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Solutions of three functional equations arising from different ways of measuring utility. (English) Zbl 0870.90019

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 \(]0,1[\)).

MSC:

91B16 Utility theory
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