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 Mat. Teor. Igr Pril., 2011, Volume 3, Issue 4, Pages 49–88 (Mi mgta69)

Decomposition algorithm of searching equilibria in the dynamical game

Nikolay A. Krasovskiy, Alexander M. Tarasyev

Institute of Mathematics and Mechanics, Ural Branch of the RAS

Abstract: A problem of noncooperative game with several players is considered, in which the players (governments of neighboring countries) make emission reduction trading. Particular attention is paid to the case of two players, one of whom is Eastern European countries, while another is countries of the former Soviet Union. A statistical analysis of the model parameters for quadratic cost functions and logarithmic benefit functions, based on the real data, is performed. The concepts of non-cooperative Nash equilibrium and cooperative Pareto maxima are introduced and linked with each other. The definition of a new concept – the market equilibrium, which combines properties of Nash and Pareto equilibria, is given. An analytic solution to the problem of finding market equilibrium is represented. This analytical solution can serve as a test for verification of numerical search algorithms. A computational algorithm of searching for market equilibrium is proposed, which shifts Nash competitive equilibrium to Pareto cooperative maximum. An algorithm is interpreted in the form of a repeated auction, in which the auctioneer has no information about cost functions and functions of environmental effect from emission reduction for the participating countries. An auctioneer strategy, which provides conditions for reaching market equilibrium, is considered. From the viewpoint of game theory, repeated auction describes the learning process in a noncooperative repeated game under uncertainty. The results of proposed computational algorithms are compared to analytical solutions. Numerical calculations of equilibrium and algorithm trajectories, converging to the equilibrium, are given.

Keywords: dynamic games, Nash equilibrium, Pareto maximum, equilibrium search algorithms, auctions modeling.

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English version:
Automation and Remote Control, 2015, 76:10, 1865–1893

UDC: 517.977
BBK: 22.1

Citation: Nikolay A. Krasovskiy, Alexander M. Tarasyev, “Decomposition algorithm of searching equilibria in the dynamical game”, Mat. Teor. Igr Pril., 3:4 (2011), 49–88; Autom. Remote Control, 76:10 (2015), 1865–1893

Citation in format AMSBIB
\Bibitem{KraTar11} \by Nikolay~A.~Krasovskiy, Alexander~M.~Tarasyev \paper Decomposition algorithm of searching equilibria in the dynamical game \jour Mat. Teor. Igr Pril. \yr 2011 \vol 3 \issue 4 \pages 49--88 \mathnet{http://mi.mathnet.ru/mgta69} \transl \jour Autom. Remote Control \yr 2015 \vol 76 \issue 10 \pages 1865--1893 \crossref{https://doi.org/10.1134/S0005117915100136} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. N. A. Krasovskii, A. V. Kryazhimskii, A. M. Tarasev, “Uravneniya Gamiltona–Yakobi v evolyutsionnykh igrakh”, Tr. IMM UrO RAN, 20, no. 3, 2014, 114–131
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
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