Small comment… From memory, Jason Brennan doesn’t do the calculations himself, but instead uses the calculation of Geoffrey Brennan and Loren Lomasky to arrive at his extremely low probability of the chance of deciding an election. So, it would be more accurate to say that this is the Geoffrey Brennan and Loren Lomasky estimate, as used by Jason Brennan in one prominent book about the ethics of voting.
(Brennan does mention the much lower Gelman estimate of the chance of being the decisive voter in a footnote, but I think dismisses it as implausible.)
Jason Brennan has a 2022 article on voting with Chris Freiman which I can’t access. perhaps he says something different there.
Something like that. Geoffrey Brennan and Lomasky indeed present the binomial formula and suggest using it in their earlier work, but I haven’t found a case of them using it in any particular way (which could get results like Jason Brennan’s or results like Banzhaf’s), so didn’t want to pin this on them. So I cited Jason Brennan whose uses it to produce these crazily low probabilities in his book. It is possible that Jason Brennan didn’t do the calculations himself and that someone else did (either Geoffrey Brennan and Lomasky or others), but I don’t know and haven’t found an earlier source for the crazy numbers.
Small comment… From memory, Jason Brennan doesn’t do the calculations himself, but instead uses the calculation of Geoffrey Brennan and Loren Lomasky to arrive at his extremely low probability of the chance of deciding an election. So, it would be more accurate to say that this is the Geoffrey Brennan and Loren Lomasky estimate, as used by Jason Brennan in one prominent book about the ethics of voting.
(Brennan does mention the much lower Gelman estimate of the chance of being the decisive voter in a footnote, but I think dismisses it as implausible.)
Jason Brennan has a 2022 article on voting with Chris Freiman which I can’t access. perhaps he says something different there.
Something like that. Geoffrey Brennan and Lomasky indeed present the binomial formula and suggest using it in their earlier work, but I haven’t found a case of them using it in any particular way (which could get results like Jason Brennan’s or results like Banzhaf’s), so didn’t want to pin this on them. So I cited Jason Brennan whose uses it to produce these crazily low probabilities in his book. It is possible that Jason Brennan didn’t do the calculations himself and that someone else did (either Geoffrey Brennan and Lomasky or others), but I don’t know and haven’t found an earlier source for the crazy numbers.
Jason Brennan discusses the background on it a bit here—https://bleedingheartlibertarians.com/2012/10/on-the-probability-of-being-decisive/
Gelman responds in the comments