F. S. Roberts [ibid. 147, No.1, 30-52 (1990; Zbl 0704.92026)] formulated several properties expected from merging functions, say, of performance scores, relative to some basic performance and asked about all merging functions which have these individual properties (and some combinations of such properties). The present paper answers many of these questions and uses the answers to give alternative proofs and generalizations of some of Robert’s results on the relative strength of these properties.
Two typical results in this paper are: U: \({\mathbb{R}}^ n_+\times {\mathbb{R}}^ n_+\to {\mathbb{R}}\) satisfies \[ \forall x,y,s,t\in {\mathbb{R}}^ n_+,\quad c\in {\mathbb{R}}_+,\exists z\in {\mathbb{R}}^ n_+:\;U(x,z)- U(s,z)=c[U(y,z)-U(t,z)]\Rightarrow \]
\[ \forall w\in {\mathbb{R}}^ n_+:\;U(x,w)-U(s,w)=c[U(y,w)-U(t,w)]\quad if,\text{ and } only\quad if, \]
\[ \exists F,H:\;{\mathbb{R}}^ n_+\to {\mathbb{R}},\quad G:\;{\mathbb{R}}^ n_+\to {\mathbb{R}}\setminus \{0\},\quad \forall x,z\in {\mathbb{R}}^ n_+:\;U(x,z)=F(x)G(z)+H(z). \] (The positivity of c causes some minor difficulties). The continuous u: \({\mathbb{R}}^ n_+\to {\mathbb{R}}_+\) \(satisfies:\)
\(\forall p,q\in {\mathbb{R}}^ n_+:\) \([u(p)>u(q)\Rightarrow (\forall r\in {\mathbb{R}}^ n:\) \(u(r,p)>u(r,q))]\) (vector-multiplication is done componentwise) if, and only if there exist a continuous, strictly monotonic h: \({\mathbb{R}}_+\to {\mathbb{R}}_+\) and constants \(c_ 1,...,c_ n\in {\mathbb{R}}\) such that: \[ u(p)=u(p_ 1,...,p_ n)=h(p_ 1^{c_ 1}p_ 2^{c_ 2}...p_ n^{c_ n}),\quad (p=(p_ 1,...,p_ n)\in {\mathbb{R}}^ n_+). \] Several results in both papers point towards the preferability of geometric means as merging functions.