Unique induced preference representations

Machina [Machina, M.J., 1984. Temporal risk and the nature of induced preferences. Journal of Economic Theory 33, 199-231] considers an individual who has to choose from a set of alternative temporal uncertain prospects, and must take an action before the uncertainty is resolved, seeking to maximize the expected value of an (action determined) von Neumann-Morgenstern utility index. It is natural to ask if the set of underlying von Neumann-Morgenstern utility indices can be uniquely recovered solely on the basis of the thus induced (ordinal) preferences over temporal prospects. Machina's conclusion is that “ordinal preferences alone will not suffice.” However, we show that it is possible to recover the action-utility set inducing the preferences uniquely if we restrict attention to action-utility sets for which no two actions induce the same preference relation on the space of temporal prospects, no action is redundant, and no action leads to a risk free outcome.