A functional equation generated by event commutativity in separable and additive utility theory

Summary.

An uncertain alternative or binary gamble is a triple (a,C;b), where C is a chance event, a and b are the consequences (they may be gambles themselves) if C or “non-C” happens, respectively. There is a (partial) preference order $\succsim$ between gambles, which generates one also between consequences. Several plausible assumptions about gambles are made, among them the “event commutativity”¶¶$((a,C;b),D;b)\sim((a,D;b),C;b)$.¶Looking for a representation of the utility of gambles, given the weights of events and the utility of consequences, this relation leads to the functional equation¶¶$\varphi(\varphi^{-1}[\varphi(xw)+\varphi(y)-\varphi(yw)]z)-\varphi(yz)$¶$= \varphi (\varphi^{-1}[\varphi(xz)+\varphi(y)-\varphi(yz)]w) -\varphi(yw)$¶$(0 \le y \le x$ < $K;\, z,w \in [0,1])$¶$(\varphi:[0,K[\, \to [0,+\infty[\,)$. In order for this equation to make sense, $\varphi(xw)+\varphi(y)-\varphi(yw)$ has to be in the codomain (set of function values) of $\varphi$. In addition it is assumed that $\varphi$ is twice differentiable and $\varphi'$ is nowhere zero. Under these assumptions all solutions are determined.