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April  2022, 9(2): 191-202. doi: 10.3934/jdg.2022003

## Stability of international pollution control games: A potential game approach

 1 Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand 2 Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand 3 Departments of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 4 Department of Mathematical Sciences, Bayero University, Kano, 700231, Nigeria 5 Mathematics Department, CINVESTAV-IPN, A. Postal 14-740, Mexico City 07000, Mexico

* Corresponding author

Received  May 2021 Revised  December 2021 Published  April 2022 Early access  February 2022

In this paper, stabilization problems for n-player noncooperative differential games of international pollution control (IPC) are analysed via the concept of the potential differential game (PDG) introduced by Fonseca-Morales and Hernández-Lerma (2018). By first identifying a game of IPC as a PDG, an associated optimal control problem (OCP) is obtained, whose optimal solution is a Nash equilibrium (NE) for the game of IPC. Thus, the problem of finding conditions for which the NE stabilizes the game of IPC reduces to finding conditions for which the optimal solution stabilizes the associated OCP. The concept not only yields mild conditions for saddle point stability analysed in the literature but also for the overtaking optimality of the NE of the game of IPC.

Citation: Jewaidu Rilwan, Poom Kumam, Onésimo Hernández-Lerma. Stability of international pollution control games: A potential game approach. Journal of Dynamics and Games, 2022, 9 (2) : 191-202. doi: 10.3934/jdg.2022003
##### References:
 [1] S. Barrett, Self-enforcing international environmental agreements, Oxford Economic Papers, 46 (1994), 878-894.  doi: 10.1093/oep/46.Supplement_1.878. [2] A. Bressan, From optimal control to non-cooperative differential games: A homotopy approach, Control and Cybernetics, 38 (2009), 1081-1106. [3] W. A. Brock and J. A. Scheinkman, Global asymptotic stability of optimal control systems with applications to the theory of economic growth, The Hamiltonian Approach to Dynamic Economics. J. Econom. Theory, 12 (1976), 164-190.  doi: 10.1016/0022-0531(76)90031-4. [4] F. Camilli, L. Grüne and F. Wirth, Control Lyapunov functions and Zubov's method, SIAM Journal on Control and Optimization, 47 (2008), 301-326.  doi: 10.1137/06065129X. [5] C. Carraro and D. Siniscalco, Strategies for the international protection of the environment, Journal of Public Economics, 52 (1993), 309-328.  doi: 10.1016/0047-2727(93)90037-T. [6] D. Cass and K. Shell, The structure and stability of competitive dynamical systems, The Hamiltonian Approach to Dynamic Economics. J. Econom. Theory, 12 (1976), 31-70.  doi: 10.1016/0022-0531(76)90027-2. [7] C. D'Aspremont, A. Jacquemin, J. J. Gabszewicz and J. A. Weymark, On the stability of collusive price leadership, Canadian Journal of Economics, 16 (1983), 17-25.  doi: 10.2307/134972. [8] E. J. Dockner and N. V. Long, International pollution control: Cooperative versus noncooperative strategies, Journal of Environmental Economics and Management, 25 (1993), 13–29. (WU Vienna University of Economics, 1991, 9), https://ideas.repec.org/p/wiw/wus005/6275.html. doi: 10.1006/jeem.1993.1023. [9] E. J. Dockner and N. V. Long, International pollution control: Cooperative versus noncooperative strategies, Journal of Environmental Economics and Management, 24 (1993), 13-29.  doi: 10.1006/jeem.1993.1023. [10] A. V. Dmitruk and N. V. Kuz'kina, Existence theorem in the optimal control problem on an infinite time interval, Mathematical Notes, 78 (2005), 466-480.  doi: 10.1007/s11006-005-0147-3. [11] C. D. Feinstein and D. G. Luenberger, Analysis of the asymptotic behavior of optimal control trajectories: The implicit programming problem, SIAM Journal on Control and Optimization, 19 (1981), 561-585.  