On comparison meaningfulness of aggregation functions
An aggregation function is a function \(M: [a,b]^n\to \mathbb R\), with \([a,b]\) an interval of reals; it is assumed continuous. One may think of \(x^A\in[a,b]^n\) as the vector of values assigned to some outcome \(A\in{\mathcal A}\) by a group of \(n\) people -- \(x^A_i\) being the value of \(A\) for subject \(i\). \(M\) is comparison meaningful if the inequalities \(M(x)\geq M(x')\) are preserved under monotonic transformations of \(x,x'\); and idempotent if \(M(x,x,\dots,x)=x\); and symmetric if it is invariant under permutations. It is known that such aggregators are all equal to some order statistic, that is \(M(x)=x_{(k)}\) for some \(k\), with \(x_{(k)}\) being the \(k\)-th larger element of \(x\). The main result of the present paper is that if one drops idempotency, the non-constant \(M\)'s are increasing transformations of order statistics.
scientific article; zbMATH DE number 1750018