Decision Making Under Uncertainty and the Two-Envelope Paradox

Abstract

If one faces a decision under uncertainty where the expected payoffs are undefined, then the fact that for some partition of the event space a specific strategy is optimal does not necessarily imply that it is an optimal strategy over whole event space. The current literature does not explain what exactly happens if one adopts that strategy repeatedly for an arbitrarily large number of times. The article provides an insight into this issue using the context of the two-envelope-paradox. If one adopts the strategy to always switch envelopes, then the average gain may be, with the same probability, a large gain or an absolute large loss. This is because for any large sample of the repeated decision-making scenario, the distribution of the maximum of the absolute gain from switching is such that, with high probability, it is unique; its effect on the average gain is massive and it is either positive or negative. In addition, we show that a strategy to switch envelopes if the amount in the first envelope does not exceed some threshold is preferred to a strategy of “no switch”, and the optimal threshold is a solution to a St-Petersburg type problem.