doi: 10.1137/0319035. [12] M. Finus and B. Rundshagen, Renegotiation-proof equilibria in a global emission game when players are impatient, Environmental and Resource Economics, 12 (1998), 275-306.  doi: 10.1023/A:1008211729093. [13] A. Fonseca-Morales and O. Hernández-Lerma, A note on differential games with Pareto-optimal Nash equilibria: Deterministic and stochastic models, Journal of Dynamics and Games, 4 (2017), 195-203.  doi: 10.3934/jdg.2017012. [14] A. Fonseca-Morales and O. Hernández-Lerma, Potential differential games, Dynamic Games and Applications, 8 (2018), 254-279.  doi: 10.1007/s13235-017-0218-6. [15] R. A. Freeman and J. A. Primbs, Control Lyapunov functions: New ideas from an old source, Proceedings of 35th IEEE Conference on Decision and Control, 4 (1996), 3926-3931.  doi: 10.1109/CDC.1996.577294. [16] V. Gaitsgory, L. Grüne and N. Thatcher, Stabilization with discounted optimal control, Systems and Control Letters, 82 (2015), 91-98.  doi: 10.1016/j.sysconle.2015.05.010. [17] D. Gale, On optimal development in a multi-sector economy, The Review of Economic Studies, 34 (1967), 1-18.  doi: 10.2307/2296567. [18] A. Haurie, Existence and global asymptotic stability of optimal trajectories for a class of infinite-horizon, nonconvex systems, Journal of Optimization Theory and Applications, 31 (1980), 515-533.  doi: 10.1007/BF00934475. [19] E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, 1967. [20] N. V. Long, A Survey of Dynamic Games in Economics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/9789814293044. [21] M. Magill, Some new results on the local stability of the process of capital accumulation, Journal of Economic Theory, 15 (1977), 174-210.  doi: 10.1016/0022-0531(77)90075-8. [22] M. J. P. Magill and J. A. Scheinkman, Stability of regular equilibria and the correspondence principle for symmetric variational problems, International Economic Review, 20 (1979), 297-315.  doi: 10.2307/2526479. [23] S. Miricǎ, Reducing a differential game to a pair of optimal control problems, Differential Equations, Chaos and Variational Problems, (2008), 269–283. doi: 10.1007/978-3-7643-8482-1_22. [24] R. T. Rockafellar, Saddle points of Hamiltonian systems in convex problems of Lagrange, Journal of Optimization Theory and Applications, 12 (1973), 367-390.  doi: 10.1007/BF00940418. [25] A. Rodriguez, On the local stability of the solution to optimal control problems, Journal of Economic Dynamics and Control, 28 (2004), 2475-2484.  doi: 10.1016/j.jedc.2003.12.003. [26] A. Rodriguez, On the local stability of the stationary solution to variational problems, Journal of Economic Dynamics and Control, 20 (1996), 415-431.  doi: 10.1016/0165-1889(94)00857-2. [27] S. J. Rubio and B. Casino, Self-enforcing international environmental agreements with a stock pollutant, Spanish Economic Review, 7 (2005), 89-109.  doi: 10.1007/s10108-005-0098-6. [28] M. E. Slade, What does an oligopoly maximize?, The Journal of Industrial Economics, 42 (1994), 45-61.  doi: 10.2307/2950588. [29] G. Sorger, The saddle point property in Hamiltonian systems, Journal of Mathematical Analysis and Applications, 148 (1990), 191-201.  doi: 10.1016/0022-247X(90)90037-G. [30] A. M. Steinberg and H. L. Stalford, On existence of optimal controls, Journal of Optimization Theory and Applications, 11 (1973), 266-273.  doi: 10.1007/BF00935195. [31] E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, Journal of Differential Equations, 258 (2015), 81-114.  doi: 10.1016/j.jde.2014.09.005. [32] A. de Zeeuw, Dynamic games of international pollution control: A selective review, Handbook of Dynamic Game Theory, (2018), 703–728. doi: 10.1007/978-3-319-44374-4_16. [33] W. Zhang, X. Lin and B-S. Chen, LaSalle-type theorem and its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems, IEEE Transactions on Automatic Control, 62 (2017), 250-261.  doi: 10.1109/TAC.2016.2558044.

