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On the Set of Balanced Games

Author

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  • Pedro Garcia-Segador

    (National Statistical Institute, 28046 Madrid, Spain)

  • Michel Grabisch

    (Université Paris I Panthéon-Sorbonne, Centre d’Economie de la Sorbonne, 75013 Paris, France; and Paris School of Economics, 75014 Paris, France)

  • Pedro Miranda

    (Complutense University of Madrid, 28040 Madrid, Spain)

Abstract

We study the geometric structure of the set of cooperative transferable utility games having a nonempty core, characterized by Bondareva and Shapley as balanced games. We show that this set is a nonpointed polyhedral cone, and we find the set of its extremal rays and facets. This study is also done for the set of balanced games whose value for the grand coalition is fixed, which yields an affine nonpointed polyhedral cone. Finally, the case of nonnegative balanced games with fixed value for the grand coalition is tackled. This set is a convex polytope, with remarkable properties. We characterize its vertices and facets, study the adjacency structure of vertices, develop an algorithm for generating vertices in a random uniform way, and show that this polytope is combinatorial and its adjacency graph is Hamiltonian. Last, we give a characterization of the set of games having a core reduced to a singleton.

Suggested Citation

  • Pedro Garcia-Segador & Michel Grabisch & Pedro Miranda, 2025. "On the Set of Balanced Games," Mathematics of Operations Research, INFORMS, vol. 50(3), pages 2047-2072, August.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:3:p:2047-2072
    DOI: 10.1287/moor.2023.0379
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