American Institute of Mathematical Sciences

Advanced
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 & 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.   Google Scholar

[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.  Google Scholar

[3]

L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013.  Google Scholar

[4]

J. Case, A class of games having Pareto optimal Nash equilibria, J Optim Theory Appl, 13 (1974), 379-385.  doi: 10.1007/BF00934872.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. J. Dockner and V. Kaitala, On efficient equilibrium solutions in dynamic games of resource management, Resour Energy, 11 (1989), 23-34.   Google Scholar

[10]

P. Dubey, Inefficiency of Nash equilibria, Math Oper Res, 11 (1986), 1-8.  doi: 10.1287/moor.11.1.1.  Google Scholar

[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.  Google Scholar

[12]

A. Fonseca-Morales and O. Hernández-Lerma, Potential differential games, Dyn. Games Appl., 7 (2017). doi: 10.1007/s13235-017-0218-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. P. Simon and L. Blume, Mathematics for Economists, Norton & Co, New York, 1994. Google Scholar

[22]

F. Van Der Ploeg and A. J. de Zeeuw, International aspects of pollution control, Environmental and Resource Economics, 2 (1992), 117-139.   Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[3]

L. D. Berkovitz and N. G. Medhin, Nonlinear Optimal Control Theory, CRC Press, Boca Raton, FL, 2013.  Google Scholar

[4]

J. Case, A class of games having Pareto optimal Nash equilibria, J Optim Theory Appl, 13 (1974), 379-385.  doi: 10.1007/BF00934872.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. J. Dockner and V. Kaitala, On efficient equilibrium solutions in dynamic games of resource management, Resour Energy, 11 (1989), 23-34.   Google Scholar

[10]

P. Dubey, Inefficiency of Nash equilibria, Math Oper Res, 11 (1986), 1-8.  doi: 10.1287/moor.11.1.1.  Google Scholar

[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.  Google Scholar

[12]

A. Fonseca-Morales and O. Hernández-Lerma, Potential differential games, Dyn. Games Appl., 7 (2017). doi: 10.1007/s13235-017-0218-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhauser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. P. Simon and L. Blume, Mathematics for Economists, Norton & Co, New York, 1994. Google Scholar

[22]

F. Van Der Ploeg and A. J. de Zeeuw, International aspects of pollution control, Environmental and Resource Economics, 2 (1992), 117-139.   Google Scholar

[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.  Google Scholar

[1]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010
[2]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
[3]

Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
[4]

Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989
[5]

John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018
[6]

Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719
[7]

Ellina Grigorieva, Evgenii Khailov. Hierarchical differential games between manufacturer and retailer. Conference Publications, 2009, 2009 (Special) : 300-314. doi: 10.3934/proc.2009.2009.300
[8]

Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082
[9]

Yu Chen. Delegation principle for multi-agency games under ex post equilibrium. Journal of Dynamics & Games, 2018, 5 (4) : 311-329. doi: 10.3934/jdg.2018019
[10]

Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics & Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555
[11]

Beatris Adriana Escobedo-Trujillo, Alejandro Alaffita-Hernández, Raquiel López-Martínez. Constrained stochastic differential games with additive structure: Average and discount payoffs. Journal of Dynamics & Games, 2018, 5 (2) : 109-141. doi: 10.3934/jdg.2018008
[12]

Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95
[13]

Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375
[14]

Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics & Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002
[15]

Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045
[16]

Lesia V. Baranovska. Pursuit differential-difference games with pure time-lag. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1021-1031. doi: 10.3934/dcdsb.2019004
[17]

Weihua Ruan. Markovian strategies for piecewise deterministic differential games with continuous and impulse controls. Journal of Dynamics & Games, 2019, 6 (4) : 337-366. doi: 10.3934/jdg.2019022
[18]

Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial & Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795
[19]

Miquel Oliu-Barton. Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights. Journal of Dynamics & Games, 2019, 6 (4) : 259-275. doi: 10.3934/jdg.2019018
[20]

Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117

 Impact Factor: 

Tools

Metrics

  • PDF downloads (39)
  • HTML views (13)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences