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On Zermelo's theorem
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 |
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, [
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. |
[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. Google Scholar |
[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. 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. |
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. |
[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. Google Scholar |
[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. 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. |
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