## The Envelope Paradox

Posted by on Friday, October 17, 2014 Under: Philosophy

Last Spring I wrote a series of articles on interesting paradoxes in mathematics and logic, and based on the response from several correspondents these were relatively well received. It would seem that many of my readers quite enjoy these thought provoking problems, and spent enjoyable hours contemplating their complexities.

And so I thought I would write another one today, just to give my reader more to ponder this week.

Today's topic is known by many names, but is usually referred to as "The Envelope Paradox". I first learned of this one in a lecture by Raymond Smullyan, although I am not sure who originally proposed it.

Suppose you are shown a table on which there are two sealed envelopes. They both contain sums of money, although you are not told the amounts. All you are told is that one of them contains twice as much as the other, but not how much or which is the greater sum. You are allowed to select one envelope and open it.

You are then given the choice of swapping envelopes or keeping the one you originally chose. Whichever envelope you end up with, you get to keep the money. So should you swap or not?

Consider the probabilities. There is a 50% chance that your first choice was the smaller sum, in which case the other envelope has twice as much money in it. So if your envelope has N dollars, then swapping gains you another N dollars. There is also a 50% chance that you chose the larger sum, and the second envelope contains only half as much, or N/2 dollars. That means the expected gain of swapping envelopes is N/4 dollars.

Except this argument is true regardless of which envelope you initially selected, and so it would seem that each envelope is better than the other, which is clearly a paradox. One choice must be better than the other, and yet the mathematics suggests that this is not so.

I will leave you to ponder this situation and its solution, primarily because the experts cannot agree on the solution (or even whether or not there is a solution). It is a curious puzzle, and one that is still the subject of much debate.

And so I thought I would write another one today, just to give my reader more to ponder this week.

Today's topic is known by many names, but is usually referred to as "The Envelope Paradox". I first learned of this one in a lecture by Raymond Smullyan, although I am not sure who originally proposed it.

Suppose you are shown a table on which there are two sealed envelopes. They both contain sums of money, although you are not told the amounts. All you are told is that one of them contains twice as much as the other, but not how much or which is the greater sum. You are allowed to select one envelope and open it.

You are then given the choice of swapping envelopes or keeping the one you originally chose. Whichever envelope you end up with, you get to keep the money. So should you swap or not?

Consider the probabilities. There is a 50% chance that your first choice was the smaller sum, in which case the other envelope has twice as much money in it. So if your envelope has N dollars, then swapping gains you another N dollars. There is also a 50% chance that you chose the larger sum, and the second envelope contains only half as much, or N/2 dollars. That means the expected gain of swapping envelopes is N/4 dollars.

Except this argument is true regardless of which envelope you initially selected, and so it would seem that each envelope is better than the other, which is clearly a paradox. One choice must be better than the other, and yet the mathematics suggests that this is not so.

I will leave you to ponder this situation and its solution, primarily because the experts cannot agree on the solution (or even whether or not there is a solution). It is a curious puzzle, and one that is still the subject of much debate.

In : Philosophy