Abstract
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.
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Notes
This was also raised to me in separate personal communications with Dick Jeffrey and Sjoerd Zwart, both in 2001.
Besides Jeffrey, similar points are discussed by others. For example, Schwitzgebel and Dever (2008) introduce the criterion of unchanging expectation. Note that they incorrectly attribute to Jeffrey the position that “one can discharge such X-for-Y substitutions only when X is a true constant”. First, they are referencing a passage in a 1995 draft of Jeffrey (2004) that did not appear in the final published book. More importantly, even in the draft passage, Jeffrey noted this only as a sufficient condition. He never claimed it was necessary. He was also primarily focused on giving a subjectivist explication of expectation. The two-envelopes problem was simply a convenient example briefly introduced as part of that.
This is related to variants of the two-envelope problem presented by many authors in which an amount is placed in one envelope, a fair coin is flipped, and half or twice that amount is placed in the other envelope. Then you are presented the pair and choose one. We discuss similar cases for analysis but consider any such specific causal stories as additional to the basic problem and not necessarily implied by it.
For another discussion of epistemic possibility and problems in trying to construe it metaphysically in the context of the two-envelope problem, see Horgan (2000). For a general discussion of the related dangers of construing epistemic modality in terms of alethic rather than epistemic possible worlds, see Syverson (2003).
In the terminology introduced in Section 5, switching dominates sticking because the game is shut below and open above.
The border we define is not to be confused with a set’s (topological) boundary. Recall, the boundary of a set is the intersection of the closure of that set with the closure of its complement. Because all open sets in this topology are clopen they all have empty boundary.
There is valid concern about the amount of money available in the world and expectations about how much could actually be offered in the presented pair (Jackson et al. 1994) and about the representation on finite sized paper or other media of the values in the envelopes. We can assume wrt money that this is just a game. Or, as is sometimes suggested, we can assume that the amount is in the form of a check. Thus, we separate what is won from any ability to collect (sort of like stock options). And, we can assume that whatever representational needs are implied by any pair of values actually chosen to offer the player in a single round of the game, the presented envelopes will be large enough to handle values enough beyond those values that the player will not be able to determine anything from the envelopes actually presented.
My familiarly assumed currency is US dollars. This point actually holds across most, and possibly all, of the currencies denominated by ‘$’ at the time of writing.
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Acknowledgements
I thank Raymond Smullyan for first introducing me to the two-envelope problem and for many lively discussions back in the 1980s. I thank Michael Jackson for reintroducing me to the problem in 1999 and for the ensuing discussions that enticed me to write this paper. For other helpful comments and discussions I thank Iliano Cervesato, Dick Jeffrey, Sjoerd Zwart, and anonymous commentators. This work supported by ONR.
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Syverson, P. Opening Two Envelopes. Acta Anal 25, 479–498 (2010). https://doi.org/10.1007/s12136-010-0096-7
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DOI: https://doi.org/10.1007/s12136-010-0096-7