Berge and Nash equilibrium in a linear-quadratic differential game

Vladislav I. Zhukovskiy ; Anton S. Gorbatov ; Konstantin N. Kudryavtsev

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 9 (2017) no. 1, pp. 62-94.

Résumé

Everyone who was concerned with the theory of stability in Liapunov`s sense can keep in mind the section: coefficient criteria of stability. The idea is in fact that without solving the differential equations and considering signs of coefficients and (or) relations among them one can judge at once about stability of nonperturbed motion. In this article we tried to apply the same approach but now with a view to choose the conceptions of equilibrium in the class of non-cooperation differential linear-quadratic games of two persons. We can answer two questions: first, are there situations of Berge equilibrium and (or) Nash equilibrium?; secondly, how we can finf them? Actually the answers «are hiding» (considering the method of dynamic programming) in possibility to judge about the existence of extendable on time interval game solution of system from two ordinary matrix differential equations Riccaty type. To answer this problem positively we had to attract the method of small parameter and theorem of Poincare on analyticity of parameter solution.

Détail

Keywords: non-cooperation differential linear-quadratic positional game, Berge and Nash equilibrium, dynamic programming, small parameter method.

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