Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Minimarg and maximarg operators

  • 62 Accesses

  • 8 Citations


Two operators on the set ofn-person cooperative games are introduced, the minimarg operator and the maximarg operator. These operators can be seen as dual to each other. Some nice properties of these operators are given, and classes of games for which these operators yield convex (respectively, concave) games are considered. It is shown that, if these operators are applied iteratively on a game, in the limit one will yield a convex game and the other a concave game, and these two games will be dual to each other. Furthermore, it is proved that the convex games are precisely the fixed points of the minimarg operator and that the concave games are precisely the fixed points of the maximarg operator.

This is a preview of subscription content, log in to check access.


  1. 1.

    Shapley, L. S.,A Value for n-Person Games, Contributions to the Theory of Games, II, Edited by H. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, pp. 307–317, 1953.

  2. 2.

    Schmeidler, D.,Cores of Exact Games, I, Journal of Mathematical Analysis and Applications, Vol. 40, pp. 214–225, 1972.

  3. 3.

    Weber, R. J.,Probabilistic Values for Games, The Shapley Value, Essays in Honor of Lloyd S. Shapley, Edited by A. E. Roth, Cambridge University Press, Cambridge, Massachusetts, pp. 101–119, 1988.

  4. 4.

    Shapley, L. S.,Cores of Convex Games, International Journal of Game Theory, Vol. 1, pp. 11–26, 1971.

  5. 5.

    Rosenmüller, J.,Extreme Games and Their Solutions, Springer-Verlag, Berlin, Germany, 1977.

  6. 6.

    Ichiishi, T.,Super-Modularity: Applications to Convex Games and to the Greedy Algorithm for LP, Journal of Economic Theory, Vol. 25, pp. 283–286, 1981.

  7. 7.

    Kikuta, K.,A Condition for a Game to Be Convex, Mathematica Japonica, Vol. 33, pp. 425–430, 1988.

  8. 8.

    Monderer, D., Samet, D., andShapley, L. S.,Weighted Values and the Core, Working Paper, University of California, Los Angeles, California, 1988.

Download references

Author information

Additional information

Communicated by P. L. Yu

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Curiel, I., Tijs, S. Minimarg and maximarg operators. J Optim Theory Appl 71, 277–287 (1991). https://doi.org/10.1007/BF00939921

Download citation

Key Words

  • Convex games
  • marginal contributions
  • duality
  • symmetric games