I got the same answer as Marcello by assuming that each of you should get the same expected utility out of the split.

Say that Nick keeps x and you keep y. Then the expected utility for Nick is

0.85 x − 0.15 ($20 − x),

while the expected utility for you is

0.8 y − 0.2 ($20 − y).

Setting these equal to each other, and using x + y = $20, yields that Nick should keep x = $9.50, leaving y = $10.50 for you.

I disagree with both of these methods. If EY were 100% sure, and NB were 50% sure, then I think the entire 20 should go to EY, and neither of the two methods have this property. I am very interested in trying to figure out what the best formula for this situation is, but I do not yet know. Here is a proposal:

Take the least amount of evidence so that you can shift both predictions by this amount of evidence, to make them the same, and split according to this probability.

Is this algorithm good?

My take on it:

You judge an odds ratio of 15:85 for the money having been yours versus it having been Nick's, which presumably decomposes into a maximum entropy prior (1:1) multiplied by whatever evidence you have for believing it's not yours (15:85). Similarly, Nick has a 80:20 odds ratio that decomposes into the same 1:1 prior plus 80:20 evidence.

In that case, the combined estimate would be the combination of both odds ratios applied to the shared prior, yielding a 1:1 * 15:85 * 80:20 = 12:17 ratio for the money being yours versus it being Nicks. Thus, you deserve 12/29 of it, and Nick deserves the remaining 17/29.