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Deterministic time-inconsistent optimal control problems — an essentially cooperative approach

  • Jiong-min YongEmail author
Article

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.

Keywords

time-inconsistency pre-committed optimal control time-consistent equilibrium control multi-level hierarchical differential games 

2000 MR Subject Classification

49L20 49N10 49N70 91A23 

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References

  1. [1]
    Basak, S., Chabakauri, G. Dynamic mean-variance asset allocation. Rev. Financ. Stud., 23(8): 2970–3016 (2010)CrossRefGoogle Scholar
  2. [2]
    Berkovitz, L.D. Optimal control theory. Springer-Verlag, New York, 1974zbMATHGoogle Scholar
  3. [3]
    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
  4. [4]
    Böhm-Bawerk, E.V. The positive theory of capital. Books for Libraries Press, Freeport, New York, 1891Google Scholar
  5. [5]
    Ekeland, I., Lazrak, A. Being serious about non-commitment: subgame perfect equilibrium in continuous time. Available online: http://arxiv.org/abs/math/0604264
  6. [6]
    Ekeland, I., Privu, T. Investment and consumption without commitment. Math. Finan. Econ., 2: 57–86 (2008)CrossRefzbMATHGoogle Scholar
  7. [7]
    Goldman, S.M. Consistent plans. Review of Economic Studies, 47: 533–537 (1980)CrossRefzbMATHGoogle Scholar
  8. [8]
    Grenadier, S.R., Wang, N. Investment under uncertainty and time-inconsistent preferences. Journal of Financial Economics, 84: 2–39 (2007)CrossRefGoogle Scholar
  9. [9]
    Herings, P.J., Rohde, K.I.M. Time-inconsistent preferences in a general equilibriub model. Econom. Theory, 29: 591–619 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Hume, D. A Treatise of Human Nature. First Edition, 1739; Reprint, Oxford Univ. Press, New York, 1978Google Scholar
  11. [11]
    Jevons, W.S. Theory of Political Economy. Mcmillan, London, 1871Google Scholar
  12. [12]
    Krusell, P., Smith, A.A.Jr. Consumption and saving decisions with quasi-geometric discounting. Econometrica, 71: 366–375 (2003)CrossRefGoogle Scholar
  13. [13]
    Laibson, D. Golden eggs and hyperbolic discounting. Quarterly J. Econ., 112: 443–477 (1997)CrossRefzbMATHGoogle Scholar
  14. [14]
    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, 1986Google Scholar
  15. [15]
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    Marshall, A. Principles of Economics. 1st ed., 1890; 8th ed., Macmillan, London, 1920Google Scholar
  17. [17]
    Miller, M., Salmon, M. Dynamic games and the time inconsistency of optimal policy in open economics. The Economic Journal, 95: 124–137 (1985)CrossRefGoogle Scholar
  18. [18]
    Palacios-Huerta, I. Time-inconsistent preferences in Adam Smith and Davis Hume. History of Political Economy, 35: 241–268 (2003)CrossRefGoogle Scholar
  19. [19]
    Pareto, V. Manuel d’économie politique. Girard and Brieve, Paris, 1909Google Scholar
  20. [20]
    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)CrossRefzbMATHGoogle Scholar
  21. [21]
    Pollak, R.A. Consistent planning. Review of Economic Studies, 35: 185–199 (1968)CrossRefGoogle Scholar
  22. [22]
    Smith, A. The Theory of Moral Sentiments. First Edition, 1759; Reprint, Oxford Univ. Press, 1976Google Scholar
  23. [23]
    Strotz, R.H. Myopia and inconsistency in dynamic utility maximization. Review of Econ. Studies, 23: 165–180 (1955)CrossRefGoogle Scholar
  24. [24]
    Tesfatsion, L. Time inconsistency of benevolent government economics. J. Public Economics, 31: 25–52 (1986)CrossRefGoogle Scholar
  25. [25]
    Yong, J., Zhou, X.Y. Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York, 1999zbMATHGoogle Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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