The Existence of Berge-Vaisman Equilibrium in a Differential Positional Game of two Persons in which the Nash-Equilibrium is absent

V. I. Zhukovskii   ; Yu. A. Bel'skikh   ; L. V. Zhukovskaya

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2024), pp. 7-17

Résumé

Every person who encountered with the research of conflict situations knows that it is very difficult (and as a rule impossible) to obtain solution in explicit analytical form. Linearquadratic differential games appear as an exception. The present article is devoted to such games. The idea of Lyapunov A. M. about possibility of investigation of behavior of solution of differential equations without solving them but using properties of Lyapunov function and its derivative here is associated with possibility of opinion about optimal properties of players strategies according to optimal properties of Bellman functions and their derivative. This approach is based on combination of dynamic programming with method of Lyapunov functions suggested by academician Krasovskii N. N. so that it allows to obtain the coefficient criteria of existence of solutions and construction their explicit form. Using such approach in the article the set of differential games is extracted in which there exists the Berge-Vaisman equilibrium. The notion of “Berge equilibrium” appeared in Russia in 1994 during the critical discussion of the published book “Theorie Generale des Jeux a n Personnes Games” by Claude Berge. Berge equilibrium removes “selfish” nature promoted in Nash equilibrium due to the altruistic approach, dictated by the concept of Berge equilibrium. In 1995 Constantin S. Vaisman (then a graduate student of Zhukovskiy) defended his Ph.D. thesis on Berge equilibrium, at the Leningrad University in 1995. Unfortunately, Vaisman died three years after the Ph.D. defense of the thesis, before the age of 36 years. During these three years he published 19 works. We observe also that Vaisman together with the first author of this article wrote individual chapters in two books. Vaisman merit lies in the fact that he presented the example that the property of individual rationality for Berge equilibrium, generally speaking, does not take place. Therefore Vaisman added this requirement in the definition of Berge equilibrium, after which Berge equilibrium was naturally called as the equilibrium by Berge–Vaisman. Berge equilibrium has not got the bright destiny. Because of Vaisman’s death, who was the greatest enthusiast of Berge equilibrium, they suspended the investigation of it in Russia. Furthermore, the publication of the book by Claude Berge aroused the acute review of Martin Shubic. However, the Algerian trainees of Zhukovskiy Radjef Mohamed Said and Larbani Moussa managed to publish the works, which caused widespread interest in the West to Berge equilibrium. Right now the research of Berge equilibrium stuck at an early stage. Namely, there are the initial accumulation of facts, the formalization of modification Berge equilibrium, a comparison with Nash equilibrium. Futhermore, basically, all the studies are limited to only finite non-cooperative games. We believe it is time to proceed to the second heuristic stage, that is to answer the following two questions: 1) Is there Berge equilibrium and how to build it? 2) How should one take into account the dynamics of the conflict? Zhukovskiy V.I. and his co-authors devoted their recently published book “Mathematical foundations of the Golden rule: altruistic way of resolving conflicts as opposed egoistic to Nash equilibrium” to answering the first question, it is true only within static version of non-cooperative games. We expect to dedicate a separate book to dynamic version of the problem (within the mathematical formalization of the positional differential game proposed by Russian acade

Détail

Keywords: Non-cooperative positional linear-quadratic differential game, dynamic programming, Berge-Vaisman equilibrium, Nash equilibrium, continuous dependence and analyticity of the solution by parameter