# American Institute of Mathematical Sciences

October  2020, 7(4): 291-302. doi: 10.3934/jdg.2020020

## On class of non-transferable utility cooperative differential games with continuous updating

 St.Petersburg State University, 7/9, Universitetskaya nab., Saint-Petersburg 199034, Russia, College of Mathematics and Computer Science, Yan'an University, 580 Shengdi Road, Yan'an, CHINA, P.C: 716099

* Corresponding author: Ovanes Petrosian

Received  April 2020 Revised  May 2020 Published  October 2020 Early access  July 2020

Fund Project: Research of the second author is supported by a grant from the Russian Science Foundation (Project No 18-71-00081)

This paper considers and describes the class of cooperative differential games with the non-transferable utility and continuous updating. It is the first detailed paper about the application of continuous updating approach to the non-transferable utility differential games. The process of how to construct Pareto optimal strategy with continuous updating and Pareto trajectory is described. Another important contribution is that the property of subgame consistency is adopted for the class of games with continuous updating. The resource extraction game model is used as an example. The Pareto optimal strategies and corresponding trajectory are constructed, and the set of Pareto optimal strategies satisfying the subgame consistency property is presented. The results of numerical simulation are demonstrated in the Matlab environment, and the conclusion is drawn.

Citation: Zeyang Wang, Ovanes Petrosian. On class of non-transferable utility cooperative differential games with continuous updating. Journal of Dynamics and Games, 2020, 7 (4) : 291-302. doi: 10.3934/jdg.2020020
##### References:
 [1] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Academic Press, Ltd., London, 1995. [2] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957. [3] E. J. Dockner, S. Jorgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511805127. [4] A. Filippov, Introduction to the Theory of Differential Equations, Editorial URSS, Moscow, 2004. [5] H. Gao, L. Petrosian, H. Qian and and A. Sedakov, Cooperation in two-stage games on undirected networks, J. Syst. Sci. Complex., 30 (2017), 680-693.  doi: 10.1007/s11424-016-5164-7. [6] A. Haurie, A note on nonzero-sum differential games with bargaining solution, J. Optim. Theory Appl., 18 (1976), 31-39.  doi: 10.1007/BF00933792. [7] A. F. Kleimenov, Nonantagonistic Positional Differential Games, Nauka Ural'skoe Otdelenie, Ekaterinburg, 1993. [8] I. Kuchkarov and O. Petrosian, On class of linear quadratic non-cooperative differential games with continuous updating, Lecture Notes in Computer Science, 11548 (2019), 635-650.  doi: 10.1007/978-3-030-22629-9_45. [9] W. H. Kwon, A. M. Bruckstein and T. Kailath, Stabilizing state-feedback design via the moving horizon method, Internat. J. Control, 37 (1983), 631-643.  doi: 10.1080/00207178308932998. [10] W. H. Kwon and A. E. Pearson, A modified quadratic cost problem and feedback stabilization of a linear system, IEEE Trans. Automat. Control, 22 (1977), 838-842.  doi: 10.1109/tac.1977.1101619. [11] D. Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 814-824.  doi: 10.1109/9.57020. [12] L. A. Petrosyan and N. V. Murzov, Game-theoretic problems in mechanics, Litovsk. Mat. Sb., 6 (1966), 423-433. [13] L. S. Pontryagin, On the theory of differential games, Uspehi Mat. Nauk, 21 (1966), 219-274. [14] L. Petrosyan, Time-consistency of solutions in multi-player differential games, Vestnik of Leningrad State University. Series 1. Mathematics. Mechanics. Astronomy, 4 (1977), 46-52. [15] L. A. Petrosyan and D. W. K. Yeung, A time-consistent solution formula for bargaining problem in differential games, Int. Game Theory Rev., 16 (2014), 11 pp. doi: 10.1142/S0219198914500169. [16] O. Petrosian and I. Kuchkarov, About the looking forward approach in cooperative differential games with transferable utility, in Frontiers of Dynamic Games, Static Dyn. Game Theory Found. Appl., Birkhäuser/Springer, Cham, 2019, 175–208. [17] O. Petrosian, L. Shi, Y. Li, and H. Gao, Moving information horizon approach for dynamic game models, Mathematics, 7 (2019), 1239. doi: 10.3390/math7121239. [18] O. Petrosian, Looking forward approach in cooperative differential games, International Game Theory Review, 18 (2016), 1-14. [19] O. Petrosian and A. Barabanov, Looking forward approach in cooperative differential games with uncertain-stochastic dynamics, J. Optim. Theory Appl., 172 (2017), 328-347.  doi: 10.1007/s10957-016-1009-8. [20] O. Petrosian, M. Nastych and D. Volf, Non-cooperative differential game model of oil market with looking forward approach, in Frontiers of Dynamic Games, Static Dyn. Game Theory Found. Appl., Birkhäuser/Springer, Cham, 2018, 189–202. [21] O. Petrosian and A. Tur, Hamilton-Jacobi-Bellman equations for non-cooperative differential games with continuous updating, in Mathematical Optimization Theory and Operations Research, Communications in Computer and Information Science, 1090, Springer, Cham, 2019, 178–191. doi: 10.1007/978-3-030-33394-2_14. [22] E. V. Shevkoplyas, Optimal solutions in differential games with random duration, J. Math. Sci. (N.Y.), 199 (2014), 715-722.  doi: 10.1007/s10958-014-1897-9. [23] D. W. K. Yeung and L. A. Petrosyan, Subgame consistent solutions of a cooperative stochastic differential game with nontransferable payoffs, J. Optim. Theory Appl., 124 (2005), 701-724.  doi: 10.1007/s10957-004-1181-0. [24] D. W. Yeung and L. A. Petrosyan, Subgame consistent cooperation, in Theory and Decision Library C, 47, Springer, Singapore, 2016. [25] D. W. K. Yeung and L. A. Petrosian, Cooperative stochastic differential games, in Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.

