A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models<sup>†</sup>

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July 2017, 4(3): 195-203. doi: 10.3934/jdg.2017012

A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models^†

Alejandra Fonseca-Morales ^, and Onésimo Hernández-Lerma

Mathematics Department, CINVESTAV-IPN, A. Postal 14-740, México City, 07000, México

* Corresponding author

† This research was partially supported by CONACyT grant 221291. The first author was also supported by a CONACyT scholarship.

Received March 2017 Revised April 2017 Published April 2017

Pareto optimality and Nash equilibrium are two standard solution concepts for cooperative and non-cooperative games, respectively. At the outset, these concepts are incompatible-see, for instance, [7] or [10]. But, on the other hand, there are particular games in which Nash equilibria turn out to be Pareto-optimal [1], [4], [6], [18], [20]. A class of these games has been identified in the context of discrete-time potential games [13]. In this paper we introduce several classes of deterministic and stochastic potential differential games [12] in which open-loop Nash equilibria are also Pareto optimal.

Keywords: Nash equilibrium, Pareto optimal, differential games, potential games.

Mathematics Subject Classification: Primary: 91A23, 49N70, 91A10; Secondary: 91A12, 91A25.

Citation: Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models^†. Journal of Dynamics and Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012

References:

[1]	R. Amir and N. Nannerup, Information structure and the tragedy of the commons in resource extraction, Journal of Bioeconomics, 8 (2006), 147-165.
[2]	T. Basar and Q. Zhu, Prices of anarchy, information, and cooperation, in differential games, Dyn Games Appl, 1 (2011), 50-73. doi: 10.1007/s13235-010-0002-3.
[3]	L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013.
[4]	J. Case, A class of games having Pareto optimal Nash equilibria, J Optim Theory Appl, 13 (1974), 379-385. doi: 10.1007/BF00934872.
[5]	C. D. Charalambous, Decentralized optimality conditions of stochastic differential decision problems via Girsanov's measure transformation Math Control Signals Syst, 28 (2016), Art 19, 55 pp. doi: 10.1007/s00498-016-0168-3.
[6]	C. Chiarella, M. C. Kemp, N. V. Long and K. Okuguchi, On the economics of international fisheries, Inter Econom Rev, 25 (1984), 85-92. doi: 10.2307/2648869.
[7]	J. E. Cohen, Cooperation and self-interest: Pareto-inefficiency of Nash equilibria in finite random games, Proc Natl Acad Sci USA, 95 (1998), 9724-9731. doi: 10.1073/pnas.95.17.9724.
[8]	E. J. Dockner, S. Jorgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, New York, 2000. doi: 10.1017/CBO9780511805127.
[9]	E. J. Dockner and V. Kaitala, On efficient equilibrium solutions in dynamic games of resource management, Resour Energy, 11 (1989), 23-34.
[10]	P. Dubey, Inefficiency of Nash equilibria, Math Oper Res, 11 (1986), 1-8. doi: 10.1287/moor.11.1.1.
[11]	J. C. Engwerda, Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games, SIAM J Control Optim, 48 (2010), 3859-3881. doi: 10.1137/080726227.
[12]	A. Fonseca-Morales and O. Hernández-Lerma, Potential differential games, Dyn. Games Appl., 7 (2017). doi: 10.1007/s13235-017-0218-6.
[13]	D. González-Sánchez and O. Hernández-Lerma, Discrete-Time Stochastic Control and Dynamic Potential Games, Springer, New York, 2013. doi: 10.1007/978-3-319-01059-5.
[14]	D. González-Sánchez and O. Hernández-Lerma, A survey of static and dynamic potential games, Sci China Math, 59 (2016), 2075-2102. doi: 10.1007/s11425-016-0264-6.
[15]	R. Josa-Fombellida and J. P. Rincón-Zapatero, Euler-Lagrange equations of stochastic differential games: application to a game of a productive asset, Economic Theory, 59 (2015), 61-108. doi: 10.1007/s00199-015-0873-z.
[16]	X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.
[17]	N. V. Long, Dynamic games in the economics of natural resources: A survey, Dyn Games Appl, 1 (2011), 115-148. doi: 10.1007/s13235-010-0003-2.
[18]	G. Martin-Herran and J. P. Rincón-Zapatero, Efficient Markov perfect Nash equilibria: Theory and application to dynamic fishery games, J Econom Dynam Control, 29 (2005), 1073-1096. doi: 10.1016/j.jedc.2004.08.004.
[19]	P. V. Reddy and J. C. Engwerda, Necessary and sufficient conditions for Pareto optimality in infinite horizon cooperative differential games, IEEE Trans Autom Control, 59 (2014), 2536-2542. doi: 10.1109/TAC.2014.2305933.
[20]	A. Seierstad, Pareto improvements of Nash equilibria in differential games, Dyn Games Appl, 4 (2014), 363-375. doi: 10.1007/s13235-013-0093-8.
[21]	C. P. Simon and L. Blume, Mathematics for Economists, Norton & Co, New York, 1994.
[22]	F. Van Der Ploeg and A. J. de Zeeuw, International aspects of pollution control, Environmental and Resource Economics, 2 (1992), 117-139.
[23]	J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

show all references

† This research was partially supported by CONACyT grant 221291. The first author was also supported by a CONACyT scholarship.

