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Game theoretic formalization of the concept of sustainable development in the hierarchical control systems. (English) Zbl 1301.91036
Summary: The resolution of numerous ecological problems on different levels must be implemented on the base of sustainable development concept that determines the conditions to the state of ecological-economic systems and impacting control actions. Those conditions can’t be realized by themselves and require special collaborative efforts of different agents using both cooperation and hierarchical control. To formalize the inevitable trade-offs it is natural to use game theoretic models. Unfortunately, the main optimality principles of hierarchical control (compulsion, impulsion) are not time consistent and therefore can’t be recommended as the direct base for collective solutions. The most prospective is the conviction method which is formalized as a transition from hierarchy to cooperation and allows a regularization that provides the time consistency. However, in current social conditions other methods of hierarchical control also keep their actuality. To ensure the time consistency of those optimality principles it is necessary to build cooperative differential games on their base. An example of the approach is considered in this paper.

##### MSC:
 91B76 Environmental economics (natural resource models, harvesting, pollution, etc.) 91A40 Other game-theoretic models 91A23 Differential games (aspects of game theory) 91A12 Cooperative games
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##### References:
 [1] Basar, T., & Olsder, G. J. (1995). Dynamic noncooperative game theory. San Diego: Academic Press. · Zbl 0828.90142 [2] Denin, K. I.; Ugolnitsky, G. A., Game theoretic model of corruption in the systems of hierarchical control, Izvestia RAS: Teoria i Systemy Upravlenia (Journal of Computer and System Sciences International), 1, 192-198, (2010) [3] Dockner, E. J.; Jorgensen, S., Cooperative and non-cooperative differential game solutions to an investment and pricing problem, The Journal of the Operational Research Society, 35, 731-739, (1984) · Zbl 0541.90010 [4] Dockner, E. J.; Long, N. V., International pollution control: cooperative versus non-cooperative strategies, Journal of Environmental Economics and Management, 24, 13-29, (1993) · Zbl 0775.90309 [5] French, J. P. R.; Raven, B.; Cartwright, D. (ed.); Zander, A. (ed.), The bases of social power, 607-623, (1960), New York [6] Fudenberg, D., & Tirole, J. (1991). Game theory. Cambridge: MIT Press. · Zbl 1339.91001 [7] Gao, L., Jakubowski, A., Klompstra, M. B., & Olsder, G. J. (1989). Time-dependent cooperation in games. Berlin: Springer. · Zbl 0677.90101 [8] Hamalainen, R. P.; Haurie, A.; Kaitala, V., Equilibria and threats in a fishery management game, Optimal Control Applications & Methods, 6, 315-333, (1986) · Zbl 0631.90017 [9] Haurie, A., A note on nonzero-sum differential games with bargaining solutions, Journal of Optimization Theory and Applications, 18, 31-39, (1976) · Zbl 0298.90075 [10] Haurie, A. (1991). Piecewise deterministic and piecewise diffusion differential games with model uncertainties. Berlin: Springer. · Zbl 0778.90100 [11] Haurie, A.; Pohjola, M., Efficient equilibria in a differential game of capitalism, Journal of Economic Dynamics & Control, 11, 65-78, (1987) · Zbl 0637.90109 [12] Haurie, A.; Zaccour, G., A differential game model of power exchange between interconnected utilizers, Athens, Greece [13] Haurie, A., & Zaccour, G. (1991). A game programming approach to efficient management of interconnected power networks. Berlin: Springer. · Zbl 0785.90106 [14] Haurie, A.; Krawczyk, J. B.; Roche, M., Monitoring cooperative equilibria in a stochastic differential game, Journal of Optimization Theory and Applications, 81, 73-95, (1994) · Zbl 0830.90143 [15] Jorgensen, S.; Zaccour, G., Time consistent side payments in a dynamic game of downstream pollution, Journal of Economic Dynamics & Control, 25, 1973-1987, (2001) · Zbl 0978.91018 [16] Kaitala, V.; Pohjola, M., Optimal