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##### References:
 [1] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Academic Press, Ltd., London, 1995. [2] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957. [3] E. J. Dockner, S. Jorgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511805127. [4] A. Filippov, Introduction to the Theory of Differential Equations, Editorial URSS, Moscow, 2004. [5] H. Gao, L. Petrosian, H. Qian and and A. Sedakov, Cooperation in two-stage games on undirected networks, J. Syst. Sci. Complex., 30 (2017), 680-693.  doi: 10.1007/s11424-016-5164-7. [6] A. Haurie, A note on nonzero-sum differential games with bargaining solution, J. Optim. Theory Appl., 18 (1976), 31-39.  doi: 10.1007/BF00933792. [7] A. F. Kleimenov, Nonantagonistic Positional Differential Games, Nauka Ural'skoe Otdelenie, Ekaterinburg, 1993. [8] I. Kuchkarov and O. Petrosian, On class of linear quadratic non-cooperative differential games with continuous updating, Lecture Notes in Computer Science, 11548 (2019), 635-650.  doi: 10.1007/978-3-030-22629-9_45. [9] W. H. Kwon, A. M. Bruckstein and T. Kailath, Stabilizing state-feedback design via the moving horizon method, Internat. J. Control, 37 (1983), 631-643.  doi: 10.1080/00207178308932998. [10] W. H. Kwon and A. E. Pearson, A modified quadratic cost problem and feedback stabilization of a linear system, IEEE Trans. Automat. Control, 22 (1977), 838-842.  doi: 10.1109/tac.1977.1101619. [11] D. Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 814-824.  doi: 10.1109/9.57020. [12] L. A. Petrosyan and N. V. Murzov, Game-theoretic problems in mechanics, Litovsk. Mat. Sb., 6 (1966), 423-433. [13] L. S. Pontryagin, On the theory of differential games, Uspehi Mat. Nauk, 21 (1966), 219-274. [14] L. Petrosyan, Time-consistency of solutions in multi-player differential games, Vestnik of Leningrad State University. Series 1. Mathematics. Mechanics. Astronomy, 4 (1977), 46-52. [15] L. A. Petrosyan and D. W. K. Yeung, A time-consistent solution formula for bargaining problem in differential games, Int. Game Theory Rev., 16 (2014), 11 pp. doi: 10.1142/S0219198914500169. [16] O. Petrosian and I. Kuchkarov, About the looking forward approach in cooperative differential games with transferable utility, in Frontiers of Dynamic Games, Static Dyn. Game Theory Found. Appl., Birkhäuser/Springer, Cham, 2019, 175–208. [17] O. Petrosian, L. Shi, Y. Li, and H. Gao, Moving information horizon approach for dynamic game models, Mathematics, 7 (2019), 1239. doi: 10.3390/math7121239. [18] O. Petrosian, Looking forward approach in cooperative differential games, International Game Theory Review, 18 (2016), 1-14. [19] O. Petrosian and A. Barabanov, Looking forward approach in cooperative differential games with uncertain-stochastic dynamics, J. Optim. Theory Appl., 172 (2017), 328-347.  doi: 10.1007/s10957-016-1009-8. [20] O. Petrosian, M. Nastych and D. Volf, Non-cooperative differential game model of oil market with looking forward approach, in Frontiers of Dynamic Games, Static Dyn. Game Theory Found. Appl., Birkhäuser/Springer, Cham, 2018, 189–202. [21] O. Petrosian and A. Tur, Hamilton-Jacobi-Bellman equations for non-cooperative differential games with continuous updating, in Mathematical Optimization Theory and Operations Research, Communications in Computer and Information Science, 1090, Springer, Cham, 2019, 178–191. doi: 10.1007/978-3-030-33394-2_14. [22] E. V. Shevkoplyas, Optimal solutions in differential games with random duration, J. Math. Sci. (N.Y.), 199 (2014), 715-722.  doi: 10.1007/s10958-014-1897-9. [23] D. W. K. Yeung and L. A. Petrosyan, Subgame consistent solutions of a cooperative stochastic differential game with nontransferable payoffs, J. Optim. Theory Appl., 124 (2005), 701-724.  doi: 10.1007/s10957-004-1181-0. [24] D. W. Yeung and L. A. Petrosyan, Subgame consistent cooperation, in Theory and Decision Library C, 47, Springer, Singapore, 2016. [25] D. W. K. Yeung and L. A. Petrosian, Cooperative stochastic differential games, in Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.
Pareto optimal trajectory with continuous updating (blue line), Pareto optimal trajectory in the initial game (red line)
Pareto optimal strategies of players $1$ and $2$ in the initial game and in the game model with continuous updating for $\alpha_1 = 0.664$, $\alpha_2 = 0.336$
Pareto optimal strategy of player $i$ with continuous updating for different weights $\alpha_i = (0.1,0.2,\dots,1)$
Payoff function (26) of player $i$ corresponding to Pareto optimal strategy profile (blue lines), payoff function (27) of player $i$ corresponding to Nash equilibrium (red lines) with continuous updating
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