Solutions to three functional equations arising from different ways of measuring utility

Abstract

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 F-1[F(X) + F(-Y)]Z = F-1[F(XZ) + F(-YZ)] (F: ] - k, k′[ → ] - K, K′[; k, k′, K, K′ > 0) (i) F(X - R)[1 - F(Y)] + F(Y) = F[F-1(F(X)[1 - F(Y)] + F(Y)) - S] (F: [0, 1[→ [0, 1[) (ii) ACZÉL, LUCE, AND MAKSA 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)]] (F: [0, 1[→ [0, 1[, P: [0, 1[×[0, 1] → [0, 1]). (iii) We determine all strictly increasing, surjective (and thus continuous) solutions of (i) and (ii) and all strictly increasing, surjective solutions of (iii) that are differentiable on [0, 1[ as are their inverses (thus, F′ ≠ 0 on ]0, 1[). © 1996 Academic Press, Inc.