Nash equilibria and bargaining solutions of differential bilinear games. (English) Zbl 1477.49005

Summary: 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.


49J15 Existence theories for optimal control problems involving ordinary differential equations
49N70 Differential games and control
49M15 Newton-type methods
35Q41 Time-dependent Schrödinger equations and Dirac equations
91A23 Differential games (aspects of game theory)


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