von Neumann-Morgenstera Utility Functions on Two Attributes

Five special forms for a von Neumann-Morgenstern utility function u on a product set A×X are examined. Each expresses u as a combination of functions defined on the separate factors A and X. The five forms include the additive and multiplicative forms, u=u₁+u₂ and u=f₁f₂, respectively, and the additive form with independent product expressed by u=u₁+u₂+f₁f₂. Necessary and sufficient preference conditions are identified for each form, along with admissible transformations on the single-argument functions.