The Bernoulli Principle and the Dirichlet Problem

Abstract

Let M be a class of lotteries where a lottery is defined by a probability measure μ on the real line R. Among all preference relations ≲ on the lotteries in M. the Bernoulli principle singles out those which are based on expected utility in the sense that

$$\mu \lesssim v$$

if and only if

$$\int {{\text{ud}}} \mu \leqslant \int {{\text{ud}}v \left( {\mu ,v \in {\text{M}}} \right)} $$

for some utility function u on R. For a parametric model M =( μ_x) _{X εE} we are thus led to consider those functions h on the parameter space E, let us call them Bernoulli functions for M, which are of the form

$${\text{h}}\left( {\text{x}} \right) = \int {{\text{ud}}{\mu _{\text{X}}}{\text{ }}\left( {{\text{x}} \in {\text{E}}} \right).} $$

.