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gjm comments on Voting is like donating thousands of dollars to charity - Less Wrong
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So, first of all, there's a factor of 2 error in there: your last equality says, in effect, Pr(CA tied | ...) = Pr(a random vote ties CA | ..., and that vote is in CA) but when CA is tied only half the votes there tie it.
I'm already late for work, so will look harder at the rest of what you're saying later. (I find myself somewhat convinced both by my toy example and by the more-detailed argument you're now making, but the two don't seem consistent with one another. I expect I'm missing something.)
I know. That's one reason why the example is a toy. But no part of your argument appeals to the correlations between states' results, and the point of the toy example is to show that the calculation you're doing produces completely wrong results in a toy example, so in the absence of anything in your argument that makes it apply to the real election and not to the toy example something's got to be wrong with the argument.
Sorry, "slightly bluish" was meant to describe vote share rather than win probability. I'm assuming that P is a win for the Blue candidate about 90% of the time, which with simple-but-unrealistic models of voters will happen if it's reasonably big and, say, 55% Blue.
I was intending to use the 538 meaning. Pr(Blue decisive) is small because in almost all elections the state that gets the winner that crucial third EV -- the one whose EV is in the middle when you line them up in order -- is Purple. What do you find wrong with this reasoning?
Nope. Half the votes prevent a Romney victory, and the other half prevent an Obama victory.
Your confusion is understandable, especially since I confused myself and started bullshitting you for a while before rederiving what I did in the first place. Sorry about that.
That's right. Sorry, I shouldn't have been stressing the high correlations between voting fluctuations in different states.
Ah, ok.
Numbers from your toy example: Pr(Blue tied) = 0.1% Pr(Blue decisive | Blue tied) = 90% Pr(Blue decisive) = 0.1% implications: Pr(Blue decisive and tied) = 0.09% Pr(Blue decisive and not tied) = 0.01% This is not plausible. Presumably Pr(Blue decisive and votes blue by 1 vote) is also roughly 0.09%, in which case Pr(Blue decisive and not tied) cannot possibly be less than that. Assuming Red never enters the picture, Blue is decisive whenever it ends up voting more reddish than Purple does. Given how often Blue ties, I would expect this to actually happen fairly frequently.
Apologies for the slow response; I've been unreasonably busy. Executive summary of what follows: Yup, you were right.
So I tried generating more realistic numbers with the general structure of my toy example, and my conclusion is: Oops, you're right and my example is no good. Sorry. And I think I agree with your simple probability-pushing argument that 538's probability for California being decisive isn't consistent with the numbers from Gelman et al being applicable in the 2012 election.
So, it seems to me that there are (at least) the following possibilities. (1) Gelman et al had a good model, and it remains reasonably applicable now, and 538 had too low a probability of California being decisive. (2) Gelman et al had a good model, but the political landscape has changed, and now California is less likely to be decisive than their model said it was in 1992. (3) Gelman et al had a screwed-up model, and their probabilities weren't right even in 1992.
I agree with you that #2 is the least likely of these, and I offer the following statistic which, if cited at the outset, might have saved us a good deal of argument :-). In 1998, California went Democratic by about 51:48. In 2012, California went Democratic by about 59:39.
I accordingly agree with you: Academian's numbers for his own case, which used the Gelman et al figures for California, likely gave much too high an expected value for his vote in California.
I assume you meant #2 is most likely? And you're right; I should have pointed that out initially (even though it was before the election, I could have used 2008 figures).
Yes, of course I meant most likely. Duh. I've edited my comment for the benefit of our thousands of future readers.