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 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.