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Trudy Inst. Mat. i Mekh. UrO RAN, 2014, Volume 20, Number 3, Pages 114–131 (Mi timm1089)  

This article is cited in 3 scientific papers (total in 3 papers)

Hamilton–Jacobi equations in evolutionary games

N. A. Krasovskiya, A. V. Kryazhimskiybc, A. M. Tarasyevdca

a Yeltsin Ural Federal University
b Steklov Mathematical Institute of Russian Academy of Sciences
c International Institute for Applied Systems Analysis
d Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Advanced methods of the theory of optimal control and generalized minimax solutions of Hamilton–Jacobi equations are applied to a nonzero sum game between two large groups of agents in the framework of economic and biological evolutionary models. Random contacts of agents from different groups happen according to a control dynamic process which can be interpreted as Kolmogorov's differential equations. Coefficients of equations are not fixed a priori and can be chosen as control parameters on the feedback principle. Payoffs of coalitions are determined by the limit functionals on infinite horizon. The notion of a dynamical Nash equilibrium is considered in the class of control feedbacks. A solution is proposed basing on feedbacks maximizing with the guarantee the own payoffs. Guaranteed feedbacks are constructed in the framework of the theory of generalized solutions of Hamilton–Jacobi equations. The analytical formulas are obtained for corresponding value functions. The equilibrium trajectory is generated and its properties are investigated. The considered approach provides new qualitative results for the equilibrium trajectory in evolutionary games.

Keywords: game theory, algorithms of equilibrium search.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-12446-офи-м2
13-01-00685-а
14-00-90408-Укр-а
14-01-00486-а
National Academy of Sciences of Ukraine 03-01-14
Russian Academy of Sciences - Federal Agency for Scientific Organizations 12-П-1-1002
12-П-1-1012
12-П-6-1038
12-С-7-1001
Ministry of Education and Science of the Russian Federation 02.А03.21.0006
International Institute for Applied Systems Analysis


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Document Type: Article
UDC: 517.977
Received: 27.02.2014

Citation: N. A. Krasovskiy, A. V. Kryazhimskiy, A. M. Tarasyev, “Hamilton–Jacobi equations in evolutionary games”, Trudy Inst. Mat. i Mekh. UrO RAN, 20, no. 3, 2014, 114–131

Citation in format AMSBIB
\Bibitem{KraKryTar14}
\by N.~A.~Krasovskiy, A.~V.~Kryazhimskiy, A.~M.~Tarasyev
\paper Hamilton--Jacobi equations in evolutionary games
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2014
\vol 20
\issue 3
\pages 114--131
\mathnet{http://mi.mathnet.ru/timm1089}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3364421}
\elib{http://elibrary.ru/item.asp?id=23503116}


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    This publication is cited in the following articles:
    1. A. V. Kryazhimskiy, A. M. Tarasyev, A. A. Usova, W. Wang, “Proportional economic growth under conditions of limited natural resources”, Proc. Steklov Inst. Math., 291 (2015), 127–145  mathnet  crossref  crossref  isi  elib
    2. Nikolay A. Krasovskii, Alexander M. Tarasyev, “Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals”, Autom. Remote Control, 79:6 (2018), 1148–1167  mathnet  crossref  isi
    3. N. A. Krasovskii, A. M. Tarasev, “Asimptoticheskoe povedenie reshenii v dinamicheskikh bimatrichnykh igrakh s diskontirovannymi indeksami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 193–209  mathnet  crossref  elib
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