Advertisement

Mixed Strategies in Vector Optimization and Germeier’s Convolution

  • N. M. Novikova
  • I. I. PospelovaEmail author
SYSTEMS ANALYSIS AND OPERATIONS RESEARCH

Abstract

The simplest two-criteria examples of a vector optimization problem and a zero-sum game are considered to study the adequacy of using mixed strategies if the linear convolution is replaced by the Germeier’s convolution (the inverse logical convolution) for parametrizing the set of optimal solutions or values of the game and also for estimating the payoffs of all participants. It is shown that the linear convolution yields different results in a comparison with the averaged inverse logical convolution. The issues of stochastic vector optimization and various conceptual formalizations for the value of multi-criteria mixed strategies games are discussed.

Notes

REFERENCES

  1. 1.
    Yu. B. Germeier, Introduction to the Theory of Operations Research (Nauka, Moscow, 1971) [in Russian].Google Scholar
  2. 2.
    V. V. Podinovskii and V. D. Nogin, Pareto Optimal Solutions for Multicriteria Problems (Nauka, Moscow, 1982) [in Russian].Google Scholar
  3. 3.
    A. V. Lotov and I. I. Pospelova, Multi-Criteria Decision Making Tasks (MAKS, Moscow, 2008) [in Russian].Google Scholar
  4. 4.
    M. Voorneveld, D. Vermeulen, and P. Borm, “Axiomatizations of Pareto equilibria in multicriteria games,” Games Econ. Behavior 28, 146–154 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    L. S. Shapley, “Equilibrium points in games with vector payoffs,” Naval Res. Log. Quart., No. 6, 57–61 (1959).Google Scholar
  6. 6.
    V. V. Morozov, “Mixed strategies in the game with vector wins,” Vestn. Mosk. Univ., Vychisl. Mat. Kibern., No. 4, 44–49 (1978).Google Scholar
  7. 7.
    M. M. Smirnov, “On the logical convolution of the vector of criteria in the problem of approximation of the Pareto set,” Zh. Vychisl. Mat. Mat. Fiz. 36 (3), 62–74 (1996).Google Scholar
  8. 8.
    N. M. Novikova, I. I. Pospelova, and A. I. Zenyukov, “Method of convolution in multicriteria problems with uncertainty,” J. Comput. Syst. Sci. Int. 56, 774 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. M. Novikova and I. I. Pospelova, “Scalarization method in multicriteria games,” Comput. Math. Math. Phys. 58, 180 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    N. M. Novikova and I. I. Pospelova, “Mixed strategies in vector game and Germeyer’s convolution,” in Proceedings of the 9th Moscow International Conference on Operation Research ORM’2018 (MAKS Press, Moscow, 2018), Vol. 2, pp. 428–432.Google Scholar
  11. 11.
    Yu. M. Ermol’ev, Stochastic Programming (Nauka, Moscow, 1976) [in Russian].zbMATHGoogle Scholar
  12. 12.
    Yu. E. Nesterov, “Algorithmic convex optimization,” Doctoral (Phys. Math.) Dissertation (Mosc. Phys.-Tech. Inst., Moscow, 2013).Google Scholar
  13. 13.
    V. V. Fedorov, Maximin Numerical Methods (Nauka, Moscow, 1979) [in Russian].Google Scholar
  14. 14.
    I. I. Pospelova, “Classification of vector optimization problems with uncertain factors,” Comput. Math. Math. Phys. 40, 820 (2000).MathSciNetzbMATHGoogle Scholar
  15. 15.
    E. M. Kreines, N. M. Novikova, and I. I. Pospelova, “Multicriteria two-person games with opposite interests,” Comput. Math. Math. Phys. 42, 1430 (2002).MathSciNetzbMATHGoogle Scholar
  16. 16.
    D. Blackwell, “An analog of the minimax theorem for vector payoffs,” Pacif. J. Math. 6 (1), 1–8 (1956).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. H. Hamel and A. Löhne, “A set optimization approach to zero-sum matrix games with multi-dimensional payoffs,” Math. Methods Operat. Res. 88, 369–397 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. M. Mármol, L. Monroy, M. A. Caraballo, et al., “Equilibria with vector-valued utilities and perfect information. The analysis of mixed duopoly,” Theory Decis. 83, 365–383 (2017).CrossRefzbMATHGoogle Scholar
  19. 19.
    M. A. Caraballo, A. M. Mármol, L. Monroy, et al., “Cournot competition under uncertainty: conservative and optimistic equilibria,” Rev. Econ. Des. 19, 145–165 (2015).MathSciNetzbMATHGoogle Scholar
  20. 20.
    S. Bade, “Nash equilibrium in games with incomplete preferences,” Econ. Theory 26, 309–332 (2005).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Dorodnicyn Computing Center, Federal Research Center for Computer Science and Control, Russian Academy of SciencesMoscowRussia
  2. 2.Faculty of Computational Mathematics and Cybernetics, Moscow State UniversityMoscowRussia

Personalised recommendations