Centroids of the core of exact capacities: a comparative study

Abstract Capacities are a common tool in decision making. Each capacity determines a core, which is a polytope formed by additive measures. The problem of eliciting a single probability from the core is interesting in a number of fields: in coalitional game theory for selecting a fair way of splitting the wealth between the players, in the transferable belief model from evidence theory or for transforming a second order into a first order model. In this paper, we study this problem when the goal is to determine the centroid of the core of a capacity, and we compare four approaches: the Shapley value, the average of the extreme points, the incenter with respect to the total variation distance and the limit of a procedure of uniform contraction. We show that these four centroids do not coincide in general, we give some sufficient conditions for their equality, and we analyse their axiomatic properties. We also discuss how to define a notion of centrality measure indicating the degree of centrality of an additive measure in the core. Finally, we also analyse these four centroids in the more general context of imprecise probabilities.