The Hypothesis of Probabilistic Sophistication

The Hypothesis of Probabilistic Sophistication
Although the subjective approach drops the assumption that uncertainty
is defined in terms of numerical probabilities, it still allows for
States of the World and the State of Decision Theory 27
individuals to possess probabilistic beliefs, with the feature that such beliefs may now differ across individuals. Formally, an individual is said to be probabilistically sophisticated, with a subjective (or personal)
probability measure μ(⋅) over the events E, if their preference function W(⋅) over subjective acts takes the form
WPS(ƒ(⋅)) = WPS(x1 on E1; ...; x on Em) = V(x1, μ(E1); ...; xm,μ(Em))
m
for some (not necessarily expected utility) preference function V(P) = V(x1, p1;...; xm, pm) over lotteries. That is to say, an individual is probabilistically
sophisticated if their uncertain beliefs can be completely summarized by a subjective probability μ(E) attached to each event E, and the individual evaluates each subjective act ƒ(⋅) = [x1 on E1;...;x
m
on Em] solely on the basis of its implied probability distribution (x1,μ(E1);...; xm,μ(Em)) over outcomes. This representation of WPS(⋅) as the composition of a preference function V(⋅) over lotteries and a subjective
probability measure μ(⋅) over events is now referred to as the classical separation of risk preferences from beliefs.