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)

A wealthy eccentric places two envelopes in front of you. She tells you that both envelopes contain money, and that one contains twice as much as the other, but she does not tell you which is which. You are allowed to choose one envelope, and to keep all the money you find inside.

I reason: (1) For any x, if I knew that A contained x, then the odds are even that B contains either 2x or x/2, so the expected amount in B would be 5x/4. So (2) for all x, if I knew that A contained x, I would have an expected gain in switching to B. So (3) I should switch to B. But this seems clearly wrong, as my information about A and B is symmetrical.

Previous claims to have resolved the two-envelope paradox have been premature. The paradoxical argument has been exposed as manifestly fallacious if there is an upper limit to the amount of money that may be put in an envelope; but the paradoxical cases which can be described if this limitation is removed do not involve mathematical error, nor can they be explained away in terms of the strangeness of infinity. Only by taking account of the partial sums of the infinite series (...) of expected gains can the paradox be resolved. (shrink)

In this paper, we present a simple axiomatic justification for indifference before opening, avoiding any expectation reasoning, which is often considered problematic in infinite cases. Although the two-envelope paradox assumes an expectation-maximizing agent, we show that analogous paradoxes arise for agents using difierent decision principles such as maximin and maximax, and that our justification for indifierence before opening applies here too.

I present in this paper a solution to the Two-Envelope Paradox. I begin with stating the paradox and describing some related experiments. I justify then the fact that choosing either envelope is indifferent. I also point out the flaw in the reasoning inherent to the two-envelope paradox.

You are given a choice between two envelopes. You are told, reliably, that each envelope has some money in it—some whole number of dollars, say—and that one envelope contains twice as much money as the other. You don’t know which has the higher amount and which has the lower. You choose one, but are given the opportunity to switch to the other. Here is an argument that it is rationally preferable to switch: Let x be the quantity of money in (...) your chosen envelope. Then the quantity in the other is either 1/2x or 2x, and these possibilities are equally likely. So the expected utility of switching is 1/2(1/2x) + 1/2(2x) = 1.25x, whereas that for sticking is only x. So it is rationally preferable to switch. There is clearly something wrong with this argument. For one thing, it is obvious that neither choice is rationally preferable to the other: it’s a tossup. For another, if you switched on the basis of this reasoning, then the same argument could immediately be given for switching back; and so on, indefinitely. For another, there is a parallel argument for the rational preferability of sticking, in terms of the quantity y in the other envelope. But the problem is to provide an adequate account of how the argument goes wrong. This is the two envelope paradox. In an earlier paper Horgan 2000) I offered a diagnosis of the paradox. I argued that the flaw in the argument is considerably more subtle and interesting than is usually believed, and that an adequate diagnosis reveals important morals about both probability and the foundations of decision theory. One moral is that there is a kind of expected utility, not previously noticed as far as I know, that I call nonstandard expected utility. I proposed a general normative principle governing the proper application of nonstandard expected utility in rational decisionmaking. But this principle is inadequate in several respects, some of which I acknowledged in note added in press and some of which I have meanwhile discovered.. (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)

This paper deals with the two-envelope paradox. Two main formulations of the paradoxical reasoning are distinguished, which differ according to the partition of possibilities employed. We argue that in the first formulation the conditionals required for the utility assignment are problematic; the error is identified as a fallacy of conditional reasoning. We go on to consider the second formulation, where the epistemic status of certain singular propositions becomes relevant; our diagnosis is that the states considered do not exhaust the possibilities. (...) Thus, on our approach to the paradox, the fallacy, in each formulation, is found in the reasoning underlying the relevant utility matrix; in both cases, the paradoxical argument goes astray before one gets to questions of probability or calculations of expected utility. (shrink)

Clark and Shackel have recently argued that previous attempts to resolve the two-envelope paradox fail, and that we must look to symmetries of the relevant expected-value calculations for a solution. Clark and Shackel also argue for a novel solution to the peeking case, a variant of the two-envelope scenario in which you are allowed to look in your envelope before deciding whether or not to swap. Whatever the merits of these solutions, they go beyond accepted decision theory, even contradicting it (...) in the peeking case. Thus if we are to take their solutions seriously, we must understand Clark and Shackel to be proposing a revision of standard decision theory. Understood as such, we will argue, their proposal is both implausible and unnecessary. (shrink)

The “exchange paradox”—also referred to in the literature by a variety of other names, notably the “two-envelopes problem”—is notoriously difficult, and experts are not all agreed as to its resolution. Some of the various expressions of the problem are open to more than one interpretation; some are stated in such a way that assumptions are required in order to fill in missing information that is essential to any resolution. In three experiments several versions of the problem were used, in each (...) of which the information given was sufficient to determine an optimal choice strategy when it exists or to justify indifference regarding keeping or trading when such a strategy does not exist. College students who were presented with the various versions of the problem tended to base their choices on simple heuristics and to give little evidence of understanding the probabilistic implications of the differences in the problem statements. (shrink)

The term “exchange paradox” refers to a situation in which it appears to be advantageous for each of two holders of an envelope containing some amount of money to always exchange his or her envelope for that of the other individual, which they know contains either half or twice their own amount. We review several versions of the problem and show that resolving the paradox depends on the specifics of the situation, which must be disambiguated, and on the player's beliefs. (...) The latter psychological variables are part and parcel of the resolution. Assuming reasonable subjective distributions, exchanging cannot always be advantageous for both players. We suggest several deep-rooted psychological reasons for the considerable difficulty people demonstrably have in dealing with this problem. Implicit widespread and compelling assumptions—that affect judgement in diverse contexts—obstruct the solution. Analysing this paradox underscores the close connection between psychology and probability theory. (shrink)

Consider this situation: Here are two envelopes. You have one of them. Each envelope contains some quantity of money, which can be of any positive real magnitude. One contains twice the amount of money that the other contains, but you do not know which one. You can keep the money in your envelope, whose numerical value you do not know at this stage, or you can exchange envelopes and have the money in the other. You wish to maximise your money. (...) What should you do?1 Here are three forms of reasoning about this situation, which we shall call.. (shrink)

The two envelopes problem has generated a significant number of publications (I have benefitted from reading many of them, only some of which I cite; see the epilogue for a historical note). Part of my purpose here is to provide a review of previous results (with somewhat simpler demonstrations). In addition, I hope to clear up what I see as some misconceptions concerning the problem. Within a countably additive probability framework, the problem illustrates a breakdown of dominance with respect to (...) infinite partitions in circumstances of infinite expected utility. Within a probability framework that is only finitely additive, there are failures of dominance with respect to infinite partitions in circumstances of bounded utility with finitely many consequences (see the epilogue). (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)

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)

Several fallacies of conditionalization are illustrated, using the two-envelope problem as a case in point.