Initially either 9 or 1 of the 10 people will have been chosen with equal likelihood, meaning I had a 50% chance of being chosen. If being chosen means I should find 90% likelihood that the coin came up tails, then not being chosen should mean I find 90% likelihood that the coin came up heads (which it does). If that were the case, I'd want nay to be what the others choose (0.9 * 100 + 0.1 * 1000 = 190 < 700). Since both branches are equally likely (initially), and my decision of what to choose in the branch in which I choose corresponds (presumably) to the others' decision of what to choose in the branch in which I don't choose, I should therefore say nay. You can't win with mere straightforward optimization when your action in 1 branch affects other (no longer reachable) branches.