The sensitivity of expected utility choice to slight variations in the description of lotteries is considered. This sensitivity is allowed to influence actual choice in what is called the expected utility with perturbed lotteries model because the slight variations are used to represent vagueness regarding the dollar-prize, probability description of a lottery. Example illustrate how this sensitivity can affect actual choice for an otherwise expected utility decision-maker and provide an explanation for some of the anomolous evidence on risky choice.
Let ~ denote a set of lotteries where each F e o ~ is represented by its dollar prizes and associated probabilities. Let Y x ~ denote the set of lottery pairs. Let C denote a mapping which takes each (F~, Fj) to one of three possible values; F~ ~> Fj, F~ < ~, F~ ~ Fj. Suppose that for each (F/, Fj) a decision-maker will either choose F~ over Fj, or Fj over F~, or be indifferent. There will then be an actual choice map, call it C*, which associates to each (F~, ~) the choice option selected by the decision-maker.
I want to consider a partition of the ~ x ~-domain into sets S and ~r according to whether or not a given C map is sensitive to slight variations in the probability numbers used to represent lotteries. The slight variations can be represented by assigning to each F~ a subset of lotteries o~'(F~) where each F~ ~ e ~(Fe) has the same dollar prizes as F~ and probabilities which differ by at most e from the F~ lottery. Given (Fi, Fj) and, say F,. > Fj we test for choice sensitivity by seeing whether C(F[, Fj ') = F~ ' > Fjf for all (F,~, ~) ~ o~(F,.) x o~(Fj). If this is the case then (Fe, Fj) is in S and the choice map is not sensitive to the specified e-variations in probabilities at (Fi, Fj). Otherwise, (F~, Fj) is in S and there is sensitivity of C at (F,., Fj) because F~ ~> Fj. and Ff < F7 for some F? and Ff whose dollar-prize, probability representations are within e of F~ and Fj respectively. A slight variation in our representation or in the decision-maker's understanding of such a lottery pair could lead to a choice reversal: