Advertisement

Mathematical Methods of Operations Research

, Volume 88, Issue 3, pp 369–397 | Cite as

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

  • Andreas H. Hamel
  • Andreas Löhne
Original Article
  • 72 Downloads

Abstract

A new solution concept for two-player zero-sum matrix games with multi-dimensional payoffs is introduced. It is based on extensions of the vector order in \(\mathbb {R}^d\) to order relations in the power set of \(\mathbb {R}^d\), so-called set relations, and strictly motivated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates for games with multi-dimensional payoffs. It is shown that–in contrast to games with one-dimensional payoffs–the corresponding strategies are different from equilibrium strategies for games with multi-dimensional payoffs. The two concepts are combined into new equilibrium notions for which existence theorems are given. Relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax and maximin solutions as well as Pareto optimal security strategies are clarified.

Keywords

Zero-sum game Multi-dimensional payoff Multi-objective programming Set relation Set optimization Incomplete preference 

Mathematics Subject Classification

Primary 91A05 Secondary 91A10 62C20 91A35 

References

  1. Aumann RJ (1962) Utility theory without the completeness axiom. Econometrica 30(3):445–462CrossRefGoogle Scholar
  2. Bade S (2005) Nash equilibrium in games with incomplete preferences. Econ Theor 26(2):309–332MathSciNetCrossRefGoogle Scholar
  3. Blackwell D (1956) An analog of the minimax theorem for vector payoffs. Pac J Math 6(1):1–8MathSciNetCrossRefGoogle Scholar
  4. Chen GY, Jahn J (1998) Special issue “Set-valued optimization”. Math Methods Oper Res 48(2)Google Scholar
  5. Cook WD (1976) Zero-sum games with multiple goals. Nav Res Logist Q 23(4):615–621MathSciNetCrossRefGoogle Scholar
  6. Corley HW (1985) Games with vector payoffs. J Optim Theory Appl 47(4):491–498MathSciNetCrossRefGoogle Scholar
  7. De Marco G, Morgan J (2007) A refinement concept for equilibria in multicriteria games via stable scalarizations. Int Game Theory Rev 9(02):169–181MathSciNetCrossRefGoogle Scholar
  8. Ehrgott M (2005) Multicriteria optimization, vol 491, 2nd edn. Lecture notes in economics and mathematical systems. Springer, BerlinzbMATHGoogle Scholar
  9. Fernández FR, Puerto J (1996) Vector linear programming in zero-sum multicriteria matrix games. J Optim Theory Appl 89(1):115–127MathSciNetCrossRefGoogle Scholar
  10. Fernández FR, Monroy L, Puerto J (1998) Multicriteria goal games. J Optim Theory Appl 99(2):403–421MathSciNetCrossRefGoogle Scholar
  11. Ghose D (1991) A necessary and sufficient condition for pareto-optimal security strategies in multicriteria matrix games. J Optim Theory Appl 68(3):463–481MathSciNetCrossRefGoogle Scholar
  12. Ghose D, Prasad UR (1989) Solution concepts in two-person multicriteria games. J Optim Theory Appl 63(2):167–189MathSciNetCrossRefGoogle Scholar
  13. Hamel AH, Heyde F, Löhne A, Rudloff B, Schrage C (eds) (2015) Set optimization–a rather short introduction. In: Set optimization and applications—the state of the art. From set relations to set-valued risk measures. Springer, Berlin, pp 65–141Google Scholar
  14. Henig MI (1986) The domination property in multicriteria optimization. J Math Anal Appl 114(1):7–16MathSciNetCrossRefGoogle Scholar
  15. Heyde F, Löhne A (2011) Solution concepts in vector optimization: a fresh look at an old story. Optimization 60(10–12):1421–1440MathSciNetCrossRefGoogle Scholar
  16. Kuroiwa D, Tanaka T, Ha TXD (1997) On cone convexity of set-valued maps. Nonlinear Anal Theory Methods Appl 30(3):1487–1496MathSciNetCrossRefGoogle Scholar
  17. Löhne A (2011) Vector optimization with infimum and supremum. Springer, BerlinCrossRefGoogle Scholar
  18. Löhne A, Weißing B (2017) The vector linear program solver Bensolve—notes on theoretical background. Eur J Oper Res 260(3):807–813MathSciNetCrossRefGoogle Scholar
  19. Luc DT, Vargas C (1992) A saddlepoint theorem for set-valued maps. Nonlinear Anal Theory Methods Appl 18(1):1–7MathSciNetCrossRefGoogle Scholar
  20. Maeda T (2015) On characterization of Nash equilibrium strategy in bi-matrix games with set payoffs. In: Hamel AH, Heyde F, Löhne A, Rudloff B, Schrage C (eds) Set optimization and applications—the state of the art. From set relations to set-valued risk measures. Springer, Berlin, pp 313–331CrossRefGoogle Scholar
  21. Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, New YorkzbMATHGoogle Scholar
  22. Nieuwenhuis JW (1983) Some minimax theorems in vector-valued functions. J Optim Theory Appl 40(3):463–475MathSciNetCrossRefGoogle Scholar
  23. Ok EA, Ortoleva P, Riella G (2012) Incomplete preferences under uncertainty: indecisiveness in beliefs versus tastes. Econometrica 80(4):1791–1808MathSciNetCrossRefGoogle Scholar
  24. Park J (2015) Potential games with incomplete preferences. J Math Econ 61:58–66MathSciNetCrossRefGoogle Scholar
  25. Puerto J, Perea F (2018) On minimax and pareto optimal security payoffs in multicriteria games. J Math Anal Appl 457(2):1634–1648MathSciNetCrossRefGoogle Scholar
  26. Shapley LS (1959) Equilibrium points in games with vector payoffs. Nav Res Logist Q 6:57–61MathSciNetCrossRefGoogle Scholar
  27. Tanaka T (1988) Some minimax problems of vector-valued functions. J Optim Theory Appl 59(3):505–524MathSciNetCrossRefGoogle Scholar
  28. Tanaka T (1994) Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. J Optim Theory Appl 81(2):355–377MathSciNetCrossRefGoogle Scholar
  29. Tanaka T (2000) Vector-valued minimax theorems in multicriteria games. In: Yong S, Milan Z (eds) New frontiers of decision making for the information technology era. World Scientific, Singapore, pp 75–99CrossRefGoogle Scholar
  30. Wierzbicki AP (1995) Multiple criteria games-theory and applications. J Syst Eng Electron 6(2):65–81MathSciNetGoogle Scholar
  31. Zeleny M (1974) Linear multiobjective programming, vol 95. Lecture notes in economics and mathematical systems. Springer, BerlinzbMATHGoogle Scholar
  32. Zeleny M (1975) Games with multiple payoffs. Int J Game Theory 4(4):179–191MathSciNetCrossRefGoogle Scholar
  33. Zhao J (1991) The equilibria of a multiple objective game. Int J Game Theory 20(2):171–182CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Economics and ManagementFree University Bozen-BolzanoBolzanoItaly
  2. 2.Department of MathematicsFriedrich Schiller UniversityJenaGermany

Personalised recommendations