Solution of a functional equation arising in an axiomatization of the utility of binary gambles

Abstract

For a new axiomatization, with fewer and weaker assumptions, of binary rank-dependent expected utility of gambles the solution of the functional equation \[ ( z / p ) γ − 1 [ z γ ( p ) ] = φ − 1 [ φ ( z ) ψ ( p ) ] ( z , p ∈ ] 0 , 1 [ ) (z/p)\gamma ^{-1}[z\gamma (p)] = \varphi ^{-1}[\varphi (z)\psi (p)] \qquad (z,p\in ]0,1[) \] is needed under some monotonicity and surjectivity conditions. We furnish the general such solution and also the solutions under weaker suppositions. In the course of the solution we also determine all sign preserving solutions of the related general equation \[ h ( u ) [ g ( u + v ) − g ( v ) ] = f ( v ) g ( u + v ) ( u ∈ R + ,   v ∈ R ) . h(u)[g(u+v)-g(v)]=f(v)g(u+v)\qquad (u\in \mathbb {R}_+,\ v\in \mathbb {R}). \]