Class of differential games with no Nash equilibrium, but with equilibrium of objections and counterobjections

V. I. Zhukovskiy ; K. N. Kudryavtsev ; S. P. Samsonov ; M. I. Vysokos ; Yu. A. Belskih

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 2, pp. 5-21

Résumé

A nonstop stream of publications is devoted to the investigation of positive and negative properties of Nash equilibrium concept prevailing in economics (as solution of noncooperative game). Mostly they are related to non-uniqueness and, as a consequence, to the lack of equivalence, interchangeability, external stability as well as instability to simultaneous deviation of such solutions of two and more players. The game "dilemma of prisoners" also revealed the property of "ability to improve". The book Equilibrium Control of Multi-criteria Dynamic Problems (V.I. Zhukovskiy and N.T. Tynyanskiy, M.: MSU, 1984) is devoted to detailed analysis of such "negative" properties for differential positional games. The authors of this book com to the following conclusion: either make use of those situations of Nash equilibrium that are simultaneously free from some of the stated disadvantages, or introduce new solutions of noncooperative game. Such solutions having the merits of Nash equilibrium situation would allow to get rid of its certain disadvantages. The present article is devoted to one of such possibilities for differential games related to concepts of objections and counterobjections. The concepts of objections and counterobjections used in it are based on the concepts of objections and counterobjections well-known classical game theory. The papers of E.I. Wilkas [1973] are devoted to theoretical questions of this concept. The term "active equilibrium" suggested R.E. Smolyakov in 1983, the notion of equilibrium of objections and counterobjections in differential games was first used apparently by E.M. Vaisbord in 1974, and then it was picked up by the first author of the present article in the above mentioned book [1984], but this concept was applied and is being applied in differential games, in our opinion, insufficiently widely. This fact "called to life" the present paper. In it the class of differential games of two persons is revealed, where the usual Nash equilibrium situation is absent, but the equilibrium of objections and counterobjections is present.

Détail

Keywords: noncooperative games, Nash equilibrium, active equilibrium, equilibrium of objections and counterobjections.

Bibliographie

[1] Owen G., Game theory, Academic Press, New York, 1995, 447 pp. | MR | Zbl

[2] Zhukovskiy V. I., Tynyanskiy N. T., Equilibrium control of multicriteria dynamic problems, Izdatel'stvo MGU Publ., M., 1984, 224 pp. (in Russ.)

[3] Vaisbord E. M., “Coalition differential games”, Differ. Uravn., 10:4 (1974), 613–623 (in Russ.) | Zbl

[4] Vaisbord E. M., Zhukovskii V. I., Introduction to Multi Player Differential Game and Their Application, Gordon and Breach, New York etc., 1988, 581 pp. | MR

[5] V.I. Zhukovskii, M.E. Salukvadze, The Vector-Valued Maximin, Academic Press, N.Y. etc., 1994, 404 pp. | DOI | MR

[6] Vilkas E. I., “Formalization of the problem of choosing a game-theoretic criterion of optimality”, Mathematical Methods in Social Sciences, 2, Institut matematiki i kibernetiki AN Lit. SSR Publ., Vil'nyus, 1972, 9–55 (in Russ.) | MR

[7] Vilkas E. I., Mayminas E. Z., Solutions: theory, information, modeling, Radio i svyaz' Publ., M., 1981, 328 pp. (in Russ.)

[8] Smol'yakov E. R., Theory of Conflict Equilibria, URSS Publ., M., 2005, 301 pp. (in Russ.)

[9] Smol'yakov E. R., Generalized optimal control and dynamic conflict problems, MAKS Press Publ., M., 1983, 232 pp. (in Russ.)

[10] Smol'yakov E. R., Equilibrium models with non-coinciding interests of participants, Nauka Publ., M., 1986, 223 pp. (in Russ.)

[11] V.I. Zhukovskii, “Some Problems of Non-Antagonistic Differential Games”, Mathematical Method in Operation Research, Academy of Sciences, Bulgaria, Sofia, 1985, 103–195

[12] S.J. Biltchev, “$\epsilon$-$Z$-Equilibrium in a Differential Game Described by a Parabolic System”, Many Players Differential Game, Technical Univ., Bulgaria, Rousse, 1984, 47–52

[13] St. A. Tersian, “On the Z-Equilibrium Points in a Differential Game”, Many Players Differential Game, Technical Univ., Bulgaria, Rousse, 1984, 106–111

[14] S.D. Gaidov, “Z-Equilibrium in Stochastic Differential Game”, Many Players Differential Game, Technical Univ., Bulgaria, Rousse, 1984, 53–63 | MR

[15] P.I. Rashkov, “Sufficient Conditions for Z-Equilibrium in a Differential Game in Banach Spase”, Many Players Differential Game, Technical Univ., Bulgaria, Rousse, 1984, 91–99

[16] D.T. Dochev, N.V. Stojanov, “Existence of Z-Equilibrium in a Differential Game with Delay”, Many Players Differential Game, Technical Univ., Bulgaria, Rousse, 1984, 64–72

[17] Zhukovskiy V. I., Introduction to differential games under uncertainty. The equilibrium of objections and counterobjections, KRASAND Publ., M., 2010, 192 pp. (in Russ.)

[18] Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer, New York, 1988, 517 pp. | MR | Zbl

[19] Gantmakher F. R., Theory of matrices, Fizmatlit Publ., M., 2004, 560 pp. (in Russ.)

[20] Zhukovskiy V. I., Chikriy A. A., Differential equations. Linear-quadratic differential games, Yurayt Publ., M., 2017, 322 pp. (in Russ.)

[21] Zhukovskiy V. I., Gorbatov A. S., Kudryavtsev K. N., “Berge and Nash equilibrium in a linear-quadratic differential game”, Mat. Teor. Igr Pril., 9:1 (2017), 62–94 (in Russ.) | Zbl

[22] Voevodin V. V., Kuznetsov Yu. A., Matrices and calculations, Nauka Publ., M., 1984, 320 pp. (in Russ.)

[23] V.I. Zhukovskiy, K.N. Kudryavtsev, “Coalition equilibrium in a three-person game”, 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA) (St. Petersburg, 22–27 May 2017), 1–4 | DOI | Zbl