on envelopes

The two-envelopes problem, via xkcd. I'm not goint to re-hash it here, because it makes me dizzy, and the wikipedia page is fine. It's a 'paradox' of probability/game theory - like Monty Hall problem but without goats, cars, or a convincing solution.

There doesn't seem to be any dispute that if you can take a 50-50 bet that will either double or halve your money, then you should - your average profit is 25%. On the other hand, looking in your envelope doesn't give you any useful information, and switching without looking can't possibly earn you money in the long run. So what's going on?

Clearly the odds bet isn't 50-50! And the problem is that in order to fill the envelopes, we have to randomly pick an amount of money $X to put in the small envelope. But what does random mean? We can't have a uniform distribution over an infinite range - it doesn't make sense. You're trying to pick a number at random between 1 and infinity, whatever you pick will be in the bottom 1%.

Once we look in the envelope and see $Y, we know X = Y or X = Y/2. It's only a 50-50 bet if those possibilities are equally likely, and there aren't any probability distributions that have P(Y) = P(Y/2) for all Y.

So the game, as stated, can never be played. Any real randomizer for the game is going to give the player worse than 50-50 odds in some cases, and they won't make profit by switching in the long run.