Iteratively Computation the Nash Equilibrium Points in the Two-Player Positive Games

AUTHOR(S):

Ivan G. Ivanov, Nikolay Netov, Vladislav Tanov

TITLE

Iteratively Computation the Nash Equilibrium Points in the Two-Player Positive Games

ABSTRACT

We consider the linear quadratic differential games for positive linear systems with the feedback information structure and two players. The accelerated Newton method to obtain the stabilizing solution of a corresponding set of Riccati equations is presented in [6], where the convergence properties are established. In addition, the Lyapunov iterative method to compute the Nash equilibrium point is presented in [7] (5th International Conference Applied and Computational Mathematics, WSEAS Conference at Mallorca, 2016). Based on these two methods we derive a new one - the accelerated Lyapunov method. Moreover, the convergence properties are proved. The performances of the proposed algorithm are illustrated on some numerical examples.

KEYWORDS

feedback Nash equilibrium, generalized Riccati equation, stabilizing solution, nonnegative solution

REFERENCES

[1] T. Azevedo-Perdicoulis and G. Jank, Linear Quadratic Nash Games on Positive Linear Systems, European Journal of Control, 11, 2005, 1– 13. [1] T. Azevedo-Perdicoulis and G. Jank, Linear Quadratic Nash Games on Positive Linear Systems, European Journal of Control, 11, 2005, 1– 13.

[2] B. Bazar and G. J. Olsder, Dynamic Noncooperative Game Theory, SIAM, (1999).

[3] S. Jorgensen, and G. Zaccour, Differential Games in Marketing. International Series in Quantitative Marketing, Kluwer Academic Publisher in 2004.

[4] E. Dockner, S. Jorgensen, N. V. Long, G. Sorger, Differential games in economics and management science, Cambridge University Press, (2000).

[5] L. Imsland, I. G.Ivanov and S.Kostova, Linear Quadratic Differential Games and Applications, Biomath Communications, 3, 2016, N 2,

[6] I. G.Ivanov, L. Imsland and B. C. Bogdanova, Iterative Algorithms for Computing the Feedback Nash Equilibrium Point for Positive Systems, accepted in International Journal of System Science,

[7] I. G.Ivanov and N. Netov, The Nash Equilibrium Point in the LQ Game on Positive Systems with Two Players, Mathematical and Computational Methods, 1, 2016, 242–246,

[8] L. Metzler, Stability of multiple markets: the Hicks condition, Econometrica, 13(4), 1945, pp. 277–292.

Cite this paper

Ivan G. Ivanov, Nikolay Netov, Vladislav Tanov. (2016) Iteratively Computation the Nash Equilibrium Points in the Two-Player Positive Games. International Journal of Mathematical and Computational Methods, 1, 378-381