Savage famously gave an axiomatic representation of expected utility when agents have subjective beliefs about the probability of events; this was extended by Machina and Schmeidler to non-expected utility with only a minor modification; M&S’ formulation can deal with the Allais paradox. Dealing with ambiguity such as that in the Ellbserg Paradox is trickier. Ellsburg’s famous two urns essentially show that people often make decisions that are incompatible with their having *any* probability measure over some events. Epstein and Zhang formalized this idea by defining probability only over events which satisfy their definition of “subjectively risky”: the set of events which have probability and which don’t are endogenous to choice, not exogenous to the model.

Kopylov extends these results in two primary ways. First, he shows that a mathematical object called a mosaic (like a sigma algebra, except that it only includes complements and finite unions of elements of a grand partition in the mosaic, rather than complements and countable unions) is useful for characterizing the subset of events for which probability can be measured. Consider Ellsberg urns where we know there are 100 balls total, 50 red plus green balls, and 50 yellow plus green balls. Presumably the agent does have a probability to assign to green balls alone, but green is the intersection of {R,G},{Y,G}, and if probability was defined on a sigma-algebra, it would have to be defined for G. This rationale led Epstein and Zhang to define probability of risky events over a lambda-system (like a sigma algebra, but closed over disjoint countable unions, not over countable unions); it turns out that a mistake in their paper implies that their definition of subjectively risky requires probability on the weaker structure called a mosaic rather than on a lambda-system.

Mosaics, unfortunately, do not allow some of the mathematical tricks used by earlier authors to show the representation; in particular, constructing the subjective probability from choice is substantially more difficult. Kopylov, with slight modifications of Savage and Machina-Schmeidler, shows that the original proofs do hold even when we are considering a mosaic that is a subset of an algebra, rather than considering the entire sigma-algebra. Because Epstein-Zhang’s definition of subjectively risky constructs a mosaic, and because the definition implies the only new axiom required, Kopylov’s result immediately implies the Epstein-Zhang representation of fully subjective utility. The proofs are not terribly difficult, but they do require a good background in set theory, and a familiarity with common concepts in decision theory that do not appear elsewhere (for instance, qualitative probability).

remind me to talk to you about my solution to the ellsburg paradox