Remarkable polyhedra related to set functions, games and capacities

Abstract

International audienceSet 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, decicion making and com-binatorial optimization. This survey paper gives an overview of these notions and studies all these polyhedra

The size of the maximum antichains in products of linear orders

“…If a game is monotone, then the set of all minimal winning profiles is an antichain. Besides, [7] shows that antichains in {1, . .…”

The size of the largest antichains in products of linear orders

“…If a game is monotone, then the set of all minimal winning profiles is an antichain. Besides, Grabisch (2016) shows that antichains in {1, . .…”