Abstract
A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N-person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.
Similar content being viewed by others
References
Basak, S., Chabakauri, G. Dynamic mean-variance asset allocation. Rev. Financ. Stud., 23(8): 2970–3016 (2010)
Berkovitz, L.D. Optimal control theory. Springer-Verlag, New York, 1974
Björk, T., Murgoci, A. A general theory of Markovian time inconsistent stochasitc control problem. working paper (September 17, 2010). Available at SSRN: http://ssrn.com/abstract=1694759
Böhm-Bawerk, E.V. The positive theory of capital. Books for Libraries Press, Freeport, New York, 1891
Ekeland, I., Lazrak, A. Being serious about non-commitment: subgame perfect equilibrium in continuous time. Available online: http://arxiv.org/abs/math/0604264
Ekeland, I., Privu, T. Investment and consumption without commitment. Math. Finan. Econ., 2: 57–86 (2008)
Goldman, S.M. Consistent plans. Review of Economic Studies, 47: 533–537 (1980)
Grenadier, S.R., Wang, N. Investment under uncertainty and time-inconsistent preferences. Journal of Financial Economics, 84: 2–39 (2007)
Herings, P.J., Rohde, K.I.M. Time-inconsistent preferences in a general equilibriub model. Econom. Theory, 29: 591–619 (2006)
Hume, D. A Treatise of Human Nature. First Edition, 1739; Reprint, Oxford Univ. Press, New York, 1978
Jevons, W.S. Theory of Political Economy. Mcmillan, London, 1871
Krusell, P., Smith, A.A.Jr. Consumption and saving decisions with quasi-geometric discounting. Econometrica, 71: 366–375 (2003)
Laibson, D. Golden eggs and hyperbolic discounting. Quarterly J. Econ., 112: 443–477 (1997)
Malthus, A. An essay on the principle of population, 1826; The Works of Thomas Robert Malthus, Vols. 2–3, Edited by E. A. Wrigley and D. Souden, W. Pickering, London, 1986
Marin-Solano, J., Navas, J. Non-constant discounting in finite horizon: the free terminal time case. J. Economic Dynamics and Control, 33: 666–675 (2009)
Marshall, A. Principles of Economics. 1st ed., 1890; 8th ed., Macmillan, London, 1920
Miller, M., Salmon, M. Dynamic games and the time inconsistency of optimal policy in open economics. The Economic Journal, 95: 124–137 (1985)
Palacios-Huerta, I. Time-inconsistent preferences in Adam Smith and Davis Hume. History of Political Economy, 35: 241–268 (2003)
Pareto, V. Manuel d’économie politique. Girard and Brieve, Paris, 1909
Peleg, B., Yaari, M.E. On the existence of a consistent course of action when tastes are changing. Review of Economic Studies, 40: 391–401 (1973)
Pollak, R.A. Consistent planning. Review of Economic Studies, 35: 185–199 (1968)
Smith, A. The Theory of Moral Sentiments. First Edition, 1759; Reprint, Oxford Univ. Press, 1976
Strotz, R.H. Myopia and inconsistency in dynamic utility maximization. Review of Econ. Studies, 23: 165–180 (1955)
Tesfatsion, L. Time inconsistency of benevolent government economics. J. Public Economics, 31: 25–52 (1986)
Yong, J., Zhou, X.Y. Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York, 1999
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported in part by the NSF grant DMS-1007514.
Rights and permissions
About this article
Cite this article
Yong, Jm. Deterministic time-inconsistent optimal control problems — an essentially cooperative approach. Acta Math. Appl. Sin. Engl. Ser. 28, 1–30 (2012). https://doi.org/10.1007/s10255-012-0120-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-012-0120-3
Keywords
- time-inconsistency
- pre-committed optimal control
- time-consistent equilibrium control
- multi-level hierarchical differential games