Skip to main content
SpringerLink
Log in
Book cover

Frontiers of Dynamic Games pp 85–102Cite as

Birkhäuser

A-Subgame Concept and the Solutions Properties for Multistage Games with Vector Payoffs

  • Chapter
  • First Online:

Part of the book series: Static & Dynamic Game Theory: Foundations & Applications ((SDGTFA))

Abstract

We deal with multistage multicriteria games in extensive form and employ so-called “A-subgame” concept to examine dynamical properties of some non-cooperative and cooperative solutions. It is proved that if we take into account only the active players at each A-subgame the set of all strong Pareto equilibria is time consistent but does not satisfy dynamical compatibility.

We construct an optimal cooperative trajectory and vector-valued characteristic function using the refined leximin algorithm. To ensure the sustainability of a cooperative agreement we design the A-incremental imputation distribution procedure for the Shapley value which provides a better incentive for cooperation than classical incremental allocation procedure. This specific payment schedule corresponds to the A-subgame concept satisfies time consistency and efficiency condition and implies non-zero current payment to the active player immediately after her move.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Bergstresser, K., Yu, P.: Domination structures and multicriteria problems in N-person games. Theor. Decis. 8, 5–48 (1977)

    Article  MathSciNet  Google Scholar 

  2. Borm, P., Tijs, S., Van Den Aarssen, J.: Pareto equilibrium in multiobjective games. Methods Oper. Res. 60, 303–312 (1990)

    MATH  Google Scholar 

  3. Climaco, J., Romero, C., Ruiz, F.: Preface to the special issue on multiple criteria decision making: current challenges and future trends. Int. Trans. Oper. Res. 25, 759–761 (2018). https://doi.org/10.1111/itor.12515

    Article  MathSciNet  Google Scholar 

  4. Finus, M.: Game Theory and International Environmental Cooperation. Edward Elgar, Cheltenham (2001)

    Book  Google Scholar 

  5. Gromova, E.V., Petrosyan, L.A.: On an approach to constructing a characteristic function in cooperative differential games. Autom. Remote. Control. 78(9), 1680–1692 (2017)

    Article  MathSciNet  Google Scholar 

  6. Gromova, E.V., Plekhanova, T.M.: On the regularization of a cooperative solution in a multistage game with random time horizon. Discret. Appl. Math. (2018). https://doi.org/10.1016/j.dam.2018.08.008

    Article  MathSciNet  Google Scholar 

  7. Haddad, M., Sanders, D.: Selection of discrete multiple criteria decision making methods in the presence of risk and uncertainty. Oper. Res. Perspect. 5, 357–370 (2018). https://doi.org/10.1016/j.orp.2018.10.003

    Article  MathSciNet  Google Scholar 

  8. Haurie, A.: A note on nonzero-sum differential games with bargaining solution. J. Optim. Theory Appl. 18, 31–39 (1976)

    Article  MathSciNet  Google Scholar 

  9. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions, 2nd edn. Springer, Berlin (2011)

    Book  Google Scholar 

  10. Kuhn, H.: Extensive games and the problem of information. Ann. Math. Stud. 28, 193–216 (1953)

    MathSciNet  MATH  Google Scholar 

  11. Kuzyutin, D.: On the problem of the stability of solutions in extensive games. Vestnik St. Petersburg Univ. Math. 4(22), 18–23 (1995) (in Russian)

    MathSciNet  MATH  Google Scholar 

  12. Kuzyutin, D.: Properties of solutions of positional games with incomplete information. J. Comput. Syst. Sci. Int. 34(1), 157–162 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Kuzyutin, D.: Reexamination of the strong time consistency concept in extensive games. In: Proceedings of the 7th International Symposium on Dynamic Games and Application, Kanagava, pp. 564–572 (1996)

    Google Scholar 

  14. Kuzyutin, D., Nikitina, M.: Time consistent cooperative solutions for multistage games with vector payoffs. Oper. Res. Lett. 45(3), 269–274 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kuzyutin, D., Nikitina, M.: An irrational behavior proof condition for multistage multicriteria games. In: Proceedings of Constructive nonsmooth analysis and related topics (dedic. to the memory of V.F. Demyanov), CNSA 2017, pp. 178–181. IEEE, Piscataway (2017)

    Google Scholar 

  16. Kuzyutin, D., Romanenko, I.: On properties of equilibrium solutions for n-person games in extensive form. Vestnik St. Petersburg Univ. Math. 3(15), 17–27 (1998) (in Russian)

    MATH  Google Scholar 

  17. Kuzyutin, D., Nikitina, M., Pankratova, Y.: O Svoystvakh Ravnovesiy v Mnogokriterialnikh Pozicionnikh Igrakh N Lic. Matematicheskaya Teoriya Igr i eyo Prilojeniya 6(1), 19–40 (2014) (in Russian)

    Google Scholar 

  18. Kuzyutin, D., Nikitina, M., Razgulyaeva, L.: On the A-equilibria properties in multicriteria extensive games. Appl. Math. Sci. 9(92), 4565–4573 (2015)

    Google Scholar 

  19. Kuzyutin, D., Gromova, E., Pankratova, Ya.: Sustainable cooperation in multicriteria multistage games. Oper. Res. Lett. 46(6), 557–562 (2018). https://doi.org/10.1016/j.orl.2018.09.004

    Article  MathSciNet  Google Scholar 

  20. Kuzyutin, D., Smirnova, N., Gromova, E.: Long-term implementation of the cooperative solution in multistage multicriteria game. Oper. Res. Perspect. 6, 100107 (2019). https://doi.org/10.1016/j.orp.2019.100107

