A functional equation arising from ranked additive and separable utility (Q2701614)

The equation in question reads NEWLINE\[NEWLINE f(v) = f(uv) + f(g^{-1}(g(v)q(w))), \quad v \in [0,k[ (0 < k \leq + \infty),\;w \in [0,1], \tag{1}NEWLINE\]NEWLINE where \(f: [0,k[ \to [0, + \infty[\), \(q: [0,1] \to [0,1],\) and \(g^{-1}\) denotes the inverse function of \(g: [0,k[ \to [0,k[.\) NEWLINENEWLINENEWLINEA more general equation than (1), written additively, viz. NEWLINE\[NEWLINE F(t) - F(t+s) = H(G(t) + Q(s)), \quad t \in I := [k, + \infty)\;(k \geq - \infty),\;s \in {\mathbb R}_+: = ]0, + \infty[, \tag{2}NEWLINE\]NEWLINE is solved under the assumptions: \(F: I \to {\mathbb R}\); \(G: I \to {\mathbb R} \) is strictly monotonic; \(Q: {\mathbb R}_+ \to {\mathbb R}\); \(H: G(I) + Q({\mathbb R}_+) \to {\mathbb R}_+ \) is strictly monotonic. NEWLINENEWLINENEWLINEHaving derived strict Jensen convexity (or concavity) of \(F\), hence the existence of the derivatives: \(F'_+, Q'\) and \(G'_+\) - everywhere; the authors reduce (2) to the functional-differential equation of the form NEWLINE\[NEWLINE \gamma(s)(\psi(t+s) - \psi(t)) = \varphi(t) \psi(t+s), \quad t \in I,\;s \in \mathbb{R}_+ \tag{3}NEWLINE\]NEWLINE (with \(\gamma: = Q'\), \(\varphi: = G'_+\), \(\psi: = F'_+\)). Equation (3) has been solved by \textit{J. Aczél, Gy. Maksa} and \textit{Zs. Palés} [``Solution of a functional equation arising in an axiomatization of the utility of binary gambles'', Proc. Am. Math. Soc. 129, No. 2, 483-493 (2001; Zbl 0963.39026)] in the case \(I ={\mathbb R}\). Here sign preserving solutions \(\varphi, \psi: I \to {\mathbb R}\), \(\gamma: {\mathbb R}_+ \to {\mathbb R}\) are determined when \(I\) is defined in (2). They are either some homographies or homographies composed with exponential functions. This yields the main result of the paper: the solution \((f,g,q)\) of (1) under the assumptions: \(f: [0,k[ \to {\mathbb R}_+\); \(g: [0,k[ \to [0,k'[\) is strictly monotonic and surjective; \(q: [0,1] \to [0,1]\) (with some \(k, k' \in ]0, + \infty]\)). NEWLINENEWLINENEWLINEFrom their main result the authors obtain an explicit expression for the expected utility \(U(x,C;y, \overline{C})\) from a gamble \((x,C;y, \overline{C})\) (the event \(C\) has consequence \(x\), the complementary event \(\overline{C}\) has consequence \(y\)), as axiomatized by \textit{R. D. Luce} and \textit{A. A. J. Marley} [``Separable and additive utility of binary gambles of gains'', Math. Social Sci., in press]. When \(g = id\) in (1) [corresponding to the case where both consequences have the same separable representation, cf. \textit{R. D. Luce}, J. Risk Uncertain. 16, No. 1, 87-114 (1998; Zbl 0911.90038)] the result is equivalent to that by \textit{J. Aczél, R. Ger} and \textit{A. Járai} [Proc. Am. Math. Soc. 127, No. 10, 2923-2929 (1999; Zbl 0930.39019)].