After explaining the well-known two-envelope paradox by indicating the fallacy involved, we consider the two-envelope problem of evaluating the factual information provided to us in the form of the value contained by the envelope chosen first. We try to provide a synthesis of contributions from economy, psychology, logic, probability theory (in the form of Bayesian statistics), mathematical statistics (in the form of a decision-theoretic approach) and game theory. We conclude that the two-envelope problem does not allow a satisfactory solution. An (...) interpretation is made for statistical science at large. (shrink)

Four variations on Two Envelope Paradox are stated and compared. The variations are employed to provide a diagnosis and an explanation of what has gone awry in the paradoxical modeling of the decision problem that the paradox poses. The canonical formulation of the paradox underdescribes the ways in which one envelope can have twice the amount that is in the other. Some ways one envelope can have twice the amount that is in the other make it rational to prefer the (...) envelope that was originally rejected. Some do not, and it is a mistake to treat them alike. The nature of the mistake is diagnosed by the different roles that rigid designators and definite descriptions play in unproblematic and in untoward formulations of decision tables that are employed in setting out the decision problem that gives rise to the paradox. The decision maker’s knowledge or ignorance of how one envelope came to have twice the amount that is in the other determines which of the different ways of modeling his decision problem is correct. Under this diagnosis, the paradoxical modeling of the Two Envelope problem is incoherent. (shrink)

In the two-envelope problem, one is offered a choice between two envelopes, one containing twice as much money as the other. After seeing the contents of the chosen envelope, the chooser is offered the opportunity to make an exchange for the other envelope. However, it appears to be advantageous to switch, regardless of what is observed in the chosen envelope. This problem has an extensive literature with connections to probability and decision theory. The literature is roughly divided between those that (...) attempt to explain what is flawed in arguments for the advantage of switching and those that attempt to explain when such arguments can be correct if counterintuitive. We observe that arguments in the literature of the two-envelope problem that the problem is paradoxical are not supported by the probability distributions meant to illustrate the paradoxical nature. To correct this, we present a distribution that does support the usual arguments. Aside from questions about the interpretation of variables, algebraic ambiguity, modal confusions and the like, most of the interesting aspects of the two-envelope problem are assumed to require probability distributions on an infinite space. Our next main contribution is to show that the same counterintuitive arguments can be reflected in finite versions of the problem; thus they do not inherently require reasoning about infinite values. A topological representation of the problem is presented that captures both finite and infinite cases, explicating intuitions underlying the arguments both that there is an advantage to switching and that there is not. (shrink)

Sutton ( 2010 ) claims that on our analysis (2007), the problem in the two-envelope paradox is an error in counterfactual reasoning. In fact, we distinguish two formulations of the paradox, only one of which, on our account, involves an error in conditional reasoning. According to Sutton, it is conditional probabilities rather than subjunctive conditionals that are essential to the problem. We argue, however, that his strategy for assigning utilities on the basis of conditional probabilities leads to absurdity. In addition, (...) we show that a crucial presupposition of Sutton’s argument — namely, that one can know that envelope A contains n simply on the basis of a stipulation — is mistaken. (shrink)

When David Lewis ( 1986 ) told us that possible worlds were a ‘paradise for philosophers’, he neglected to add that they are a minefield for decision theorists. Possibilities — be they nomological, metaphysical, or epistemic possibilities — have little to do with subjective probabilities, and it is these latter that matter most to decision theory. Bernard Katz and Doris Olin ( 2007 ) have tried to solve the two-envelope problem by appealing to possible worlds and counterfactual conditionals. In this (...) article, I explain why any such attempt is misguided, and why we, qua decision theorists, must focus on the probable rather than the possible. (shrink)