Sure.Spoiler for D wins
Pocket stamps are in parenthesis. in all below combinations D wins in the first round. All have equal probability of 1/126.
A B C DBB BB BR RR (RR)BB BB RR RR (BR)BB BR BB RR (RR)BB RR BB RR (BR)RR RR BR BB (BB)RR RR BB BB (BR)RR BR RR BB (BB)RR BB RR BB (BR)BR BB BB RR (RR)BR RR RR BB (BB)
A B C D
BB BB BR RR (RR) *
BB BB RR RR (BR) *
BB BR BB RR (RR)
BB RR BB RR (BR) *
RR RR BR BB (BB) *
RR RR BB BB (BR) *
RR BR RR BB (BB)
RR BB RR BB (BR) *
BR BB BB RR (RR)
BR RR RR BB (BB)
You may want to check the distribution with the the star next to them. For instance, for the second distribution
BB BB RR RR (BR)
On his turn, B will reason that his hat is either BB or BR. If A saw [BB *BR* RR RR (BB)], he would have called his hat at turn 1. But A didn't, so B knows his hat.
Let's look at the first distribution, [ BB BB BR RR (RR) ]. On his turn, C reasons that his hat is either RR or BR. If it is RR, then B would have won on turn 2 due to the reasoning for distribution 2. There on his turn C would know his color.
You're right. I messed up. Here are the corrected probabilities...
Based on these odds, B is still the pest position with the expected payout of almost $1.27.