Positive-difference structures and bilinear utility functions

Let (M₁, f), (M₂, g) be mixture sets and let ? be a binary preference relation on M₁ × M₂. By using the concept of positive-difference structures, necessary and sufficient conditions are given for the existence of a real-valued utility function u on M₁ × M₂ which represents ? and possesses the bilinearity property

$\text{u(?(α, x}{}_1\text{,x}{}_2\text{),g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{αu(x}{}_1\text{, g(βy}{}_1\text{, y}{}_2\text{))}\text{+}\text{(1 ? α) u(x}{}_2\text{, g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{βu(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_1\text{)}\text{+}\text{(1 ? β) u(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_2\text{)}$

, for all α, β ∈ [0, 1], all x₁, x₂ ∈ M₁ and all y₁, y₂ ∈ M₂. Moreover, uniqueness up to positive linear transformations can be proved for those utility functions. Finally an outline is given of applications of these results in expected utility theory.