Utility of a Distribution

LET U(G|F) denote the utility of the assertion that the distribution of a random vector x is G when it is really F. Although the utility must depend on the context it is interesting to consider what form the functional U might take if it is some form of generalized expectation of v(x,y), defined as the utility of asserting that the value of the random vector is y when it is really x. We assume that v(x,y)≤v(x,x). Natural axioms are (i) if a constant is added to v then the same constant is added to U; (ii) additivity for mutually irrelevant vectors (iii) invariance under non-singular transformations of x: U(G|F) is unchanged if a non-singular transformation x = ψ(x′), y = ψ(y′) is made, subject to the obvious desideratum that the transformed form of v is v(ψ(x′), ψ(y′)).