Mathematical Methods of Operations Research

, Volume 64, Issue 1, pp 155–163 | Cite as

Characterizations of the Random Order Values by Harsanyi Payoff Vectors

  • Jean Derks
  • Gerard van der Laan
  • Valery Vasil’ev
Original Article


A Harsanyi payoff vector (see Vasil’ev in Optimizacija Vyp 21:30–35, 1978) of a cooperative game with transferable utilities is obtained by some distribution of the Harsanyi dividends of all coalitions among its members. Examples of Harsanyi payoff vectors are the marginal contribution vectors. The random order values (see Weber in The Shapley value, essays in honor of L.S. Shapley, Cambridge University Press, Cambridge, 1988) being the convex combinations of the marginal contribution vectors, are therefore elements of the Harsanyi set, which refers to the set of all Harsanyi payoff vectors.

The aim of this paper is to provide two characterizations of the set of all sharing systems of the dividends whose associated Harsanyi payoff vectors are random order values. The first characterization yields the extreme points of this set of sharing systems and is based on a combinatorial result recently published (Vasil’ev in Discretnyi Analiz i Issledovaniye Operatsyi 10:17–55, 2003) the second characterization says that a Harsanyi payoff vector is a random order value iff the sharing system is strong monotonic.


TU-games Harsanyi set Weber set Selectope Monotonic sharing systems 

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  1. Derks J (2005) A new proof for Weber’s characterization of the random order values. Math Soc Sci 49:327–334CrossRefzbMATHMathSciNetGoogle Scholar
  2. Derks J, Peters H (2002) A note on a consistency property for permutations. Discrete Math 250:241–244CrossRefzbMATHMathSciNetGoogle Scholar
  3. Derks J, Haller H, Peters H (2000) The selectope for cooperative games. Int J Game Theory 29:23–38CrossRefzbMATHMathSciNetGoogle Scholar
  4. Dragan I (1994) Multiweighted Shapley values and random order values. In: Proceedings of the 10th conference on applied mathematics, U.C. Oklahoma, pp 33–47Google Scholar
  5. Hammer PL, Peled UN, Sorensen S (1977) Pseudo-boolean functions and game theory. I. Core elements and Shapley value. Cahiers Centre Etudes Rech Oper 19:159–176zbMATHMathSciNetGoogle Scholar
  6. Harsanyi JC (1959) A bargaining model for cooperative n-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games vol IV. Princeton University Press, Princeton, pp 325–355Google Scholar
  7. Harsanyi JC (1963) A simplified bargaining model for the n-person game. Int Econ Rev 4:194–220zbMATHCrossRefGoogle Scholar
  8. Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, vol II. Princeton University Press, Princeton, pp 307–317Google Scholar
  9. Vasil’ev VA (1978) Support function of the core of a convex game. Optimizacija Vyp 21:30–35 (in Russian)Google Scholar
  10. Vasil’ev VA (1980) On the H-payoff vectors for cooperative games. Optimizacija Vyp 24:18–32 (in Russian)Google Scholar
  11. Vasil’ev VA (1981) On a class of imputations in cooperative games. Soviet Math Dokl 23:53–57zbMATHGoogle Scholar
  12. Vasil’ev VA (1988) Characterization of the cores and generalized NM-solutions for some classes of cooperative games. Proc Inst Math Novosibirsk Nauk 10:63–89 (in Russian)Google Scholar
  13. Vasil’ev VA (2003) Extreme points of the Weber polyhedron. Discretnyi Analiz i Issledovaniye Operatsyi Ser.1 10:17–55 (in Russian)Google Scholar
  14. Vasil’ev VA, Laan, G van der (2002) The Harsanyi set for cooperative TU-games. Siberian Adv Math 12:97–125zbMATHGoogle Scholar
  15. Weber RJ (1988) Probabilistic values for games. In: Roth AE (eds) The Shapley value, Essays in honor of L.S. Shapley. Cambridge University Press, Cambridge, pp 101–119Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Jean Derks
    • 1
  • Gerard van der Laan
    • 2
  • Valery Vasil’ev
    • 3
  1. 1.Department of MathematicsUniversiteit MaastrichtMaastrichtThe Netherlands
  2. 2.Department of Econometrics and Tinbergen InstituteVrije UniversiteitAmsterdamThe Netherlands
  3. 3.Sobolev Institute of MathematicsNovosibirskRussia

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