Nested identification of subjective probabilities

The theory of games against nature relies on complete preferences among all conceivable acts (case 1). Aumann and Drèze (Am Econ J Microecon 1(1):1–16, 2009) consider situations where preferences are defined only for a given set of acts (case 2). We extend these results to situations where (i) only the set of optimal elements from a given set of acts is known (case 3); (ii) only a single optimal act is known (case 4). To these four cases correspond four nested sets of admissible subjective probabilities. Cases 3 and 4 define the extent to which probabilities must be specified to solve a decision problem.