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

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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. 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 these properties in terms of the subjacent poset, a problem that is usually simpler to solve. 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.
Bandlow, J.2008. An elementary proof of the hook formula. The Electronic Journal of Combinatorics,(15). Research paper 45. Choquet, G.1953. Theory of capacities. Annales de l'Institut Fourier, (5):131-295. Combarro, E. F. and Miranda, P. 2008. On the polytope of non-additive measures. Fuzzy Sets andSystems, 159(16):2145-2162. Combarro, E. F. and Miranda, P. 2010. Adjacency on the order polytope with applications to the theory of fuzzy measures. Fuzzy Sets and Systems, 180:384-398. Cook, W., Cunningham, W., Pulleyblank, W. and Schrijver, A. 1988. Combinatorial Optimization,Wiley Series in Discrete Mathematics and Optimization. Wiley. Davey, B. A. and Priestley, H. A. 2002. Introduction to lattices and order. Cambridge University Press. Denneberg, D. 1994. Non-additive measures and integral. Dordrecht (The Netherlands): KluwerAcademic. Faigle, U. 1989. Cores of games with restricted cooperation. Zeitschrift f�ur Operations Research, 33(6):405-422. Friedl, T. 2017. Double posets and real invariant varieties: Two interactions between combinatorics and geometry. PhD thesis, Fachbereich Mathematik und Informatik der Freien Universit�at Berlin- Tokyo Institute of Technology. García-Segador, P. and Miranda, P. 2020. Applying young diagrams to 2-symmetric fuzzy measures with an application to general fuzzy measures. Fuzzy Sets and Systems, 379:20-36. Grabisch, M. 2011. Ensuring the boundedness of the core of games with restricted cooperation. Annals of Operations Research, 191:137-154. Grabisch, M. 2013. The core of games on ordered structures and graphs. Annals of Operations Research, 204:33{64. doi: 10.1007/s10479-012-1265-4. Grabisch, M. 2016. Set functions, games and capacities in Decision Making, volume 46 of Theory and Decision Library. Springer. Grabisch, M. and Kroupa, T. 2019. The cone of supermodular games on �nite distributive lattices. Discrete Applied Mathematics, 260:144-154. Katsev, I. and Yanovskaya, E. 2013. The prenucleolus for games with restricted cooperation. Mathematical Social Sciences, 66(1):56-65. Miranda, P., Grabisch, M. and Gil, P. 2002. p-symmetric fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10 (Suppl.):105-123. Pulido, M. and S�anchez-Soriano, J. 2006. Characterization of the core in games with restricted cooperation. European Journal of Operational Research, 175(2):860-869. Shapley, L. S.1971. Cores of convex games. International Journal of Game Theory, 1:11-26. Stanley, R.1986. Two poset polytopes. Discrete Comput. Geom., 1(1):9-23. Sugeno, M. 1974. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology.