I posted about a year ago on the Two Envelope Paradox, and my paper with Josh Dever on the topic has finally been published (as of Monday, in Sorites).
In 1993, when Josh and I were both graduate students in Berkeley, he introduced me to the paradox, which is very simple to formulate:
You are presented with the choice between two envelopes, Envelope A and Envelope B. You know that one envelope has half as much money as the other, but you don't know which has more. Arbitrarily, you choose Envelope A. Then you think to yourself, should I switch to Envelope B instead? There's a 50-50 chance it has twice as much money and a 50-50 chance it has half as much money. And since double or nothing is a fair bet, double or half should be more than fair! Using the tools of formal decision theory, you might call "X" the amount of money in Envelope A and then calculate the expectation of switching as (.5)*.5(X) + (.5)*2X = 5/4 X. So you switch. (But of course that's absurd.)For some reason, the problem completely took hold of me. I found myself waking in the middle of the night and writing equations. Josh and I bothered just about every graduate student at Berkeley and about half the faculty with the problem. It seemed to me, to us, that the core problem was in the use of a variable with different expectations in different terms of the expected value equation (in the first term, where Envelope A has more, the expectation of X, the value in Envelope A, is higher than it is in the second term, which represents the possibility that Envelope A has less). Just about everyone we spoke to was eventually won over by our reasoning on this, and I presented a paper on it at a graduate student conference later that year.
For a while, I flirted with the idea of writing my dissertation on decision theory, but when I decided to work on connections between philosophy and developmental psychology instead, it seemed the practical decision to set the essay aside. (Berkeley had at the time, and maybe still has, a culture of discouraging graduate students from attempting to publish essays based on anything other than a virtually completed dissertation.)
A couple years later, one of our professors, Charles Chihara, published a paper on the problem (in which he generously thanks me) with a solution similar to ours but also in some important ways different -- and not, it seemed to me, very mathematically precise. Other approaches to the problem came out through the mid- and late 1990s, when it was briefly trendy, but all of them seemed to me to miss the point.
In 2002, I had a long conversation about the problem with Terry Horgan, who had published a couple of papers on it, and I felt myself almost convincing him that my solution was better than his own. (He might not agree with this description of our conversation!) He advised that I seek publication again, so I teamed up with Josh and wrote a new version of the essay.
In 2003, we submitted to Mind, which had published other essays on the problem, but which we thought was a longshot. The referee report came back saying that our solution, though correct, was too technical -- although we felt our paper less technical (and maybe, too, of broader general interest, though that's harder to judge) than the other papers Mind had published on the topic. We received the same reply -- with even less justification, we think, from Analysis, which has published more essays on the topic than any other journal. We then sent the essay to Theory and Decision, whose referee gave us the first substantive criticism we had received (a helpful simplication of our proof) but who recommended rejecting our solution as "obvious" -- despite the fact that in ten years no other essay on the topic had offered it! We considered Synthese, but Springer is such a noxious and expensive publisher, that we decided to send it to an open-access journal instead. We chose the Australian Journal of Logic, which, when we received no reply to several queries sent through various media over the course of a year, we decided had folded. (Though now I see they have a 2007 issue. Hm!) So we withdrew the paper from there to send it to another open-access online journal, Sorites, which we also started to worry about when we got no replies over the course of six months. Finally, when we were about to withdraw the article, we received an apologetic email from the editor -- now, after fifteen years! -- and five journals, and only one minor substantive criticism, it's finally in print.
Anyone out there with a more convoluted publication story?