You have been chosen to receive one of our A-series prizes, worth at least $100. All you need do is reply to this email saying you want to receive the money and it will be yours, with no strings attached.
But wait! We have an even more generous offer. We are willing to upgrade your prize to a B-series prize. The value of the B-series prize is calculated as follows. If the A-series prize is worth x then the B-series prize will be worth 2x in 3/7 of cases and x/2 in 4/7 of cases. So the gain of switching from A to B is 1/7( 3(2x) + 4(x/2) - 7x ) = x/7. In other words, switching to the B-series prize gives you a gain of at least $100/7 = $14.29. We charge a fee of $5 to upgrade your prize. So by paying just $5 you will receive, on average $114.29 if the A-series prize was the smallest in our range, and possibly much more. So please consider this swap, which is favourable to you.
One more thing. We have a third range of prizes, the C-series. The value of the C-series prize is calculated as follows. If the B-series prize is worth y then the C-series prize is worth 2y in 4/7 of cases, and y/2 in 3/7 of cases. So by upgrading from B to C you gain 1/7( 4(2y) + 3y/2 - 7y ) = 5y/14. Naturally, there is a small fee for switching from the B-series to the C-series prize. As a one-off special, we are able to offer you this swap for just $5. It follows that the C-series prize must be worth, at the very least, the value of B plus an amount of $5(100)/14 = $35.71. So this is a real bargain.
If you take up the two generous bonus offers described above then you will, on average, receive $14.29 + $35.71 = $50 more than if you don't. This for an investment of only $10. That's assuming the original prize was the minimum of $100. If it was more then you gain more than $50. This is on top of the value of the A-series prize itself. If you choose not to benefit from the two offers above, you may receive just the A-series prize and pay nothing.
Please check our arithmetic and let us know asap what you have decided.
Yours truthfully
Robert A. Mambo-Jawbo
For the benefit of this exercise let us assume that everything in the above email is 100% accurate and that there are no hidden pitfalls or tricks. It seems you should pay $10 in order to upgrade your prize from the A-series to the C-series. But before you go for your credit card, just consider one thing.If you look at how the three prizes are defined you will notice that the C-series prize is identical to the A-series prize. So the email is suggesting that you pay $10 in order to end up with the prize you started out with. On the other hand, the arithmetic shows that there is a clear benefit to swapping A for B and then B for C.
Should you pay $10 or not?
If you find the above too complicated, have a look at a stripped-down version of the paradox.
This puzzle is a version of a paradox that appears in a draft paper, "Bayesianism, Infinite Decisions, and Binding" by Arntzenius, Elga, and Hawthorne, October, 2003. The answer given in that paper is wrong.