Subjective probabilities on "small" domains
The classic choice-theoretic construction of subjective probability (Savage, 1954) does not apply to preferences, like those in the Ellsberg Paradox, that reflect a distinction between risk and ambiguity. We formulate two representation results – one for expected utility, the other for probabilistic sophistication – that derive subjective probabilities but only on a “small” domain of risky events. Risky events can be either specified exogenously or in terms of choice behavior; in the latter case, both the values and the domain of probability are subjective. The analysis identifies a mathematical structure – called a mosaic – that is intuitive for both exogenous and behavioral specifications of risky events. This structure is weaker than an algebra or even a lambda-system.