**Hi Lucky Person!**

**
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.