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On inexact versions of a quasi-equilibrium problem: a Cournot duopoly perspective

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Abstract

This paper has two parts. In the mathematical part, we present two inexact versions of the proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. Under mild assumptions, we prove that the methods find a solution to the quasi-equilibrium problem with an approximated computation of each iteration or using a perturbation of the regularized bifunction. In the behavioral part, we justify the choice of the new perturbation, with the help of the main example that drives quasi-equilibrium problems: the Cournot duopoly model, which founded game theory. This requires to exhibit a new QEP reformulation of the Cournot model that will appear more intuitive and rigorous. It leads directly to the formulation of our perturbation function. Some numerical experiments show the performance of the proposed methods.

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Acknowledgements

We express our gratitude to the referees for their valuable comments, which have enabled us to enhance the quality of our work.

Funding

The first author is partially supported by CAPES. The fourth author is partially supported by CNPq grant 424169/2018-5 and 313901/2020-1. The project leading to this publication has received funding from the French government under the “France 2030” investment plan managed by the French National Research Agency (reference: ANR-17-EURE-0020) and from Excellence Initiative of Aix-Marseille University - A*MIDEX.

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Authors and Affiliations

  1. Department of Mathematics, CCN, Federal University of Piauí, Teresina, PI, 64049-550, Brazil

    E. L. Dias Júnior

  2. Universidade Federal do Delta do Parnaíba, Parnaíba, PI, 64202-020, Brazil

    P. J. S. Santos

  3. CNRS & EHESS, Aix-Marseille School of Economics, Aix-Marseille University, 13007, Marseille, France

    A. Soubeyran

  4. CNRS & EHESS, Aix-Marseille School of Economics, Aix-Marseille University, France and Department of Mathematics, CCN, Federal University of Piauí, Teresina, PI, 64049-550, Brazil

    J. C. O. Souza

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  1. E. L. Dias Júnior
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Júnior, E.L.D., Santos, P.J.S., Soubeyran, A. et al. On inexact versions of a quasi-equilibrium problem: a Cournot duopoly perspective. J Glob Optim 89, 171–196 (2024). https://doi.org/10.1007/s10898-023-01341-5

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