References:

[1]	R. Amir and N. Nannerup, Information structure and the tragedy of the commons in resource extraction, Journal of Bioeconomics, 8 (2006), 147-165.
[2]	T. Basar and Q. Zhu, Prices of anarchy, information, and cooperation, in differential games, Dyn Games Appl, 1 (2011), 50-73. doi: 10.1007/s13235-010-0002-3.
[3]	L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013.
[4]	J. Case, A class of games having Pareto optimal Nash equilibria, J Optim Theory Appl, 13 (1974), 379-385. doi: 10.1007/BF00934872.
[5]	C. D. Charalambous, Decentralized optimality conditions of stochastic differential decision problems via Girsanov's measure transformation Math Control Signals Syst, 28 (2016), Art 19, 55 pp. doi: 10.1007/s00498-016-0168-3.
[6]	C. Chiarella, M. C. Kemp, N. V. Long and K. Okuguchi, On the economics of international fisheries, Inter Econom Rev, 25 (1984), 85-92. doi: 10.2307/2648869.
[7]	J. E. Cohen, Cooperation and self-interest: Pareto-inefficiency of Nash equilibria in finite random games, Proc Natl Acad Sci USA, 95 (1998), 9724-9731. doi: 10.1073/pnas.95.17.9724.
[8]	E. J. Dockner, S. Jorgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, New York, 2000. doi: 10.1017/CBO9780511805127.
[9]	E. J. Dockner and V. Kaitala, On efficient equilibrium solutions in dynamic games of resource management, Resour Energy, 11 (1989), 23-34.
[10]	P. Dubey, Inefficiency of Nash equilibria, Math Oper Res, 11 (1986), 1-8. doi: 10.1287/moor.11.1.1.
[11]	J. C. Engwerda, Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games, SIAM J Control Optim, 48 (2010), 3859-3881. doi: 10.1137/080726227.
[12]	A. Fonseca-Morales and O. Hernández-Lerma, Potential differential games, Dyn. Games Appl., 7 (2017). doi: 10.1007/s13235-017-0218-6.
[13]	D. González-Sánchez and O. Hernández-Lerma, Discrete-Time Stochastic Control and Dynamic Potential Games, Springer, New York, 2013. doi: 10.1007/978-3-319-01059-5.
[14]	D. González-Sánchez and O. Hernández-Lerma, A survey of static and dynamic potential games, Sci China Math, 59 (2016), 2075-2102. doi: 10.1007/s11425-016-0264-6.
[15]	R. Josa-Fombellida and J. P. Rincón-Zapatero, Euler-Lagrange equations of stochastic differential games: application to a game of a productive asset, Economic Theory, 59 (2015), 61-108. doi: 10.1007/s00199-015-0873-z.
[16]	X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.
[17]	N. V. Long, Dynamic games in the economics of natural resources: A survey, Dyn Games Appl, 1 (2011), 115-148. doi: 10.1007/s13235-010-0003-2.
[18]	G. Martin-Herran and J. P. Rincón-Zapatero, Efficient Markov perfect Nash equilibria: Theory and application to dynamic fishery games, J Econom Dynam Control, 29 (2005), 1073-1096. doi: 10.1016/j.jedc.2004.08.004.
[19]	P. V. Reddy and J. C. Engwerda, Necessary and sufficient conditions for Pareto optimality in infinite horizon cooperative differential games, IEEE Trans Autom Control, 59 (2014), 2536-2542. doi: 10.1109/TAC.2014.2305933.
[20]	A. Seierstad, Pareto improvements of Nash equilibria in differential games, Dyn Games Appl, 4 (2014), 363-375. doi: 10.1007/s13235-013-0093-8.
[21]	C. P. Simon and L. Blume, Mathematics for Economists, Norton & Co, New York, 1994.
[22]	F. Van Der Ploeg and A. J. de Zeeuw, International aspects of pollution control, Environmental and Resource Economics, 2 (1992), 117-139.
[23]	J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

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