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

Author

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  • P. García-Segador

    (National Statistics Institute)

  • P. Miranda

    (Complutense University of Madrid)

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.

Suggested Citation

  • P. García-Segador & P. Miranda, 2020. "Order cones: a tool for deriving k-dimensional faces of cones of subfamilies of monotone games," Annals of Operations Research, Springer, vol. 295(1), pages 117-137, December.
    • Handle: RePEc:spr:annopr:v:295:y:2020:i:1:d:10.1007_s10479-020-03712-7
    DOI: 10.1007/s10479-020-03712-7
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    References listed on IDEAS

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    1. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    2. Michel Grabisch & Tomáš Kroupa, 2018. "The core of supermodular games on finite distributive lattices," Documents de travail du Centre d'Economie de la Sorbonne 18010, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    3. Michel Grabisch, 2016. "Set Functions, Games and Capacities in Decision Making," Theory and Decision Library C, Springer, number 978-3-319-30690-2, March.
    4. Michel Grabisch, 2011. "Ensuring the boundedness of the core of games with restricted cooperation," Annals of Operations Research, Springer, vol. 191(1), pages 137-154, November.
    5. Katsev, Ilya & Yanovskaya, Elena, 2013. "The prenucleolus for games with restricted cooperation," Mathematical Social Sciences, Elsevier, vol. 66(1), pages 56-65.
    6. Pulido, Manuel A. & Sanchez-Soriano, Joaquin, 2006. "Characterization of the core in games with restricted cooperation," European Journal of Operational Research, Elsevier, vol. 175(2), pages 860-869, December.
    7. Pedro Miranda & Michel Grabisch & Pedro Gil, 2002. "p-symmetric fuzzy measures," Post-Print hal-00273960, HAL.
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