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Uniqueness of Nash Equilibrium in Continuous Weighted Potential Games

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Abstract

The literature results about existence of Nash equilibria in continuous potential games (Monderer and Shapley, 1996) exploits the property that any maximum point of the potential function is a Nash equilibrium of the game (the vice versa being not true) and those about uniqueness use strict concavity of the potential function. Therefore, the following question arises: can we find sufficient conditions on the data of the game which guarantee one and only one Nash equilibrium when existence of a maximum of the potential function is not ensured and the potential function in not strictly concave? The paper positively answers this question for two-player weighted potential games when the strategy sets are not bounded sets of not necessarily finite dimensional spaces. Significative examples infinite dimensional spaces are provided, together with an application in infinite dimensional ones.

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  • 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.
    • Handle: RePEc:sef:csefwp:471
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    1. Fuente,Angel de la, 2000. "Mathematical Methods and Models for Economists," Cambridge Books, Cambridge University Press, number 9780521585293, January.
    2. 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.
    3. Efe A. Ok, 2007. "Preliminaries of Real Analysis, from Real Analysis with Economic Applications," Introductory Chapters, in: Real Analysis with Economic Applications, Princeton University Press.
    4. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, January.
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    Cited by:

    1. 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.
    2. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2018. "An Adjustment Process-based Algorithm with Error Bounds for Approximating a Nash Equilibrium," CSEF Working Papers 502, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, revised 23 Mar 2020.

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    Keywords

    Non-cooperative game; weighted potential game; uniqueness of Nash equilibrium; fixed point.;
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