Decomposition algorithm of searching equilibria in the dynamical game

Nikolay A. Krasovskiy ; Alexander M. Tarasyev

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 3 (2011) no. 4, pp. 49-88.

Résumé

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.

Détail

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

Bibliographie

[1] Vorobev N. N., Teoriya igr dlya ekonomistov-kibernetikov, Nauka, M., 1985, 271 pp. | MR

[2] Germeier Yu. B., Igry s neprotivopolozhnymi interesami, Nauka, M., 1976, 328 pp. | MR

[3] Kleimenov A. F., Neantagonisticheskie pozitsionnye differentsialnye igry, Nauka, Ekaterinburg, 1993, 184 pp. | MR

[4] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR | Zbl

[5] Kryazhimskii A. V., Osipov Yu. S., “O differentsialno-evolyutsionnykh igrakh”, Tr. Mat. in-ta RAN, 211, 1995, 257–287 | MR | Zbl

[6] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 390 pp. | MR | Zbl

[7] Axelrod R., The evolution of cooperation, Basic Books, New York, 1984

[8] Barrett S., “International environmental agreements as games”, Conflicts and Cooperation in Managing Environmental Resources, ed. R. Pethig, Springer-Verlag, Berlin, 1990, 11–37

[9] Basar T., Olsder G. J., Dynamic noncooperative game theory, Academic Press, London, 1982, 519 pp. | MR | Zbl

[10] Chander P., Tulkens H., “Theoretical foundations of negotiations and cost sharing in transfrontier pollution problems”, European Economic Review, 36 (1992), 388–398 | DOI

[11] Ehtamo H., Hamalainen R. P., “A cooperative incentive equilibrium for a resource management problem”, J. Econom. Dynam. Control, 17 (1993), 659–678 | DOI | MR | Zbl

[12] Ellerman A. D., Decaux A., Analysis of post-Kyoto $CO2$ emissions trading using marginal abatement curves, Joint Program Report Series, No 40, Massachusetts Institute of Technology, Cambridge, 1998, 32 pp.

[13] Fudenberg D., Kreps D. M., “Learning mixed equilibria”, Games and Econ. Behavior, 5 (1993), 320–367 | DOI | MR | Zbl

[14] Hoel M., “Global environmental problems: the effect of unilateral actions taken by one country”, J. of Environmental Economics and Management, 20 (1991), 55–70 | DOI

[15] Hoffbauer J., Sigmund K., Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 2001, 323 pp. | MR

[16] Kaniovski Yu. M., Young H. P., “Learning dynamics in games with stochastic perturbations”, Games and Econ. Behavior, 11 (1995), 330–363 | DOI | MR | Zbl

[17] Krasovskii A. N., Krasovskii N. N., Control under lack of information, Birkhauser, Boston etc., 1994, 320 pp. | Zbl

[18] Kryazhimskii A. V., Tarasyev A. M., Equilibrium and guaranteeing solutions in evolutionary nonzero sum games, IIASA Interim Report IR-98-003, IIASA, Laxenburg, 1998, 56 pp.

[19] Mäler K. G., “The acid rain game”, Valuation Methods and Policy Making, Environmental Economics, eds. H. Folmer, E. van Ireland, Elsevier, Amsterdam, 1989, 231–252

[20] Nentjes A., “An economic model of transfrontier pollution abatement”, Public finance, trade and development, ed. V. Tanzi, Wayne State University Press, Detroit, Mich., 1993, 243–261

[21] Nentjes A., “Control of reciprocal transboundary pollution and joint implementation”, Economic Instruments for Air Pollution Control, eds. G. Klaassen, F. Foersund, Kluwer, 1994, 209–230 | DOI

[22] Nowak M., Sigmund K., “The evolution of stochastic strategies in the Prisoner's Dilemma”, Acta Applic. Math., 20 (1992), 247–265 | DOI | MR

[23] Smale S., “The prisoner's dilemma and dynamical systems associated to non-cooperative games”, Econometrica, 48:7 (1980), 1617–1634 | DOI | MR | Zbl

[24] Tol R., The benefits of greenhouse gas emission reduction: an application of FUND, Working Paper FNU-64, Hamburg University, Hamburg, 2005, 33 pp.

[25] Vasin V. V., Ageev A. L., Ill-posed problems with a priori information, Inverse and Ill-Posed Probl. Ser., VSP, Utrecht, 1995, 255 pp. | MR | Zbl

[26] Zangwill W. I., Garcia C. B., Pathways to solutions, fixed points, and equilibria, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1981 | Zbl