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Journal of Optimization Theory and Applications

, Volume 172, Issue 1, pp 328–347 | Cite as

Looking Forward Approach in Cooperative Differential Games with Uncertain Stochastic Dynamics

  • Ovanes Petrosian
  • Andrey Barabanov
Article

Abstract

In this study, a novel approach for defining and computing a solution for a differential game is presented for a case, wherein players do not have complete information about the game structure for the full time interval. At any instant in time, players have certain information about the motion equations and payoff functions for a current subinterval, and a forecast about the game structure for the rest of the time interval. The forecast is described by stochastic differential equations. The information about the game structure updates at fixed instants of time and is completely unknown in advance. A new solution is defined as a recursive combination of sets of imputations in the combined truncated subgames that are analyzed by the Looking Forward Approach. An example with a resource extraction game is presented to demonstrate a comparison of payoff functions without a forecast and that with stochastic and deterministic forecasts.

Keywords

Differential game Looking Forward Approach Imputation distribution procedure Time consistency Strong time consistency 

Mathematics Subject Classification

49N70 91A12 

Notes

Acknowledgments

The first author acknowledges Saint-Petersburg State University for the research Grant No. 9.38.205.2014. The work of the second author was supported by Saint-Petersburg State University, Project 6.37.349.2015 and 6.38.230.2015.

References

  1. 1.
    Haurie, A.: A note on nonzero-sum differential games with bargaining solutions. J. Optim. Theory Appl. 18, 31–39 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Petrosyan, L.A.: Time-consistency of solutions in multi-player differential games. Vestn. Leningr. State Univ. 4, 46–52 (1977)Google Scholar
  3. 3.
    Goodwin, G.C., Seron, M.M., Dona, J.A.: Constrained Control and Estimation: An Optimisation Approach. Springer, New York (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    Kwon, W.H., Han, S.H.: Receding Horizon Control: Model Predictive Control for State Models. Springer, New York (2005)Google Scholar
  5. 5.
    Wang, L.: Model Predictive Control System Design and Implementation Using MATLAB. Springer, New York (2005)Google Scholar
  6. 6.
    Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI (2009)Google Scholar
  7. 7.
    Petrosyan, L.A., Danilov, N.N.: Stability of solutions in non-zero sum differential games with transferable payoffs. Vestn. Leningr. Univ. 1, 52–59 (1979)zbMATHGoogle Scholar
  8. 8.
    Petrosian, O.L.: Looking forward approach in cooperative differential games. Int Game Theory Rev (2016). doi: 10.1142/S0219198916400077 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Petrosian, O.L.: Looking Forward approach in cooperative differential games with infinite-horizon. Vestn. Leningr. Univ. (2016) (to be published)Google Scholar
  10. 10.
    Gromova, E.V., Petrosian, O.L.: Control of informational horizon for cooperative differential game of pollution control. IEEE (2016). doi: 10.1109/STAB.2016.7541187
  11. 11.
    Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)zbMATHGoogle Scholar
  12. 12.
    Petrosyan, L.A., Zaccour, G.: Time-consistent Shapley value allocation of pollution cost reduction. J. Econ. Dyn. Control. 3, 381–398 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. Academic Press, London (1995)zbMATHGoogle Scholar
  14. 14.
    Petrosyan, L.A., Yeung, D.W.K.: Dynamically stable solutions in randomly-furcating differential games. Trans. Steklov Inst. Math. 1, 208–220 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jorgensen, S., Martin-Herran, G., Zaccour, G.: Agreeability and time consistency in linear-state differential games. J. Optim. Theory Appl. 1, 49–63 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Petrosjan, L.A.: Strongly time-consistent differential optimality principles. Vestn. St. Petersb. Univ. Math. 4, 40–46 (1993)MathSciNetGoogle Scholar
  17. 17.
    Cassandras, C.J., Lafortune, S.: Introduction to Discrete Event Systems. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    Jorgensen, S., Yeung, D.W.K.: Inter- and intragenerational renewable resource extraction. Ann. Oper. Res. 88, 275–289 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yeung, D.W.K., Petrosyan, L.A.: Subgame-consistent Economic Optimization. Springer, New York (2012)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Saint-Petersburg State UniversitySaint-PetersburgRussia

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