Geodesic
Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 8 (2016) no. 2, pp. 58-90.

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Models of evolutionary nonzero-sum games are considered on the infinite time interval. Methods of differential games theory are used for the analysis of game interactions between two groups of participants. It is assumed, that participants in groups are subject to control through signals for the behavior change. Payoffs of coalitions are determined as average integral functionals on the infinite horizon. The problem of constructing a dynamical Nash equilibrium is posed for the considered evolutionary game. Ideas and approaches of non-antagonistic differential games are applied for the determination of the Nash equilibrium solutions. The results are based on dynamic constructions and methods of evolutionary games. The great attention is paid to the formation of the dynamical Nash equilibrium with players strategies, that maximize the corresponding payoff functions and have the guaranteed properties according to the minimax approach. The application of the minimax approach for constructing optimal control strategies synthesizes trajectories of the dynamical Nash equilibrium that provide better results in comparison to static solutions and evolutionary models with the replicator dynamics. The dynamical Nash equilibrium trajectories for evolutionary games with the average integral quality functionals are compared with trajectories for evolutionary games based on the global terminal quality functionals on the infinite horizon.
Mots-clés : dynamical bimatrix games
Keywords: average integral payoffs, characteristics of Hamilton–Jacobi equations, equilibrium trajectories. 
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Nikolay A. Krasovskii; Alexander M. Tarasyev. Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 8 (2016) no. 2, pp. 58-90. http://geodesic.mathdoc.fr/item/MGTA_2016_8_2_a2/

[1] Arnold V. I., “Optimizatsiya v srednem i fazovye perekhody v upravlyaemykh dinamicheskikh sistemakh”, Funktsionalnyi analiz i ego prilozheniya, 36:2 (2002), 1–11 | DOI | MR

[2] Vorobev N. N., Teoriya igr dlya ekonomistov-kibernetikov, Nauka, M., 1985 | MR

[3] Davydov A. A., Mena Matosh E., “Tipichnye fazovye perekhody i osobennosti vygody v modeli Arnolda”, Matematicheskii sbornik, 198:1 (2007), 21–42 | DOI | MR | Zbl

[4] Kleimenov A. F., Neantagonisticheskie pozitsionnye differentsialnye igry, Nauka, Ekaterinburg, 1993

[5] Kolmogorov A. N., “Ob analiticheskikh metodakh v teorii veroyatnostei”, Uspekhi matematicheskikh nauk, 5 (1938), 5–41

[6] Krasovskii N. A., Kryazhimskii A. V., Tarasev A. M., “Uravneniya Gamiltona–Yakobi v evolyutsionnykh igrakh”, Trudy Instituta matematiki i mekhaniki UrO RAN, 20, no. 3, 2014, 114–131 | MR | Zbl

[7] Krasovskii N. A., Tarasev A. M., “Poisk tochek maksimuma vektornogo kriteriya s dekompozitsionnymi svoistvami”, Trudy Instituta matematiki i mekhaniki UrO RAN, 15, no. 4, 2009, 167–182

[8] Krasovskii N. A., Tarasev A. M., “Dekompozitsionnyi algoritm poiska ravnovesiya v dinamicheskoi igre”, Matematicheskaya teoriya igr i ee prilozheniya, 3:4 (2011), 49–88

[9] Krasovskii N. A., Tarasev A. M., Ravnovesnye resheniya v dinamicheskikh igrakh, UrGAU, Ekaterinburg, 2015

[10] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974 | MR

[11] Kryazhimskii A. V., Osipov Yu. S., “O differentsialno-evolyutsionnykh igrakh”, Trudy matematicheskogo instituta im. V. A. Steklova RAN, 211, 1995, 257–287 | MR | Zbl

[12] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977 | MR

[13] Mazalov V. V., Rettieva A. N., “Asimmetriya v kooperativnoi zadache upravleniya bioresursami”, Upravlenie bolshimi sistemami, 55 (2015), 280–325

[14] Mazalov V. V., Tokareva Yu. S., “Ravnovesie v zadache o sdelkakh s neravnomernym raspredeleniem rezervnykh tsen”, Matematicheskaya teoriya igr i ee prilozheniya, 3:2 (2011), 37–49 | MR

[15] Petrosyan L. A., Zakharov V. V., Matematicheskie modeli v ekologii, SPbGU, SPb., 1997

[16] Petrosyan L. A., Zenkevich N. A., “Printsipy ustoichivoi kooperatsii”, Matematicheskaya teoriya igr i ee prilozheniya, 1:1 (2009), 106–123

[17] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1969

[18] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona–Yakobi, Nauka, M., 1991 | MR

[19] Subbotin A. I., Tarasev A. M., “Sopryazhennye proizvodnye funktsii tseny differentsialnoi igry”, Doklady AN SSSR, 283:3 (1985), 559–564 | MR | Zbl

[20] Subbotina N. N., “Metod kharakteristik Koshi i obobschennye resheniya uravnenii Gamiltona–Yakobi–Bellmana”, Doklady AN SSSR, 320:3 (1991), 556–561 | Zbl

[21] Ushakov V. N., “K zadache postroeniya stabilnykh mostov v differentsialnoi igre sblizheniya–ukloneniya”, Izvestiya AN SSSR. Tekhnicheskaya kibernetika, 1980, no. 4, 29–36 | MR

[22] Ushakov V. N., Uspenskii A. A., Lebedev P. D., “Metody negladkogo analiza pri postroenii approksimatsii reshenii differentsialnykh igr i zadach upravleniya”, Vestnik SPbGU. Seriya «Matematika, mekhanika, astronomiya». Ser. 10, 2013, no. 3, 157–167 | MR

[23] Aubin J.-P., “A Survey of Viability Theory”, SIAM Journal on Control and Optimization, 28:4 (1990), 749–788 | DOI | MR | Zbl

[24] Basar T., Olsder G. J., Dynamic Noncooperative Game Theory, Academic Press, London, 1982 | MR | Zbl

[25] Crandall M. G., Lions P.-L., “Viscosity Solutions of Hamilton–Jacobi Equations”, Transactions of the American Mathematical Society, 277:1 (1983), 1–42 | DOI | MR | Zbl

[26] Friedman D., “Evolutionary Games in Economics”, Econometrica, 59:3 (1991), 637–666 | DOI | MR | Zbl

[27] Hofbauer J., Sigmund K., The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988 | MR | Zbl

[28] Intriligator M., Mathematical Optimization and Economic Theory, Prentice-Hall, New York, 1971 | MR

[29] Krasovskii A. N., Krasovskii N. N., Control Under Lack of Information, Birkhauser, Boston, 1995 | MR

[30] Tarasyev A. M., A Differential Model for a $2 \times 2$ Evolutionary Game Dynamics, IIASA Working Paper, WP-94–63, Laxenburg, Austria, 1994

[31] CNN Money, http://money.cnn.com/

[32] Grafiki dokhodnosti obligatsii, Rynok Foreks, http://www.fxstreet.ru.com/charts/bond-yield/