Core-based criterion for extreme supermodular functions

Abstract

We give a necessary and sufficient condition for extremality of a supermodular function based on its min-representation by means of (vertices of) the corresponding core polytope. The condition leads to solving a certain simple linear equation system determined by the combinatorial core structure. This result allows us to characterize indecomposability in the class of generalized permutohedra. We provide an in-depth comparison between our result and the description of extremality in the supermodular/submodular cone achieved by other researchers.

The intermediate set and limiting superdifferential for coalitional games: between the core and the Weber set

“…, |C i \ C i−1 |. Sincev i is linear on very particular domains and since y lies in the convex hull of the above points, we obtain that formula (22) holds also for y. This finishes the proof.…”

“…The family of all coalitional chains in the player set N is in one-to-one correspondence with the set of all nonempty faces of the permutohedron of order n; see [26]. An idea is to look at the relation between the algebraic structure of the corresponding face The coincidence of the core with the Weber set is essential for the characterization of extreme rays of the cone of supermodular games presented in [22]. There can be a large gap between the core and the Weber set outside the family of supermodular games.…”

Coxeter submodular functions and deformations of Coxeter permutahedra

“…For example, the matroid polytope P M of any connected matroid M on [d] is a ray of Nef(Σ Φ ). [39,52]. Therefore the number of rays of this nef cone is doubly exponential, because the asymptotic proportion of matroids that are connected is at least 1/2 and conjecturally equal to 1 [36] and the number m d of matroids on [d] satisfies log log m d ≥ d − 3 2 log d − O(1).…”

Set Functions, Games and Capacities in Decision Making

“…Later, Studený and Kroupa [317] find out another criterion for finding the extremal rays of G Þ .X/, based on the fact that supermodular games are exact and their core coincides with the Weber set (see Chap. 3 and especially Sect.…”