Join Coherent Previsions

Abstract We prove that the theory of coherent previsions by Bruno de Finetti can be framed in the theory of join systems, where a join system is a set with a hyperoperation satisfying a given set of “geometric” properties. A prevision by Bruno de Finetti is a point of a particular Euclidean space and the conditions for a prevision is coherent are the conditions for such prevision belongs to a suitable convex subset. In this paper, by considering an abstract join system we give a geometrical generalization of the coherent prevision. Precisely we assume a join prevision is a point of a join system and we introduce the concept of join coherent prevision by considering some geometrical conditions that reduce to the ones of de Finetti coherence conditions if the join system considered is an Euclidean space. We emphasize that the concept of join coherent prevision is meaningful if we consider join systems having particular “intuitive” geometrical properties, as join spaces and join geometries. The join coherence conditions can have many applications: they can be utilized to have criteria for a rational assessment of scores and their aggregation in multi-objective decision making problems. In particular we obtain useful tools to introduce reasonable consistence conditions in a fuzzy ambit as a generalization of the coherence conditions in the probabilistic ambit.