Order cones: a tool for deriving k-dimensional faces of cones of subfamilies of monotone games

Abstract

In this paper we introduce the concept of order cone. This concept is inspired by the concept of order polytopes, a well-known object coming from Combinatorics with which order cones share many properties. Similarly to order polytopes, order cones are a special type of polyhedral cones whose geometrical structure depends on the properties of a partially ordered set (brief poset). This allows to study the geometrical properties of these cones in terms of the subjacent poset, a problem that is usually simpler to solve. Besides, for a given poset, the corresponding order polytope and order cone are deeply related. From the point of view of applicability, it can be seen that many cones appearing in the literature of monotone TU-games are order cones. Especially, it can be seen that the cones of monotone games with restricted cooperation are order cones, no matter the structure of the set of feasible coalitions and thus, they can be studied in a general way applying order cones.

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Notes

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    We are very grateful to an anonymous reviewer for focusing our attention on braid cones.

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Correspondence to P. Miranda.

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This paper has been supportesd by the Ministry of Economy and Competitiveness of Spain under Grant PGC2018-095194-B-100 and by the Interdisciplinary Mathematical Institute of Complutense University.

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García-Segador, P., Miranda, P. Order cones: a tool for deriving k-dimensional faces of cones of subfamilies of monotone games. Ann Oper Res 295, 117–137 (2020). https://doi.org/10.1007/s10479-020-03712-7

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Keywords

  • Monotone games
  • Restricted cooperation
  • Order polytope
  • Cone