions on the utility functions for such assumptions to be satisfied; in particular, we find necessary and sufficient conditions for the isotonicity and concavity∖chain-concavity of best reply functions. 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 convex∖chain-convex best reply functions. A final extension to Bayesian games is exhibited.

Suggested Citation

  • Ceparano, Maria Carmela & Quartieri, Federico, 2017. "Nash equilibrium uniqueness in nice games with isotone best replies," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 154-165.
    • Handle: RePEc:eee:mateco:v:70:y:2017:i:c:p:154-165
    DOI: 10.1016/j.jmateco.2017.02.011
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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; Strategic complementarity; Nice game; Concave best reply function; Chain-concave best reply function;
    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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