Below is very unpolished chain of thoughts, which is based on vague analogy with symmetrical state of two indistinguishable quantum particles.

When participant is said ze is decider, ze can reason: let's suppose that before coin was flipped I changed places with someone else, will it make difference? If coin came up heads, than I'm sole decider and there are 9 swaps which make difference in my observations. If coin came up tails then there's one swap that makes difference. But if it doesn't make difference it is effectively one world, so there's 20 worlds I can distinguish, 10 correspond to my observations, 9 have probability (measure?) 0.5 * 0.1 (heads, I'm decider), 1 have probability 0.5 * 0.9 (tails, I'm decider). Consider following sentence as edited out. What I designated as P(heads) is actually total measure (?) of worlds participant is in. All this worlds are mutually exclusive, thus P(heads)=9 * 0.5 * 0.1+1 * 0.5 * 0.9=0.9.

What is average benefit of "yea"? (9 * 0.5 * 0.1 * $100 + 1 * 0.5 * 0.9 * $1000)=$495

Same for "nay": (9 * 0.5 * 0.1 * $700+1 * 0.5 * 0.9 * $700)=$630