I came across a YouTube video featuring another well-known paradoxical problem that is complete bogus. The video is done in a very stressful style with a sort of speaking that I can’t stand very long. I am still not sure, but I think the author just wanted to confuse.
The problem is rather simple. Assume you pick two strangers on the streets of New York and ask them to play a game: They should open their purses and count the money they are carrying. The one with the smaller sum wins the money of the one with the larger sum.
The confusion is now spread by claiming that each of the two strangers must be interested in the deal because the possible loss is always less than the possible win by the very nature of the game.
That’s nonsense. The argument falls apart if you think of a person with an empty purse, who would be overjoyed to play this game, while someone who just picked up a large amount of money from the bank should be very hesitant to make the deal.
The reason for the confusion is that it does not make sense to talk about „expected“ outcome without the assumption of an a priori distribution. I have written about this before in the two-envelopes-problem. This is just a variation.
The best way to avoid this error is to think of a simulation of the experiment on a computer. For this, you would have to fix a distribution which determines the amount of money that you put into the purses of the two strangers. With that knowledge, you can compute the expected outcome of the experiment. Without that knowledge, it does not make sense to talk about „expectation“.