Automation and Remote Control

, Volume 79, Issue 10, pp 1912–1928 | Cite as

Strong Time-Consistent Subset of the Core in Cooperative Differential Games with Finite Time Horizon

  • O. L. PetrosianEmail author
  • E. V. Gromova
  • S. V. Pogozhev
Mathematical Game Theory and Applications


Time consistency is one of the most important properties of solutions in cooperative differential games. This paper uses the core as a cooperative solution of the game. We design a strong time-consistent subset of the core. The design method of this subset is based on a special class of imputation distribution procedures (IDPs).


cooperative differential games time consistency strong time consistency imputation distribution procedure core 


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  1. 1.
    Vorob’ev, N.N., Teoriya igr dlya ekonomistov-kibernetikov (The Theory of Games for Economists-Cyberneticians), Moscow: Nauka, 1985.zbMATHGoogle Scholar
  2. 2.
    Gromova, E.V. and Petrosyan, L.A., On an Approach to Constructing a Characteristic Function in Cooperative Differential Games, Autom. Remote Control, 2017, vol. 78, no. 9, pp. 1680–1692.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gromova, E.V. and Petrosyan, L.A., Strong Time-Consistent Cooperative Solution for a Differential Game of Pollution Control, Upravlen. Bolsh. Sist., 2015, no. 55, pp. 140–159.zbMATHGoogle Scholar
  4. 4.
    Petrosyan, L.A., Stability of the Solutions of Differential Games with Several Players, Vestn. Leningrad. Univ., 1977, no. 19, pp. 46–52.MathSciNetGoogle Scholar
  5. 5.
    Petrosyan, L.A., Strong Time-Consistent Differential Optimality Principles, Vestn. Leningrad. Univ., 1993, no. 4, pp. 35–40.Google Scholar
  6. 6.
    Petrosyan, L.A. and Danilov, N.N., Stable Solutions in Nonantagonistic Differential Games with Transferable Payoffs, Vestn. Leningrad. Univ., 1979, no. 1, pp. 52–79.zbMATHGoogle Scholar
  7. 7.
    Petrosian, O.L., Looking Forward Approach in Cooperative Differential Games with Infinite Horizon, Vestn. St. Peterb. Gos. Univ., Ser. 10, Prikl. Mat. Inform. Prots. Upravlen., 2016, no. 4, pp. 18–30.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Pecherskii, S.L. and Yanovskaya, E.E., Kooperativnye igry: resheniya i aksiomy (Cooperative Games: Solutions and Axioms), St. Petersburg: Evrop. Univ., 2004.Google Scholar
  9. 9.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Fizmatlit, 1961.Google Scholar
  10. 10.
    Sedakov, A.A., On the Strong Time Consistency of the Core, Autom. Remote Control, 2018, vol. 79, no. 4, pp. 757–767.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Smirnova, E., Stable Cooperation in One Linear-Quadratic Differential Game, in Proc. Student Conf. “Control Processes and Stability” (CPS’13), St. Petersburg, 2013, pp. 666–672.Google Scholar
  12. 12.
    Basar, T. and Olsder, G., Dynamic Noncooperative Game Theory, London: Academic, 1995.zbMATHGoogle Scholar
  13. 13.
    Bellman, R., Dynamic Programming, Princeton: Princeton Univ. Press, 1957.zbMATHGoogle Scholar
  14. 14.
    Breton, M., Zaccour, G., and Zahaf, M., A Differential Game of Joint Implementation of Environmental Projects, Automatica, 2005, vol. 41, no. 10, pp. 1737–1749.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dockner, E.J., Jorgensen, S., van Long, N., and Sorger, G., Differential Games in Economics and Management Science, New York: Cambridge Univ. Press, 2001.zbMATHGoogle Scholar
  16. 16.
    Gromova, E., The Shapley Value as a Sustainable Cooperative Solution in Differential Games of 3 Players, in Recent Advances in Game Theory and Applications, Static and Dynamic Game Theory: Foundations and Applications, Chapter: IV, New York: Springer, 2016. DOI: 10.1007/978-3-319-43838-2_4Google Scholar
  17. 17.
    Gromova, E. and Petrosyan, O., Control of Informational Horizon for Cooperative Differential Game of Pollution Control, 2016 Int. Conf. on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference), 2016. DOI: 10.1109/STAB.2016.7541187Google Scholar
  18. 18.
    Haurie, A., A Note on Nonzero-Sum Differential Games with Bargaining Solutions, J. Optimiz. Theory Appl., 1976, vol. 18, no. 1, pp. 31–39.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Haurie, A. and Zaccour, G., Differential Game Models of Global Environmental Management, in Control Game-Theor. Model Environment, 1995, vol. 2, pp. 3–23.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Petrosian, O.L., Looking Forward Approach in Cooperative Differential Games, Int. Game Theory Rev., 2016. DOI: 10.1142/S0219198916400077CrossRefGoogle Scholar
  21. 21.
    Petrosian, O.L. and Barabanov, A.E., Looking Forward Approach in Cooperative Differential Games with Uncertain-Stochastic Dynamics, J. Optimiz. Theory Appl., 2016. DOI: 10.1007/s10957-016-1009-8Google Scholar
  22. 22.
    Petrosjan, L. and Zaccour, G., Time-consistent Shapley Value Allocation of Pollution Cost Reduction, J. Econom. Dynamics Control, 2003, vol. 27, no. 3, pp. 381–398.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shapley, L.S., A Value for n-Person Games, in Contributions to the Theory of Games, vol. II, Kuhn, W. and Tucker, A.W., Eds., Princeton: Princeton Univ. Press, 1953, pp. 307–317.Google Scholar
  24. 24.
    Shapley, L.S., Cores of Convex Games, Int. J. Game Theory, 1971, vol. 1, no. 1, pp. 11–26.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yeung, D.W.K., An Irrational-Behavior-Proofness Condition in Cooperative Differential Games, Int. J. Game Theory Rev., 2007, vol. 9, no. 1, pp. 256–273.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • O. L. Petrosian
    • 1
    Email author
  • E. V. Gromova
    • 1
  • S. V. Pogozhev
    • 1
  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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