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Remarkable polyhedra related to set functions, games and capacities

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Abstract

Set functions are widely used in many domains of operations research (cooperative game theory, decision under risk and uncertainty, combinatorial optimization) under different names (TU-game, capacity, nonadditive measure, pseudo-Boolean function, etc.). Remarkable families of set functions form polyhedra, e.g., the polytope of capacities, the polytope of p-additive capacities, the cone of supermodular games, etc. Also, the core of a set function, defined as the set of additive set functions dominating that set function, is a polyhedron which is of fundamental importance in game theory, decision-making and combinatorial optimization. This survey paper gives an overview of these notions and studies all these polyhedra.

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Notes

  1. In general, normalization is not required. We assume it throughout this paper for convenience.

  2. As far as possible, we stick to these notations: \(\xi \) for set functions, v for games and \(\mu \) for capacities.

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Authors and Affiliations

  1. Paris School of Economics, University of Paris I, 106-112, Bd de l’Hôpital, 75013, Paris, France

    Michel Grabisch

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  1. Michel Grabisch
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Correspondence to Michel Grabisch.

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This invited paper is discussed in the comments available at doi:10.1007/s11750-016-0418-z, doi:10.1007/s11750-016-0419-y, doi:10.1007/s11750-016-0420-5.

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Grabisch, M. Remarkable polyhedra related to set functions, games and capacities. TOP 24, 301–326 (2016). https://doi.org/10.1007/s11750-016-0421-4

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