    Article  MathSciNet  Google Scholar 

  21. Madani, K., Lund, J.R.: A Monte-Carlo game theoretic approach for multi-criteria decision making under uncertainty. Adv. Water Resour. 34, 607–616 (2011)

    Article  Google Scholar 

  22. Mendoza, G.A., Martins, H.: Multi-criteria decision analysis in natural resource management: A critical review of methods and new modelling paradigms. For. Ecol. Manag. 230, 1–22 (2006)

    Article  Google Scholar 

  23. Pankratova, Y., Tarashnina, S., Kuzyutin, D.: Nash equilibria in a group pursuit game. Appl. Math. Sci. 10(17), 809–821 (2016)

    Google Scholar 

  24. Petrosyan, L.: Stable solutions of differential games with many participants. Vestn. Leningr. Univ. 19, 46–52 (1977) (in Russian)

    Google Scholar 

  25. Petrosyan, L., Danilov, N.: Stability of the solutions in nonantagonistic differential games with transferable payoffs. Vestn. Leningr. Univer. 1, 52–59 (1979) (in Russian)

    MathSciNet  MATH  Google Scholar 

  26. Petrosyan, L.A., Kuzyutin, D.V.: On the stability of E-equilibrium in the class of mixed strategies. Vestn. St.Petersburg Univ. Math. 3(15), 54–58 (1995) (in Russian)

    Google Scholar 

  27. Petrosyan, L., Kuzyutin, D.: Games in Extensive Form: Optimality and Stability. Saint Petersburg University Press, Saint Petersburg (2000) (in Russian)

    MATH  Google Scholar 

  28. Petrosyan, L., Zaccour, G.: Time-consistent Shapley value allocation of pollution cost reduction. J. Econ. Dyn. Control. 27(3), 381–398 (2003)

    Article  MathSciNet  Google Scholar 

  29. Pieri, G., Pusillo, L.: Interval values for multicriteria cooperative games. AUCO Czech. Econ. Rev. 4, 144–155 (2010)

    Google Scholar 

  30. Pieri, G., Pusillo, L.: Multicriteria partial cooperative games. Appl. Math. 6, 2125–2131 (2015)

    Article  Google Scholar 

  31. Podinovskii, V., Nogin, V.: Pareto-optimal solutions of multicriteria problems. Nauka (1982) (in Russian)

    Google Scholar 

  32. Puerto, J., Perea, F.: On minimax and Pareto optimal security payoffs in multicriteria games. J. Math. Anal. Appl. 457(2), 1634–1648 (2018). https://doi.org/10.1016/j.jmaa.2017.01.002

    Article  MathSciNet  Google Scholar 

  33. Reddy, P., Shevkoplyas, E., Zaccour, G.: Time-consistent Shapley value for games played over event trees. Automatica 49(6), 1521–1527 (2013)

    Article  MathSciNet  Google Scholar 

  34. Rettieva, A.: Cooperation in dynamic multicriteria games with random horizons. J. Glob. Optim. (2018). https://doi.org/10.1007/s10898-018-0658-6.

  35. Shapley, L.: A value for n-person games. In: Contributions to the Theory of Games, II, Kuhn, H., Tucker, A.W. (eds.), pp. 307–317. Princeton University Press, Princeton (1953)

    Google Scholar 

  36. Shapley, L.: Equilibrium points in games with vector payoffs. Nav. Res. Logist. Q. 6, 57–61 (1959)

    Article  MathSciNet  Google Scholar 

  37. Voorneveld, M., Vermeulen, D., Borm, P.: Axiomatizations of Pareto equilibria in multicriteria games. Games Econom. Behav. 28, 146–154 (1999)

    Article  MathSciNet  Google Scholar 

  38. Voorneveld, M., Grahn, S., Dufwenberg, M.: Ideal equilibria in noncooperative multicriteria games. Math. Methods Oper. Res. 52, 65–77 (2000)

    Article  MathSciNet  Google Scholar 

  39. Wang, Y.: Existence of a Pareto equilibrium. J. Optim. Theory Appl. 79, 373–384 (1993)

    Article  MathSciNet  Google Scholar 

  40. Yeung, D.: An irrational-behavior-proof condition in cooperative differential games. Int. Game Theory Rev. 8(4), 739–744 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research of the first and the second author was funded by RFBR under the research project 18-00-00727 (18-00-00725). The research of the third author was funded by RFBR under the research project 18-00-00727 (18-00-00628).

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    Denis Kuzyutin & Yaroslavna Pankratova

  2. National Research University Higher School of Economics at St. Petersburg, St. Petersburg, Russia

    Denis Kuzyutin

  3. The Herzen State Pedagogical University of Russia, St. Petersburg, Russia

    Roman Svetlov

Authors
  1. Denis Kuzyutin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaroslavna Pankratova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roman Svetlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denis Kuzyutin .

Editor information

Editors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    Leon A. Petrosyan

  2. Institute of Applied Mathematical Research, Karelia Research Center of RAS, Petrozavodsk, Russia

    Vladimir V. Mazalov

  3. Graduate School of Management, St. Petersburg State University, St. Petersburg, Russia

    Nikolay A. Zenkevich

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kuzyutin, D., Pankratova, Y., Svetlov, R. (2019). A-Subgame Concept and the Solutions Properties for Multistage Games with Vector Payoffs. In: Petrosyan, L., Mazalov, V., Zenkevich, N. (eds) Frontiers of Dynamic Games. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23699-1_6

Download citation

Publish with us

Policies and ethics

Search

Navigation