recovery of a shared resource stock: a differential game with efficient memory equilibria, Natural Resource Modeling, 3, 91-118, (1988) · Zbl 0850.90163 [17] Kaitala, V.; Pohjola, M., Sustainable international agreements on greenhouse warming: a game theory study, Annals of the International Society of Dynamic Games, 2, 67-87, (1995) · Zbl 0836.90044 [18] Kaitala, V.; Maler, K. G.; Tulkens, H., The acid rain game as a resource allocation process with an application to the international cooperation among Finland, Russia, and Estonia, The Scandinavian Journal of Economics, 97, 325-343, (1995) · Zbl 0915.90078 [19] Kidland, F. E.; Prescott, E. C., Rules rather than decisions: the inconsistency of optimal plans, Journal of Political Economy, 85, 473-490, (1977) [20] Ledyaev, V. G. Power: a conceptual analysis. Nova Science Publishers, New York (1998). [21] Leitmann, G. (1974). Cooperative and non-cooperative many players differential games. New York: Springer. · Zbl 0358.90085 [22] Leitmann, G. (1975) Cooperative and non-cooperative differential games. Amsterdam: Reidel. · Zbl 0317.90069 [23] Maler, K. G.; Zeeuw, A. D., The acid rain differential game, Environmental & Resource Economics, 12, 167-184, (1998) [24] Mazalov, V., & Rettieva, A. (2004). A fishery game model with migration: reserved territory approach. Game Theory and Applications, 10. · Zbl 1077.91012 [25] Mazalov, V., & Rettieva, A. (2005). Dynamic game methods for reserved territory optimization. Survey in Applied and Industrial Math., $$3$$(12) (in Russian). · Zbl 1086.91010 [26] Mazalov, V., & Rettieva, A. (2006). Nash equilibrium in bioresource management problem. Mathematical Modeling, $$5$$(18) (in Russian). · Zbl 1099.91077 [27] Mazalov, V., & Rettieva, A. (2007). Cooperative incentive equilibrium. In L. A. Petrosjan & N. A. Zenkevich (Eds.), Contributions to game theory and management. Collected papers presented on the international conference game theory and management. SPb, Graduate School of Management SPbU (pp. 316-325). · Zbl 0978.91018 [28] Ougolnitsky, G. A.; Petrosjan, L. A. (ed.); Mazalov, V. V. (ed.), Game theoretic modeling of the hierarchical control of sustainable development, No. 8, 82-91, (2002), New York [29] Ougolnitsky, G. A. (2009). A generalized model of hierarchically controlled dynamical system. In L. A. Petrosjan & N. A. Zenkevich (Eds.), Contributions to game theory and management. Vol. II. Collected papers presented on the second international conference game theory and management. SPb, Graduate School of Management SPbU (pp. 320-333). · Zbl 0298.90075 [30] Ougolnitsky, G. A.; Usov, A. B.; Cracknell, A. P. (ed.); Krapivin, V. P. (ed.); Varotsos, C. A. (ed.), Problems of the sustainable development of ecological-economic systems, 427-444, (2009), Berlin [31] Petrosyan, L. A., The stability of solutions in differential games with many players, Vestnik Leningradskogo Universiteta, 1, 19, 46-52, (1977) [32] Petrosyan, L. A.; Hamalainen, R. P. (ed.); Ehtamo, H. K. (ed.), The time-consistency of the optimality principles in non-zero sum differential games, 299-311, (1991), Berlin · Zbl 0770.90094 [33] Petrosyan, L. A., Strongly time-consistent differential optimality principles, Vestnik Saint Petersburg University Mathematics, 26, 40-46, (1993) · Zbl 0838.90150 [34] Petrosyan, L. A. (1993a). The time consistency in differential games with a discount factor. Game Theory and Applications, 1. [35] Petrosyan, L. A., Agreeable solutions in differential games, International Journal of Mathematics, Game Theory, and Algebra, 7, 165-177, (1997) · Zbl 0905.90194 [36] Petrosyan, L. A.; Danilov, N. N., Stability of solutions in non-zero sum differential games with transferable payoffs, Vestnik of Leningrad University, 1, 52-59, (1979) · Zbl 0419.90095 [37] Petrosyan, L. A., & Danilov, N. N. (1985). Cooperative differential games and their applications. Tomsk: Tomsk State University (in Russian). · Zbl 0585.90096 [38] Petrosyan, L. A., & Zenkevich, N. A. (1996). Game theory. Singapore: World Scientific. · Zbl 0863.90145 [39] Petrosyan, L. A., & Zenkevich, N. A. (2007). Time consistency of cooperative solutions. In L. A. Petrosjan & N. A. Zenkevich (Eds.), Contributions to game theory and management. Collected papers presented on the international conference game theory and management. SPb, Graduate School of Management SBbU (pp. 413-440). · Zbl 0541.90010 [40] Petrosyan, L. A., & Zenkevich, N. A. (2009). Conditions for sustainable cooperation. In L. A. Petrosyan & N. A. Zenkevich (Eds.), Contributions to game theory and management. Vol. II. Collected papers presented on the second international conference game theory and management. SPb, Graduate School of Management SPbU (pp. 344-354). [41] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., & Mishchenko, E. F. (1962). The mathematical theory of optimal processes. New York: Wiley. [42] Rubio, S. J.; Casino, B., A note on cooperative versus non-cooperative strategies in international pollution control, Resource and Energy Economics, 24, 251-261, (2002) [43] Rybasov, E. A.; Ugolnitsky, G. A.; Ugolnitsky, G. A. (ed.), Mathematical modeling of the hierarchical control of ecological-economic systems considering corruption, 46-65, (2004), Moscow [44] Sorokin, P. (1957). Social and cultural dynamics. A study of change in major systems of art, truth, ethics, law and social relationships. Boston: Porter Sargent Publisher. [45] Tahvonen, O., Carbon dioxide abatement as a differential game, European Journal of Political Economy, 10, 685-705, (1994) [46] Tolwinsky, B.; Haurie, A.; Leitmann, G., Cooperative equilibria in differential games, Journal of Mathematical Analysis and Applications, 119, 182-202, (1986) · Zbl 0607.90097 [47] Ugolnitsky, G. A. (1999). Control of ecological-economic systems. Moscow: Vuzovskaya Kniga (in Russian). [48] Ugolnitsky, G. A., Game theoretic investigation of some methods of hierarchical control, Izvestia RAS: Teoria i Systemy Upravlenia (Journal of Computer and System Sciences International), 1, 97-101, (2002) [49] Ugolnitsky, G. A., Hierarchical control of the sustainable development of social organizations, Obshchestvennye Nauki i Sovremennost’ (Social Sciences and Modernity), 3, 133-140, (2002) [50] Ugolnitsky, G. A., Game theoretic principles of optimality of the hierarchical control of sustainable development, Izvestia RAS: Teoria i Systemy Upravlenia (Journal of Computer and System Sciences International), 4, 72-78, (2005) [51] Ugolnitsky, G. A.; Usov, A. B., Mathematical formalization of the methods of hierarchical control of ecological-economic systems, Problemy Upravleniya (Control Problems), 4, 64-69, (2007) [52] Ugolnitsky, G. A.; Usov, A. B., Hierarchical systems of diamond-shaped structure for river water quality control, Upravlenie Bol’shimi Systemami (Control of Big Systems), 19, 187-203, (2007) [53] Ugolnitsky, G. A.; Usov, A. B., An information-analytical system of control of the ecological-economic objects, Izvestia RAS: Teoria i Systemy Upravlenia (Journal of Computer and System Sciences International), 2, 168-176, (2008) [54] Ugolnitsky, G. A.; Usov, A. B., Control of the complex ecological-economic systems, Avtomatika i Telemechanika (Automation and Remote Control), 5, 169-179, (2009) [55] Uugolnitsky, G. A.; Usov, A. B., Three level Fan-like systems of control of the ecological-economic objects, Problemy Upravleniya (Control Problems), 1, 26-32, (2010) [56] United Nations, (1987). Report of the world commission on environment and development. General Assembly Resolution 42/187, 11 December. · Zbl 0830.90143 [57] Yeung, D. W. K. (2009). Cooperative game-theoretic mechanism design for optimal resource use. In L. A. Petrosyan & N. A. Zenkevich (Eds.), Contributions to game theory and management. Vol. II. Collected papers presented on the second international conference game theory and management. SPb, Graduate School of Management SPbU (pp. 483-513). · Zbl 0836.90044 [58] Yeung, D. W. K., & Petrosyan, L. A. (2001). Proportional time-consistent solution in differential games. Saint Petersburg: Saint Petersburg State University.
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