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Satisficing behavior, Brouwer’s Fixed Point Theorem and Nash Equilibrium

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  • Robert Becker
  • Subir Chakrabarti

Abstract

We show that Nash Equilibrium points can be obtained by using response maps or reply functions that simply use better responses rather than best responses. We demonstrate the existence of a Nash Equilibrium as the fixed point of a better response map and since the better response map is continuous the fixed point can be established by simply using Brouwer’s fixed point theorem. The proof applies to games with a finite number of strategies as well as to games with a continuum of strategies. In case the games have a continuum of strategies the payoff functions have to be continuous on the action sets and quasi concave on the player’s action set. Copyright Springer-Verlag Berlin/Heidelberg 2005

Suggested Citation

  • Robert Becker & Subir Chakrabarti, 2005. "Satisficing behavior, Brouwer’s Fixed Point Theorem and Nash Equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(1), pages 63-83, July.
    • Handle: RePEc:spr:joecth:v:26:y:2005:i:1:p:63-83
    DOI: 10.1007/s00199-004-0519-z
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    Citations

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    Cited by:

    1. Haake, Claus-Jochen & Su, Francis Edward, 2011. "A simplicial algorithm approach to Nash equilibria in concave games," Center for Mathematical Economics Working Papers 382, Center for Mathematical Economics, Bielefeld University.
    2. Roy Allen & John Rehbeck, 2021. "A Generalization of Quantal Response Equilibrium via Perturbed Utility," Games, MDPI, vol. 12(1), pages 1-16, March.
    3. Alexis Toda, 2015. "Bayesian general equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(2), pages 375-411, February.
    4. Kairui Zuo & Jiancheng Guan, 2017. "Measuring the R&D efficiency of regions by a parallel DEA game model," Scientometrics, Springer;Akadémiai Kiadó, vol. 112(1), pages 175-194, July.
    5. Salvador Barberà & Geoffroy de Clippel & Alejandro Neme & Kareen Rozen, 2022. "Order-k rationality," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(4), pages 1135-1153, June.
    6. Chakrabarti, Subir K., 2014. "On the robustness of the competitive equilibrium: Utility-improvements and equilibrium points," Journal of Mathematical Economics, Elsevier, vol. 55(C), pages 36-47.
    7. P. Baecker & G. Grass & U. Hommel, 2010. "Business value and risk in the presence of price controls: an option-based analysis of margin squeeze rules in the telecommunications industry," Annals of Operations Research, Springer, vol. 176(1), pages 311-332, April.
    8. Liang Liang & Jie Wu & Wade D. Cook & Joe Zhu, 2008. "The DEA Game Cross-Efficiency Model and Its Nash Equilibrium," Operations Research, INFORMS, vol. 56(5), pages 1278-1288, October.

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