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##### References:
 [1] S. Barrett, Self-enforcing international environmental agreements, Oxford Economic Papers, 46 (1994), 878-894.  doi: 10.1093/oep/46.Supplement_1.878. [2] A. Bressan, From optimal control to non-cooperative differential games: A homotopy approach, Control and Cybernetics, 38 (2009), 1081-1106. [3] W. A. Brock and J. A. Scheinkman, Global asymptotic stability of optimal control systems with applications to the theory of economic growth, The Hamiltonian Approach to Dynamic Economics. J. Econom. Theory, 12 (1976), 164-190.  doi: 10.1016/0022-0531(76)90031-4. [4] F. Camilli, L. Grüne and F. Wirth, Control Lyapunov functions and Zubov's method, SIAM Journal on Control and Optimization, 47 (2008), 301-326.  doi: 10.1137/06065129X. [5] C. Carraro and D. Siniscalco, Strategies for the international protection of the environment, Journal of Public Economics, 52 (1993), 309-328.  doi: 10.1016/0047-2727(93)90037-T. [6] D. Cass and K. Shell, The structure and stability of competitive dynamical systems, The Hamiltonian Approach to Dynamic Economics. J. Econom. Theory, 12 (1976), 31-70.  doi: 10.1016/0022-0531(76)90027-2. [7] C. D'Aspremont, A. Jacquemin, J. J. Gabszewicz and J. A. Weymark, On the stability of collusive price leadership, Canadian Journal of Economics, 16 (1983), 17-25.  doi: 10.2307/134972. [8] E. J. Dockner and N. V. Long, International pollution control: Cooperative versus noncooperative strategies, Journal of Environmental Economics and Management, 25 (1993), 13–29. (WU Vienna University of Economics, 1991, 9), https://ideas.repec.org/p/wiw/wus005/6275.html. doi: 10.1006/jeem.1993.1023. [9] E. J. Dockner and N. V. Long, International pollution control: Cooperative versus noncooperative strategies, Journal of Environmental Economics and Management, 24 (1993), 13-29.  doi: 10.1006/jeem.1993.1023. [10] A. V. Dmitruk and N. V. Kuz'kina, Existence theorem in the optimal control problem on an infinite time interval, Mathematical Notes, 78 (2005), 466-480.  doi: 10.1007/s11006-005-0147-3. [11] C. D. Feinstein and D. G. Luenberger, Analysis of the asymptotic behavior of optimal control trajectories: The implicit programming problem, SIAM Journal on Control and Optimization, 19 (1981), 561-585.  doi: 10.1137/0319035. [12] M. Finus and B. Rundshagen, Renegotiation-proof equilibria in a global emission game when players are impatient, Environmental and Resource Economics, 12 (1998), 275-306.  doi: 10.1023/A:1008211729093. [13] A. Fonseca-Morales and O. Hernández-Lerma, A note on differential games with Pareto-optimal Nash equilibria: Deterministic and stochastic models, Journal of Dynamics and Games, 4 (2017), 195-203.  doi: 10.3934/jdg.2017012. [14] A. Fonseca-Morales and O. Hernández-Lerma, Potential differential games, Dynamic Games and Applications, 8 (2018), 254-279.  doi: 10.1007/s13235-017-0218-6. [15] R. A. Freeman and J. A. Primbs, Control Lyapunov functions: New ideas from an old source, Proceedings of 35th IEEE Conference on Decision and Control, 4 (1996), 3926-3931.  doi: 10.1109/CDC.1996.577294. [16] V. Gaitsgory, L. Grüne and N. Thatcher, Stabilization with discounted optimal control, Systems and Control Letters, 82 (2015), 91-98.  doi: 10.1016/j.sysconle.2015.05.010. [17] D. Gale, On optimal development in a multi-sector economy, The Review of Economic Studies, 34 (1967), 1-18.  doi: 10.2307/2296567. [18] A. Haurie, Existence and global asymptotic stability of optimal trajectories for a class of infinite-horizon, nonconvex systems, Journal of Optimization Theory and Applications, 31 (1980), 515-533.  doi: 10.1007/BF00934475. [19] E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, 1967. [20] N. V. Long, A Survey of Dynamic Games in Economics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/9789814293044. [21] M. Magill, Some new results on the local stability of the process of capital accumulation, Journal of Economic Theory, 15 (1977), 174-210.  doi: 10.1016/0022-0531(77)90075-8. [22] M. J. P. Magill and J. A. Scheinkman, Stability of regular equilibria and the correspondence principle for symmetric variational problems, International Economic Review, 20 (1979), 297-315.  doi: 10.2307/2526479. [23] S. Miricǎ, Reducing a differential game to a pair of optimal control problems, Differential Equations, Chaos and Variational Problems, (2008), 269–283. doi: 10.1007/978-3-7643-8482-1_22. [24] R. T. Rockafellar, Saddle points of Hamiltonian systems in convex problems of Lagrange, Journal of Optimization Theory and Applications, 12 (1973), 367-390.  doi: 10.1007/BF00940418. [25] A. Rodriguez, On the local stability of the solution to optimal control problems, Journal of Economic Dynamics and Control, 28 (2004), 2475-2484.  doi: 10.1016/j.jedc.2003.12.003. [26] A. Rodriguez, On the local stability of the stationary solution to variational problems, Journal of Economic Dynamics and Control, 20 (1996), 415-431.  doi: 10.1016/0165-1889(94)00857-2. [27] S. J. Rubio and B. Casino, Self-enforcing international environmental agreements with a stock pollutant, Spanish Economic Review, 7 (2005), 89-109.  doi: 10.1007/s10108-005-0098-6. [28] M. E. Slade, What does an oligopoly maximize?, The Journal of Industrial Economics, 42 (1994), 45-61.  doi: 10.2307/2950588. [29] G. Sorger, The saddle point property in Hamiltonian systems, Journal of Mathematical Analysis and Applications, 148 (1990), 191-201.  doi: 10.1016/0022-247X(90)90037-G. [30] A. M. Steinberg and H. L. Stalford, On existence of optimal controls, Journal of Optimization Theory and Applications, 11 (1973), 266-273.  doi: 10.1007/BF00935195. [31] E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, Journal of Differential Equations, 258 (2015), 81-114.  doi: 10.1016/j.jde.2014.09.005. [32] A. de Zeeuw, Dynamic games of international pollution control: A selective review, Handbook of Dynamic Game Theory, (2018), 703–728. doi: 10.1007/978-3-319-44374-4_16. [33] W. Zhang, X. Lin and B-S. Chen, LaSalle-type theorem and its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems, IEEE Transactions on Automatic Control, 62 (2017), 250-261.  doi: 10.1109/TAC.2016.2558044.
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