The core of bicapacities and bipolar games

Abstract

Bicooperative games generalize classical cooperative games in the sense that a player is allowed to play in favor or against some aim, besides non participation. Bicapacities are monotonic bicooperative games, they are useful in decision making where underlying scales are of bipolar nature, i.e., they distinguish between good/satisfactory values and bad/unsatisfactory ones. We propose here a more general framework to represent such situations, called bipolar game. We study the problem of finding the core of such games, i.e., the set of additive dominating games.

m-Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets

“…m -polar fuzzy sets can be used in cooperative games (cf. [ 51 ]). Let X = { x 1 , x 2 ,…, x n } be the set of n agents or players ( n ≥ 1), m = {0,1,…, m − 1} be the set of the grand coalitions, and A : X → [0,1] m be an m -polar fuzzy set, where p i ∘ A ( x ) is the degree of player x participating in coalition i ( x ∈ X , i ∈ m ).…”

“…(1) (a public good game; compare with [ 51 , Example 6.5]) Suppose n agents x 1 , x 2 ,…, x n want to create a facility for joint use. The cost of the facility depends on the sum of the participation levels (or degrees) of the agents and it is described by where k : [0, n ] → R is a continuous monotonic increasing function with k (0) = 0 and B : X → [0,1] is a mapping.…”

Pointed order polytopes: Studying geometrical aspects of the polytope of bi-capacities

“…In the definition of bi-capacities, it is taken into account that for an object some criteria might be completely satisfactory, some completely unsatisfactory and some have a neutral score. Hence, they are a suitable option for dealing with these situations and several papers based on this concept have appeared since then [22,18,27,19,1].…”