, Volume 6, Issue 2, pp 119–123 | Cite as

Permutation games: Another class of totally balanced games

  • S. H. Tijs
  • T. Parthasarathy
  • J. A. M. Potters
  • V. Rajendra Prasad
Theoretical Papers


A class of cooperative games in characteristic function form arising from certain sequencing problems and assignment problems, is introduced. It is shown that games of this class are totally balanced. In the proof of this fact we use the Birkhoff-von Neumann theorem on doubly stochastic matrices and the Bondareva-Shapley theorem on balanced games. It turns out that this class of permutation games coincides with the class of totally balanced games if the number of players is smaller than four. For larger games the class of permutation games is a nonconvex subset of the convex cone of totally balanced games.


Characteristic Function Assignment Problem Convex Cone Function Form Cooperative Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Wir führen eine Klasse von kooperativen Spielen in charakteristischer Funktionsform ein, die bei gewissen Folgeproblemen und Zuordnungsproblemen auftreten. Wir zeigen, daß diese Spiele vollständig balanciert sind. Zum Beweis verwenden wir den Satz von Birkhoff-von Neumann über doppelt stochastische Matrizen und den Satz von Bondareva-Shaplex über balancierte Spiele. Es zeigt sich, daß diese Klasse von Permutationsspielen mit der Klasse von vollständig balancierten Spiele übereinstimmt, falls die Zahl der Spieler kleiner als vier ist. Für größere Spiele ist die Klasse der Permutationsspiele eine nichtkonvexe Teilmenge des konvexen Kegels der vollständig balancierten Spiele.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Billera LJ (1981) Economic market games. Game theory and its applications. Proceedings of Symposia in Applied Mathematics 24:37–53CrossRefGoogle Scholar
  2. 2.
    Birkhoff G (1948) Lattice theory. Revised edition Am Math Soc Coll Series 25:266Google Scholar
  3. 3.
    Bondareva ON (1963) Some applications of linear programming methods to the theory of cooperative games (in Russian). Probl Kibern 10:119–139Google Scholar
  4. 4.
    Dubey P, Shapley L (1982) Totally balanced games arising from controlled programming problems. Discussion paperGoogle Scholar
  5. 5.
    Granot D, Huberman G (1981) Minimum cost spanning tree games. Math Programming 21:1–18CrossRefGoogle Scholar
  6. 6.
    Kalai E, Zemel E (1982) Totally balanced games and games of flow. Math Oper Res 7:476–478CrossRefGoogle Scholar
  7. 7.
    Kalai E, Zemel E (1982) Generalized network problems yielding totally balanced games. Oper Res 30:998–1008CrossRefGoogle Scholar
  8. 8.
    Neumann J von (1953) A certain zero-sum two-person game equivalent to the optimal assignment problem. Contributions to the Theory of Games II:5–12Google Scholar
  9. 9.
    Owen G (1975) On the core of linear production games. Math Programming 9:358–370CrossRefGoogle Scholar
  10. 10.
    Shapley LS (1967) On balanced sets and cores. Nav Res Logistics Q 14:453–460CrossRefGoogle Scholar
  11. 11.
    Shapley LS (1983) Private CommunicationGoogle Scholar
  12. 12.
    Shapley L, Scarf H (1974) On cores and indivisibility. J Math Econom 1:23–37CrossRefGoogle Scholar
  13. 13.
    Shapley LS, Shubik M (1969) On market games. J Econom Theory 1:9–25CrossRefGoogle Scholar
  14. 14.
    Shapley LS, Shubik M (1972) The assignment game I: The core. Int J Game Theory 1:111–130CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • S. H. Tijs
    • 1
  • T. Parthasarathy
    • 2
  • J. A. M. Potters
    • 1
  • V. Rajendra Prasad
    • 2
  1. 1.Department of MathematicsCatholic UniversityED NijmegenThe Netherlands
  2. 2.Indian Statistical InstituteNew DelhiIndia

Personalised recommendations