## 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.

### MSC:

 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)

### Software:

ReHaG; LONE; SKRYN
Full Text:

### References:

 [1] Adams, RA; Fournier, J., Sobolev spaces (2003), Amsterdam: Elsevier/Academic Press, Amsterdam · Zbl 1098.46001 [2] Assémat, E.; Lapert, M.; Zhang, Y.; Braun, M.; Glaser, SJ; Sugny, D., Simultaneous time-optimal control of the inversion of two spin-$$\frac{1}{2}$$ particles, Phys Rev A, 82, 013415 (2010) [3] Balder, EJ, A unifying approach to existence of Nash equilibria, Int J Game Theory, 24, 79-94 (1997) · Zbl 0837.90137 [4] Borzì, A.; Ciaramella, G.; Sprengel, M., Formulation and numerical solution of quantum control problems (2017), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1402.81006 [5] Bressan, A., Noncooperative differential games, Milan J Math, 79, 2, 357-427 (2011) · Zbl 1238.49055 [6] Broom, M.; Rychtar, J., Game-theoretical models in biology. Mathematical and computational biology (2013), Boca Raton: Chapman & Hall/CRC Press, Boca Raton · Zbl 1264.92002 [7] Ciaramella, G.; Borzì, A., SKRYN: a fast semismooth-Krylov-Newton method for controlling Ising spin systems, Comput Phys Commun, 190, 213-223 (2015) · Zbl 1344.81011 [8] Ciaramella, G.; Borz.ì, A., A LONE code for the sparse control of quantum systems, Comput Phys Commun, 200, 312-323 (2016) · Zbl 1351.81005 [9] Ciaramella, G.; Borzì, A.; Dirr, G.; Wachsmuth, D., Newton methods for the optimal control of closed quantum spin systems, SIAM J Sci Comput, 37, 1, A319-A346 (2015) · Zbl 1367.70057 [10] Ciarlet, P., Linear and nonlinear functional analysis with applications (2013), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1293.46001 [11] Dockner, E.; Jorgensen, S.; Van Long, N.; Sorger, G., Differential games in economics and management science (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0996.91001 [12] Ehtamo, H.; Ruusunen, J.; Kaitala, V.; Hämäläinen, RP, Solution for a dynamic bargaining problem with an application to resource management, J Optim Theory Appl, 59, 3, 391-405 (1988) · Zbl 0628.90101 [13] Engwerda, J., LQ dynamic optimization and differential games (2005), Chichester: Wiley, Chichester [14] Engwerda, J., Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games, SIAM J Control Optim, 48, 6, 3859-3881 (2010) · Zbl 1206.49020 [15] Friedman, A., Differential games (1971), Hoboken: Wiley-Interscience, Hoboken · Zbl 0229.90060 [16] Friesz, T., Dynamic optimization and differential games. International series in operations research & management science (2010), New York: Springer, New York [17] Ge X, Ding H, Rabitz H, Wu R (2019) Robust quantum control in games: an adversarial learning approach. arXiv e-prints. arXiv:1909.02296 [18] Isaacs, R., Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization (1965), Hoboken: Wiley, Hoboken · Zbl 0125.38001 [19] Jørgensen, S.; Zaccour, G., Differential games in marketing. International series in quantitative marketing (2003), New York: Springer, New York [20] Krawczyk, JB; Uryasev, S., Relaxation algorithms to find Nash equilibria with economic applications, Environ Model Assess, 5, 1, 63-73 (2000) [21] Murray, J., Mathematical biology. Biomathematics texts (1989), Berlin: Springer, Berlin · Zbl 0682.92001 [22] Mylvaganam, T.; Sassano, M.; Astolfi, A., Constructive $$\epsilon$$-Nash equilibria for nonzero-sum differential games, IEEE Trans Autom Control, 60, 4, 950-965 (2015) · Zbl 1360.91031 [23] Nash, JF, The bargaining problem, Econometrica, 18, 2, 155-162 (1950) · Zbl 1202.91122 [24] Nash, JF, Equilibrium points in n-person games, Proc Natl Acad Sci, 36, 1, 48-49 (1950) · Zbl 0036.01104 [25] Nash, JF, Non-cooperative games, Ann Math, 54, 2, 286-295 (1951) · Zbl 0045.08202 [26] Pardalos, P.; Yatsenko, V., Optimization and control of bilinear systems: theory, algorithms, and applications. Springer optimization and its applications (2010), New York: Springer, New York · Zbl 1181.93001 [27] Tolwinski, B., On the existence of Nash equilibrium points for differential games with linear and non-linear dynamics, Control Cybern, 7, 3, 57-69 (1978) · Zbl 0389.90095 [28] Varaiya, P., N-person nonzero sum differential games with linear dynamics, SIAM J Control, 8, 4, 441-449 (1970) · Zbl 0229.90063 [29] Zeidler, E., Nonlinear functional analysis and its application III: variational methods and optimization. Nonlinear functional analysis and its applications (1985), New York: Springer, New York · Zbl 0583.47051
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