I'm having trouble understanding how you (and buybuydandavis) see this puzzle as illustrating (or evidencing?) a subjective approach to probability. Wouldn't it be perfectly solvable in the frequency/propensity approaches in just the same way? Conditional probability and the Bayes rule work the same way everywhere.
(I haven't read Jaynes yet) (Also enjoyed working out your puzzle, and reposted it in my blog, hope you don't mind)
Certainly don't mind. It's certainly solvable with a propensity approach, it's just that the problem description points you toward the wrong kind of propensity: there really is an absolute proportion of coins to envelopes that has strictly decreased, but that's not the relevant value.
This went over well in the xkcd logic puzzle forum (my hand was not removed), so I thought I'd try it here. It came to me in a dream, so by solving it you may be helping to summon an elder god or something.
Bob replies, "That depends on what random function you used to choose how many envelopes to fill. If you, say, flipped m coins and put each one that came up heads in an envelope, the expected value is $.50."
Alice explains what her random function was, and Bob calculates the expected value. For kicks, he pays her that amount, and she lets him pick a random envelope. It has a coin in it! Bob pockets the coin. Alice then takes the now-empty envelope back, and shuffles it into the others. "Congratulations," she says. "So, what's the expected value of playing the game again, now that there's one fewer coin?"
"Same as before," Bob replies.
Problem 1: Give a value for m and a random function for which this makes sense (there are many).