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CCC comments on Confidence levels inside and outside an argument - Less Wrong
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The easy-but-not-very-rigorous method is to use the principle of indifference, since there's no particular reason a tie +/-1 should be much less likely than any other result.
If the election is balanced (the mean of the distribution is a tie), and the distribution looks anything like normal or binomial, 1/X is an underestimate of P(tie | election is within vote margin of X), since a tie is actually the most likely result. A tie +/- 1 is right next to the peak of the curve, so it should also be more than 1/X.
The 10^-90 figure cited in the paper was an example of how the calculation is very sensitive to slight imbalances - a 50/50 chance for each voter gave a .00006 chance of tie, while 49.9/50.1 gave the 10^-90. But knowing that an election will be very slightly imbalanced in one direction is a hard epistemic state to get to. Usually we just know something like "it'll be close", which could be modeled as a distribution over possible near-balances. If that distribution is not itself skewed either direction, then we again find that individual results near the mean should be at least 1/X.
That's an important and non-obvious assumption to make.
So, in short, the 10^-90 figure is based on the explicit assumption that the election is not balanced?
That's why the two methods you mention produce such wildy different figures; they base their calculations on different basic assumptions. One can argue back and forth about the validity or lack thereof of a given set of assumptions, of course...
Yes, I agree.
I'm much more sympathetic to the 10^-90 estimate in the paper than Gelman's quote is; I think he misrepresents the authors in claiming they asserted that probability, when actually they offered it as a conditional (if you model it this way, then it's 10^-90).
That is why I posted it as a comment on this particular post, after all. It's clear that our subjective probability of casting a tie-breaking vote is going to be far less extreme than 10^-90 because our belief in the binomial idealization being correct puts a much less extreme bound on the tie-breaking vote probability than just taking 10^-90 at face value.