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

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

Zeyang Wang and  Ovanes Petrosian ,

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  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.

Keywords: Non-transferable, Cooperative games, Differential games, Continuous updating, Subgame consistency.
Mathematics Subject Classification: Primary:91A12, 91A23;Secondary:91A80.
Citation: Zeyang Wang, Ovanes Petrosian. On class of non-transferable utility cooperative differential games with continuous updating. Journal of Dynamics & 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.   Google Scholar
[2] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.   Google Scholar
[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.  Google Scholar
[4]

A. Filippov, Introduction to the Theory of Differential Equations, Editorial URSS, Moscow, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. F. Kleimenov, Nonantagonistic Positional Differential Games, Nauka Ural'skoe Otdelenie, Ekaterinburg, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. A. Petrosyan and N. V. Murzov, Game-theoretic problems in mechanics, Litovsk. Mat. Sb., 6 (1966), 423-433.   Google Scholar

[13]

L. S. Pontryagin, On the theory of differential games, Uspehi Mat. Nauk, 21 (1966), 219-274.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

O. Petrosian, Looking forward approach in cooperative differential games, International Game Theory Review, 18 (2016), 1-14.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. W. Yeung and L. A. Petrosyan, Subgame consistent cooperation, in Theory and Decision Library C, 47, Springer, Singapore, 2016.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Academic Press, Ltd., London, 1995.   Google Scholar
[2] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.   Google Scholar
[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.  Google Scholar
[4]

A. Filippov, Introduction to the Theory of Differential Equations, Editorial URSS, Moscow, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. F. Kleimenov, Nonantagonistic Positional Differential Games, Nauka Ural'skoe Otdelenie, Ekaterinburg, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. A. Petrosyan and N. V. Murzov, Game-theoretic problems in mechanics, Litovsk. Mat. Sb., 6 (1966), 423-433.   Google Scholar

[13]

L. S. Pontryagin, On the theory of differential games, Uspehi Mat. Nauk, 21 (1966), 219-274.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

O. Petrosian, Looking forward approach in cooperative differential games, International Game Theory Review, 18 (2016), 1-14.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. W. Yeung and L. A. Petrosyan, Subgame consistent cooperation, in Theory and Decision Library C, 47, Springer, Singapore, 2016.  Google Scholar

[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.  Google Scholar

Figure 1.  Pareto optimal trajectory with continuous updating (blue line), Pareto optimal trajectory in the initial game (red line)
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Figure 2.  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 $
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Figure 3.  Pareto optimal strategy of player $ i $ with continuous updating for different weights $ \alpha_i = (0.1,0.2,\dots,1) $
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Figure 4.  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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