# American Institute of Mathematical Sciences

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†

 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.

Citation: Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models. Journal of Dynamics & 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.

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##### 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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