Nash Equilibria and Bargaining Solutions of Differential Bilinear Games

Francesca Calà Campana (), Gabriele Ciaramella () and Alfio Borzì ()
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Francesca Calà Campana: Universität Würzburg
Gabriele Ciaramella: Universität Konstanz
Alfio Borzì: Universität Würzburg

Dynamic Games and Applications, 2021, vol. 11, issue 1, No 1, 28 pages

Abstract: 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.

Keywords: Bilinear evolution models; Nash equilibria; Nash bargaining problem; Optimal control theory; Quantum evolution models; Lotka–Volterra models; Newton methods; 49J15; 49N70; 49M15; 35Q41 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13235-020-00351-2

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