“That means that for our posterior from above, if you choose blue, the probability of the majority choosing red is about 0.1817. If you choose red, it’s the opposite: the probability that the majority will choose red is 1-0.1817 = 0.8183, over 4 times higher.”
It may well be that I don’t understand Beta priors, or maybe I just see myself as unique or non-representative. But intuitively, I would not think that my own choice can justify anywhere close to that large of a swing in my estimate of the probabilities of the majority outcome. Maybe I am subconsciously conflating this in my mind with the objective probability that I actually cast the tiebreaking vote, which is very small.