Sometimes it seems like there are too many thoughts to think and too little time in which to think them. Here’s an example: I really ought to think of a detailed response to Brian Weatherson’s comments on my two envelopes paper co-authored with Graham, but that detailed response will have to wait a little while.
However, let me write a little enigmatic response while it’s on my mind:
There is an ambiguity in asking whether one ‘ought to switch’ or not. One answer will depend on which action maximises expected utility, given the process which places money in each envelope. Another will depend on which action maximises expected utility, given what I know about the process placing money in each envelope. If the circumstance is as Brian describes (I have envelope A, and the process has resulted $n in A and either $(0.5)n or $2n in envelope B with equal likelihood), then switching will maximise utility expected in sense 2. If I don’t know whether I have envelope A or envelope B, and I do know that I have equal probability of having envelope A or having envelope B (and the rest of the process is as Brian describes), then we are indifferent between switching and keeping.
That seems right to me, but I can’t confirm if that’s the view of the authors of the paper, as we haven’t met to talk about this one yet. Graham presented the paper at the ANU (the subtitle the definitive solution to the two-envelope paradox was not my idea!) and we have some other comments to work through, too.
But I have other pressing business also in the queue, including the final touches on a grant application, referee’s reports, overdue book reviews, a too big pile of email to read and preparation for upcoming lectures.