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Subgame Consistent Cooperative Solution of Dynamic Games with Random Horizon

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Abstract

In cooperative dynamic games, a stringent condition—that of subgame consistency—is required for a dynamically stable cooperative solution. In particular, under a subgame-consistent cooperative solution an extension of the solution policy to a subgame starting at a later time with a state brought about by prior optimal behavior will remain optimal. This paper extends subgame-consistent solutions to dynamic (discrete-time) cooperative games with random horizon. In the analysis, new forms of the Bellman equation and the Isaacs–Bellman equation in discrete-time are derived. Subgame-consistent cooperative solutions are obtained for this class of dynamic games. Analytically tractable payoff distribution mechanisms, which lead to the realization of these solutions, are developed. This is the first time that subgame-consistent solutions for cooperative dynamic games with random horizon are presented.

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References

  1. Yeung, D.W.K., Petrosyan, L.A.: Subgame consistent cooperative solutions in stochastic differential games. J. Optim. Theory Appl. 120, 651–666 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Yeung, D.W.K., Petrosyan, L.A.: Cooperative Stochastic Differential Games. Springer, New York (2006)

    MATH  Google Scholar 

  3. Petrosyan, L.A., Yeung, D.W.K.: Subgame-consistent cooperative solutions in randomly-furcating stochastic differential games. Int. J. Math. Comput. Model. 45, 1294–1307 (2007) (Special Issue on Lyapunov’s Methods in Stability and Control)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yeung, D.W.K.: Dynamically Consistent cooperative solution in a differential game of transboundary industrial pollution. J. Optim. Theory Appl. 134, 143–160 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Yeung, D.W.K., Petrosyan, L.A.: A Cooperative stochastic differential game of transboundary industrial pollution. Automatica 44(6), 1532–1544 (2008)

    Article  MathSciNet  Google Scholar 

  6. Başar, T., Ho, Y.C.: Informational properties of the Nash solutions of two stochastic nonzero-sum games. J. Econ. Theory 7(4), 370–387 (1974)

    Article  Google Scholar 

  7. Başar, T.: Decentralized multicriteria optimization of linear stochastic systems. IEEE Trans. Autom. Control AC-23(2), 233–243 (1978)

    Google Scholar 

  8. Krawczyk, J.B., Tidball, M.: A discrete-time dynamic game of seasonal water allocation. J. Optim. Theory Appl. 128(2), 411–429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Nie, P., Chen, L., Fukushima, M.: Dynamic programming approach to discrete time dynamic Feedback Stackelberg games with independent and dependent followers. Eur. J. Oper. Res. 169, 310–328 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dockner, E., Nishimura, K.: Transboundary pollution in a dynamic game model. Jpn. Econ. Rev. 50(4), 443–456 (1999)

    Article  Google Scholar 

  11. Rubio, S., Ulph, A.: An infinite-horizon model of dynamic membership of international environmental agreements. J. Environ. Econ. Manag. 54(3), 296–310 (2007)

    Article  MATH  Google Scholar 

  12. Dutta, P.-K., Radner, R.: Population growth and technological change in a global warming model. Econ. Theory 29(2), 251–270 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ehtamo, H., Hamalainen, R.: A cooperative incentive equilibrium for a resource management. J. Econ. Dyn. Control 17(4), 659–678 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yeung, D.W.K., Petrosyan, L.A.: Subgame consistent solutions for cooperative stochastic dynamic games. J. Optim. Theory Appl. 145(3), 579–596 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Petrosyan, L.A., Murzov, N.V.: Game theoretic problems in mechanics. Litov. Math. Sb. N6, 423–433 (1966)

    Google Scholar 

  16. Petrosyan, L.A., Shevkoplyas, E.V.: Cooperative solutions for games with random duration. Game Theory Appl. IX, 125–139 (2003)

    MathSciNet  Google Scholar 

  17. Shevkoplyas, E.V.: The Shapley value in cooperative differential games with random duration. Ann. Dyn. Games 11, 359–373 (2011). doi:10.1007/978-0-8176-8089-3_18

    Article  Google Scholar 

  18. Nash, J.F.Jr.: Noncooperative games. Ann. Math. 54, 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  19. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Wiley, New York (1944)

    MATH  Google Scholar 

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Authors and Affiliations

  1. SRS Consortium for Advanced Study, Hong Kong Shue Yan University, North Point, Hong Kong

    D. W. K. Yeung

  2. Centre for Game Theory, St. Petersburg State University, St. Petersburg, Russia

    D. W. K. Yeung

  3. Faculty of Applied Mathematics & Control Processes, Centre for Game Theory, St. Petersburg State University, St. Petersburg, 198904, Russia

    L. A. Petrosyan

Authors
  1. D. W. K. Yeung
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  2. L. A. Petrosyan
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Corresponding author

Correspondence to D. W. K. Yeung.

Additional information

Communicated by George Leitmann.

This research was supported by the EU TOCSIN Project.

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Yeung, D.W.K., Petrosyan, L.A. Subgame Consistent Cooperative Solution of Dynamic Games with Random Horizon. J Optim Theory Appl 150, 78–97 (2011). https://doi.org/10.1007/s10957-011-9824-4

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  • DOI: https://doi.org/10.1007/s10957-011-9824-4

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