I had an email from Noga Alon a couple of days ago, who told me about a much better version of the paradox I discussed in an earlier post. Some of the comments relating to that post also allude to this better version. The reason it is better is that one can no longer object to it on the grounds that it assumes the existence of a probability distribution with impossible properties.
Here is the version as Noga described it (except that if I say anything accidentally stupid, then that is my own personal contribution). You are presented with two envelopes, but now you are told that the amount of money in the envelopes is dollars and dollars, where the positive integer is chosen with probability .
Suppose that you open one of the envelopes. If it contains 10 dollars, then trivially you should switch: the other contains 100 dollars. Now suppose that it contains 100 dollars. This could have happened in one of two ways: with probability we have , so the other envelope contains 10 dollars and you chose the envelope with more money; and with probability we have , so the other envelope contains 1000 dollars and you chose the envelope with less money. So the conditional probabilities are and . But the amount you gain by switching is so great in the case that your expected gain if you switch is certainly positive.
It is not hard to see that this argument works for any amount of money you find in the envelope, as long as it is not 10 dollars. But in that case you still switch—it’s just that the reasons are different. So whatever amount you discover in the envelope when you open it, you improve your expected gain if you switch. So it should surely follow that you don’t need to look in the envelope before deciding to switch. And that is the paradox.
You might object by saying that the expected amount in the envelopes is infinite, so in practice you would always know that the situation was not truly as I have just described it. But that objection will not do: the situation is at least logically possible, and one can easily modify the scenario. For example, suppose that just after your death you find that you are still conscious, but that none of the world’s major religions have got it quite right about life after death. Instead, you are greeted by the great god (at which point you finally understand why it was that you had been so mysteriously obsessed by mathematics). tells you that you will live a life of eternal bliss if and only if you manage to solve all the Clay millennium problems within a time that is specified in one of two envelopes. Moreover, the amount of time (in centuries) is chosen randomly according to the distribution described above for the dollars. And you are warned that one of the problems is pretty hard, even for someone with mathematical training and ten thousand years to do it, so you are advised to try to maximize your expected time. Then, once again, one argument says you should switch envelopes and another says it makes no difference.