Nigerian Prize Swap Paradox resolution


Is the C-series prize really the A-series prize? It isn't because the possible values of the B-series prize are 2x and x/2, so that the possible values of the C-series prize are 4x, x or x/4, ie not just x. If the C-series prize were the same as the A-series prize then the B-series prize would be worth 2 times the value of C in 3/7 of cases and 1/2 the value of C in the remaining 4/7. Yet B has the possible values 2x and x/2, while C has the possible values 4x, x or x/4. It just does not compute.

So why does it seem that C is A? The error arises because we are erroneously trying to extend the valid rule that if prize X is worth two times prize Y and prize Z is worth half of prize X, then Y and Z are equal. This rule does not apply when the value of each prize in the chain of three is probabilistically contingent on the one before, as in the paradox. The intuitive explanation is that when two forking processes are linked up then the four paths will not cancel each other to return to the original value, at any rate not in general, though there may be some unusual cases where they do so. (I suspect they never do.)

The value of the C-series prize is actually 76x/49, which is the product of the individual gains, ie 8/7 and 19/14. Alternatively, we can work it out by adding a 12/49 chance of 4x, a (9/49 + 16/49) chance of x, and a 12/49 chance of x/4. This adds up to 48/49 + 25/49 + 3/49 = 76/49 chance of x.

If you still find this hard to see, have a look at this numerical example. Say A is worth 196. Then B is worth 392 x 3/7 + 98 x 4/7 = 168 + 56 = 224. So C is worth 448 x 4/7 + 112 x 3/7 = 256 + 48 = 304. This is confirmed by the gain calculation ie 76/49 x 196 = 304. So the C-series prize is indeed worth more than the A-series prize. Nigerian emails can be trustworthy!




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