Axioms for measuring utility on partial mixture sets

A mixture set is path-connected via a suitable collection of paths, the most common example being a convex set. We introduce partial mixture sets (where certain paths may be missing or excluded) and provide a formal extension of the well-known axiomatisation of cardinal linear utility by Herstein and Milnor. In order to compensate for the weaker structure of partial mixture sets, it is necessary to strengthen the Herstein-Milnor axioms. We show that partial mixture sets feature in a variety of benchmark models in the literature and especially where utility may be cardinal nonlinear on a richer set of paths or a larger set of prospects. For instance, in the theory of ambiguity: the canonical partial mixture set consists of the full set of acts together with the set of convex paths that connect comonotonic acts.