If you change it so that in the tails case, rather than taking the consensus decision, and giving nothing if there is not consensus, the experimenter randomly selects one of the nine decision makers as the true decision maker (restating to make sure I understand), then this analysis is obviously correct. It is not clear to me which decision theories other than UDT recognize that this modified problem should have the same answer as the original.
Meanwhile, that forumulation is equivalent to just picking one decider at random and then flipping heads or tails to determine what a “yea” is worth! So in that case of course you choose “nay”.
If you change it so that in the tails case, rather than taking the consensus decision, and giving nothing if there is not consensus, the experimenter randomly selects one of the nine decision makers as the true decision maker (restating to make sure I understand), then this analysis is obviously correct. It is not clear to me which decision theories other than UDT recognize that this modified problem should have the same answer as the original.
Meanwhile, that forumulation is equivalent to just picking one decider at random and then flipping heads or tails to determine what a “yea” is worth! So in that case of course you choose “nay”.
The equivalence is not obvious to me. Learning that you’re one of the “potential deciders” still makes it more likely that the coin came up tails.