I think the basic error with the “vote Y” approach is that it throws away half of the outcome space. If you make trustworthy precommitments the other people are aware of, it should be clear that once two people have committed to Y the best move for everyone else is to vote Y. Likewise, once two people have committed to N the best move for everyone else is to vote N.
But, since the idea of updating on evidence is so seductive, let’s take it another step. We see that before you know whether or not you’re a decider, E(N)>E(Y). One you know you’re a decider, you naively calculate E(Y)>E(N). But now you can ask another question- what are P(N->Y|1) and P(N->Y|9)? That is, the probability you change your answer from N to Y given that you are the only decider and given that you are one of the nine deciders.
It should be clear there is no asymmetry there- both P(N->Y|1) and P(N->Y|9)=1. But without an asymmetry, we have obtained no actionable information. This test’s false positive and false negative rates are aligned exactly as to do nothing for us. Even though it looks like we’re preferentially changing our answer in the favorable circumstance, it’s clear from the probabilities that there’s no preference, and we’re behaving exactly as if we precommitted to vote Y, which we know has a lower EV.
It doesn’t make much sense to say “I precommit to answering ‘nay’ iff I am not selected as a decider.”
But then … hmm, yeah. Maybe I have this the wrong way around. Give me half an hour or so to work on it again.
edit: So far I can only reproduce the conundrum. Damn.
I think the basic error with the “vote Y” approach is that it throws away half of the outcome space. If you make trustworthy precommitments the other people are aware of, it should be clear that once two people have committed to Y the best move for everyone else is to vote Y. Likewise, once two people have committed to N the best move for everyone else is to vote N.
But, since the idea of updating on evidence is so seductive, let’s take it another step. We see that before you know whether or not you’re a decider, E(N)>E(Y). One you know you’re a decider, you naively calculate E(Y)>E(N). But now you can ask another question- what are P(N->Y|1) and P(N->Y|9)? That is, the probability you change your answer from N to Y given that you are the only decider and given that you are one of the nine deciders.
It should be clear there is no asymmetry there- both P(N->Y|1) and P(N->Y|9)=1. But without an asymmetry, we have obtained no actionable information. This test’s false positive and false negative rates are aligned exactly as to do nothing for us. Even though it looks like we’re preferentially changing our answer in the favorable circumstance, it’s clear from the probabilities that there’s no preference, and we’re behaving exactly as if we precommitted to vote Y, which we know has a lower EV.