It Is Important to Take All Available Information into Account When Making a Decision: Case of the Two Envelopes Problem

In situations when we know the probabilities of all possible consequences, traditional decision theory recommends selecting the action that maximizes expected utility. In many practical situations, however, we only have partial information about the corresponding probabilities. In this case, for different possible probability distributions, we get different values of expected utility. In general, possible values of expected utility form an interval. One way to approach this situation is to use the optimism-pessimism approach proposed by Nobelist Leo Hurwicz. Another approach is to select one of the possible probability distributions -- e.g., the one that has the largest possible entropy. Both approaches have led to many good practical applications. Usually, we get reasonable conclusions even when we ignore some of the available information -- e.g., because this information is too vague to be easily formalized. In this paper, we show, on the example of the two envelopes problem, that ignoring available information can lead to counter-intuitive recommendations