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Nash Equilibria and Bargaining Solutions of Differential Bilinear Games

Author

Listed:
  • Francesca Calà Campana

    (Universität Würzburg)

  • Gabriele Ciaramella

    (Universität Konstanz)

  • Alfio Borzì

    (Universität Würzburg)

Abstract

This paper is devoted to a theoretical and numerical investigation of Nash equilibria and Nash bargaining problems governed by bilinear (input-affine) differential models. These systems with a bilinear state-control structure arise in many applications in, e.g., biology, economics, physics, where competition between different species, agents, and forces needs to be modelled. For this purpose, the concept of Nash equilibria (NE) appears appropriate, and the building blocks of the resulting differential Nash games are different control functions associated with different players that pursue different non-cooperative objectives. In this framework, existence of Nash equilibria is proved and computed with a semi-smooth Newton scheme combined with a relaxation method. Further, a related Nash bargaining (NB) problem is discussed. This aims at determining an improvement of all players’ objectives with respect to the Nash equilibria. Results of numerical experiments successfully demonstrate the effectiveness of the proposed NE and NB computational framework.

Suggested Citation

  • Francesca Calà Campana & Gabriele Ciaramella & Alfio Borzì, 2021. "Nash Equilibria and Bargaining Solutions of Differential Bilinear Games," Dynamic Games and Applications, Springer, vol. 11(1), pages 1-28, March.
    • Handle: RePEc:spr:dyngam:v:11:y:2021:i:1:d:10.1007_s13235-020-00351-2
    DOI: 10.1007/s13235-020-00351-2
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    References listed on IDEAS

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    1. Terry L. Friesz, 2010. "Dynamic Optimization and Differential Games," International Series in Operations Research and Management Science, Springer, number 978-0-387-72778-3, April.
    2. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    3. 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, September.
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