Imperfect information as a source of non-symmetry in the two envelope problem

摘要

We present new views on the two envelope problem (also called the exchange paradox). We consider it as a game of a single player (and a banker) or a constant-sum game of two players, each of them having different imperfect information.We discuss models with a known probability distribution. We demonstrate several levels of the paradox, depending on a single parameter of our model. In usual situations, the expected gain is clearly zero. However, there are scenarios in which the exchange of envelopes is supported by probabilistic arguments based on the information obtained by the player(s). The explanation of the higher levels of the paradox is essentially that, when the expected gain does not exist and the players have different information, probabilistic arguments may support contradictory strategies (which do not bring a positive pay-off). As a consequence of the non-existence of the expectation of the underlying distribution, we lack a criterion of optimality of a strategy. We support our conclusions also by computer simulation.Then we discuss the case when the amounts in envelopes are drawn from an unknown distribution. It is known that there is a strategy for one player which, opening one envelope before the decision, gives always a positive expected pay-off. Such a strategy is necessarily random. We contribute to its description and show that this result again fails if the distribution does not have an expectation. Even if the expectation exists, random strategies do not bring an advantage in the game of two players.The two envelope problem is not only a mathematical curiosity; it is a model of decisions which are really done in economics, where erroneous reasoning described here could be of great influence. (C) 2019 Elsevier Inc. All rights reserved.