Functional equations arising in a theory of rank dependence and homogeneous joint receipts

Abstract

This paper focuses on a class of utility representations of uncertain alternatives with two possible consequences (binary gambles) when they are linked via a distributivity property called segregation to an operation of joint receipt, which may be non-commutative. The assumption that the gambling structure and the joint receipt operation both have homogeneous representations that are order preserving leads to a functional equation that has too many solutions to be useful for characterizing a reasonably specific utility representation. A plausible restriction on the form of the utility of gambles leads to the functional equation H[pw,qK(w)]=qθ −1 ψ(w)θ p q (w∈[0,1],q∈]0,k[,p∈[q,k[) whose solution shows that it is equivalent to the widely studied rank-dependent representation. That representation is related to segregation via a particular homogeneous representation of joint receipt. In that case the above functional equation simplifies to G[vF(z)]=A(v)G(vz)+G(v) (v>0,z⩾0), and it is solved under smoothness conditions. Two families of solutions arise, one commutative and associative and the other non-commutative and bisymmetric, but not associative. These are then interpreted in utility terms. The former is well studied. Parallel to earlier results for the commutative case, an axiomatic treatment of the non-commutative case is provided. An application to psychophysics favors non-commutativity. Several open problems are mentioned.