Random Sets

Abstract

In this chapter we address the problem of aggregating a number of expert opinions which have been expressed by vague data. In our approach a meta-expert attaches nonnegative weights of importance to each of his or her experts θ_i ∈ Θ i = 1,..., n. Therefore we have to discuss the problem of finding suitable weights, integrating pieces of certain knowledge and giving methods for decision making. To clarify our basic idea let us consider a set Ξ (which is for the moment assumed to be finite) containing n “atomic agents” each of which is equally important. If we imagine these atomic agents voting on some motion, then the weight of some “party” A ⊆ E is given by |A|. So selecting agents ξ ∈ Ξ randomly would yield, in a long run, members of A in $(\tfrac{1}{n}\left| A \right|\bullet 100) \frac{\hbox{$\scriptstyle 0$}}{\hbox{$\scriptstyle 0$}} P(A) = \frac{1}{n}\left| A \right|$ of all cases. Thus we consider $P(A) = \frac{1}{n}\left| A \right|$ to be the probability of A. Unfortunately, in general not the space E but only a projection Θ is accessible, where φ may denote the point-wise projection mapping, i.e. φ: Ξ → Θ. On the intuitive level 0 represents an assembly of “floor leaders”. Thus the weight of each θ ∈ Θ is given by the “probability” of φ^-(θ) = {ξ ∈ Ξ | φ(ξ) = θ}, which means it is determined by the number of supporters. So we can interpret the importance weights here as a probability (with a classical or frequentistic interpretation).