Aczél, János; Luce, R. Duncan; Maksa, Gyula Solutions of three functional equations arising from different ways of measuring utility. (English) Zbl 0870.90019 J. Math. Anal. Appl. 204, No. 2, 451-471 (1996). Summary: Utility of gains (losses) can be measured in four distinct ways: riskless vs risky choices and gains (losses) alone vs the gain-loss trade-off Conditions forcing these measures all to be the same lead to functional equations, three of which are(i) \(F:]-k,k'[\to ]- K,K'[;\;k,k',K,K'>0\);\(F^{-1}[F(X)+ F(-Y)]Z= F^{-1}[F(XZ)+ F(-YZ)]\),(ii) \(F:[0,1[\to [0,1[\);\(F(X- R)[1- F(Y)]+ F(Y)= F[F^{-1}(F(X)[1- F(Y)]+ F(Y))-S]\),(iii) \(F:[0,1[\to [0,1[,\;P:[0,1[\times [0,1]\to [0,1]\);\(F^{-1}[F(X)+ F(Y)- F(X)F(Y)]Z= F^{-1}[F(XZ)+ F[YP(X,Z)]- F(XY)F[YP(X,Z)]]\).We determine all strictly increasing, surjective (and thus continuous) solutions of (i) and (ii) and all strictly increasing, subjective solutions of (iii) that are differentiable on \([0,1[\) as are their inverses (thus, \(F'\neq 0\) on \(]0,1[\)). Cited in 2 ReviewsCited in 4 Documents MSC: 91B16 Utility theory Keywords:utility of gains PDFBibTeX XMLCite \textit{J. Aczél} et al., J. Math. Anal. Appl. 204, No. 2, 451--471 (1996; Zbl 0870.90019) Full Text: DOI