Two-level cooperation in technological alliance differential games

N. V. Kolabutin

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2015), pp. 42-63

Résumé

The paper is devoted to two-level cooperation in differential games. Cooperative differential games are currently one of most important parts of game theory. They mathematically describe the conflict-controlled processes in management and economics. The solution of a differential game is a cooperative agreement, and the selected principle of optimality, according to which the received payoff is distributed. Studies showed that initially selected cooperative solution often loses its optimality over time. Therefore, the question arose about the stability of the co-operative solutions. The issue of dynamic stability or time consistency is considered. This concept was formalized by L. A. Petrosyan. Cooperative solution is dynamically stable if the principle of optimality selected early in the game stays consistent throughout the gameplay. For dynamic stability it is necessary at each moment of time to carry out the regularization of the chosen principle of optimality. For this regularization L. A. Petrosyan proposed to use the redistribution of received payoff in accordance with the “imputation distribution procedure”. In some cases coalitional solutions in differential games are studied, where the coalitions act as individual players. Coalitions play with each other in a noncooperative game, and payoff of each coalition is distributed among its members in accordance with some principle of optimality. In this paper we began to investigate the model in which participants form coalitions, acting as individual players, but the coalitions may also cooperate to increase joint payoff. In this case, the coalition play their cooperative game, maximizing overall benefits and distributing them among themselves according to some principle of optimality. Then, payoff of each coalition is distributed among its members as well according to maybe a different principle of optimality. Such cooperation is called two-level cooperation. Principles of profit distribution between coalitions and within coalition may be different. To solve such models which is required at both levels of the cooperation it is necessary to build the characteristic function and imputation distribution procedure. This paper describes a model of a two-level cooperation on the example of a technological alliance differential game. Participants in the game are the firms with the technology that brings profit. On the first (lower) level firms form coalitions to increase joint profit. On the second (upper) level coalitions act as individual players and also form the one grand coalition to increase the joint profit. The resulting top-level payoff is distributed between coalitions-participants. Thus, each coalition party may get more than it would receive by playing individually. Then each coalition distributes the resulting share of payoff among its member firms. This article also presented a stable cooperative solution in this model. For its implementation at every level of cooperation we build the characteristic function and prove its superadditivity. As a sharing the dynamic Shapley value is selected. The results are illustrated by a quantitative example. Bibliogr. 8. Tables 2.

Détail

Keywords: differential game, cooperation, imputation distribution procedure.

Bibliographie

[1] Kostyunin S. Y., Shevkoplyas E. V., “On the simplification of the integral gain in differential games with random duration”, Vestn. of St. Petersburg University. Serie 10: Applied mathematics, computer science, control processes, 2011, no. 4, 47–56

[2] Yeung D. W. K., Petrosyan L. A., Cooperative Stochastic Differential Games, Springer, New York, 2006, 253 pp.

[3] Petrosyan L. A., Kozlovskaya N. V., Ilyina A. V., “Coalition solution for the problem of reducing emissions”, Vestn. of St. Petersburg University. Serie 10: Applied mathematics, computer science, control processes, 2010, no. 2, 46–59

[4] Petrosyan L. A., Sedakov A. A., Syurin A. N., “Multistage games with coalition structure”, Vestn. of St. Petersburg University. Serie 10: Applied mathematics, computer science, control processes, 2006, no. 4, 97–110

[5] Denisov A. M., Razgulin A. V., The ordinary differential equations, Handbook for 2nd year students, Publisher Moscow University, M., 2008, 70 pp.

[6] Petrosyan L. A., Zaccour G., “Time-consistent Shapley value allocation of pollution cost reduction”, Journal of economic dynamics and control, 27:3 (2003), 381–398 | DOI

[7] Petrosyan L. A., “Stability of solutions of differential games with many participants”, Vestn. of St. Leningrad University, 1977, no. 19, 46–52

[8] Zenkevich N. A., Petrosyan L. A., “The problem of time consistency of cooperative solutions in management”, Vestn. of St. Petersburg University. Serie 8: Management, 2007, no. 1, 7–42