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Tournament Prize Money Conundrum
Posted By: Timothy Chow
Date: Saturday, 15 October 2016, at 9:09 p.m.
In Response To: Tournament Prize Money Conundrum (Rick Janowski)
This is known as the two envelopes problem.
One reason this problem is so controversial is that, despite appearances, it is not actually a welldefined probability problem. One way to see that it isn't welldefined is to imagine playing the game over and over again. How much money would be in the opened envelope each time? There is no clear answer to this question. If, hypothetically, one were to postulate a certain prior probability distribution over prize values, then the paradox would evaporate and one could calculate the answer mathematically. However, different prior probability distributions lead to different answers. In the problem as stated, the prior probability distribution is unknown, and so you can't calculate a specific answer.
In some puzzles of this type, even though no prior probability distribution is specified, there's an intuitively plausible choice of prior. For example, we can ask this question: If I flip a coin 100 times and it comes up heads 75 times, then what is the probability that the coin is fair (by which we mean, let's say, that the probability of heads is between 49% and 51%)? An intuitively plausible choice of prior would be that the unknown probability p of heads is uniformly distributed between 0 and 1.
For the two envelopes, however, most people seem to want the prior probability of the prize money to be uniformly distributed between 0 and infinity. But this is mathematically impossible, so we can't take this easy road out of the paradox either. Now, you might say, we can approximate such a distribution. Perhaps the smaller prize amount is some multiple m of £100, where m is an integer uniformly chosen from the range 1 to 100. When you observe £2000, then you have now reduced to two cases that are equally likely, m=20 and m=10. In this case you should switch. But the "price" you pay for producing this justification for switching is that you have introduced some weird boundary effects. Specifically, under these assumptions, it follows that if you open the envelope and observe an odd multiple of £100, then you will know that you have picked the smaller amount. Similarly if you observe an amount greater than £10000 then you should definitely not switch. You can try to wriggle out of these weirdnesses by adjusting the prior probability distribution, but no matter how you twist and turn, something weird is going to happen, and so there's no "canonical" answer to the problem.
Common sense suggests that you should use your judgment and experience to assess the chances that the bigger prize is £4000 versus the chances that the bigger prize is £2000. And this is one of those cases where the tools of probability theory don't have anything to add to what common sense tells you. In fact, trying to invoke probability theory will likely just confuse the issue.

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