Thanks for writing up this post, @Eric Neyman . I’m just finding it now, but want to share some of my thoughts while they’re still fresh in my mind before next election season.
This means that one extra vote for Harris in Pennsylvania is worth 0.3 μH. Or put otherwise, the probability that she wins the election increases by 1 in 3.4 million
My independent estimate from the week before the election was that Harris getting one extra vote in PA would increase her chance of winning the presidential election by about 1 in 874,000.
My methodology was to forecast the number of votes that Harris and Trump would each receive in PA, calculate the probability of a tie in PA given my probability distributions for the number of votes they would each get, then multiply the probability of the PA tie by the probability that PA is decisive (conditional on a tie).
I used normal distributions to model the expected number of votes Harris and Trump would each get for simplicity so I could easily model the outcomes in Google Sheets (even though my credence/PDF did not perfectly match a normal distribution). These were my parameters:
Harris Mean
Harris SD
Trump Mean
Trump SD
3450000
80000
3480000
90000
Simulating 10,000 elections in Google Sheets with these normal distributions found that about 654 elections per 10,000 were within 20,000 votes, which translates to a 1 in ~306,000 chance of PA being tied. I then multiplied this by a ~35%[1] chance that PA would be decisive (conditional on it being tied), to get a 1 in ~874,000 chance of an extra vote for Harris in PA changing who won overall.
99% of the votes in PA are in right now, with the totals currently at: 3,400,854 for Harris and 3,530,234 for Trump.
This means that the vote totals for Harris and Trump are both within 1 standard deviation of my mean expectation. Harris is about half an SD low and Trump was about half an SD high.
From this, it’s not clear that my SDs of 80,000 votes for Harris and 90,000 votes for Trump were too narrow, as your (or Nate’s) model expected.
So I think my 1 in ~874,000 of an extra vote for Harris determining the president might have been more reasonable than your 1 in ~3.4 million.
[1] Note: I mistakenly privately thought models and prediction markets might be wrong about the chance of PA being decisive, and thought that maybe it was closer to ~50% rather than ~25-35%, but the reason I thought this was bad and I didn’t realize until after the election: I made a simple pattern matching mistake by assuming “the election is a toss-up” meant “it will be a close election”. I failed to consider other possibilities like “the election will not be close, but we just don’t know which side will win by a lot.” (In retrospect, this was a very silly mistake for me to make, especially since I had seen that as of some late-October date The Economist said that the two most likely outcomes of the 128 swing-state combinations was 20% that Trump swings all seven and 7% that Harris swings all 7.)
I haven’t looked at your math, but I actually agree, in the sense that I also got about 1 in 1 million when doing the estimate again a week before the election!
I think my 1 in 3 million estimate was about right at the time that I made it. The information that we gained between then and 1 week before the election was that the election remained close, and that Pennsylvania remained the top candidate for the tipping point state.
The information that we gained between then and 1 week before the election was that the election remained close
I’m curious if by “remained close” you meant “remained close to 50/50″?
(The two are distinct, and I was guilty of pattern-matching “~50/50” to “close” even though ~50/50 could have meant that either Trump or Harris was likely to win by a lot (e.g. swing all 7 swing states) and we just had no idea which was more likely.)
Note that I also made five Manifold Markets questions to also help evaluate my PA election model (Harris and Trump means and SDs) and the claim that PA is ~35% likely to be decisive.
(Note: I accidentally resolved my Harris questions (#4 & #5) to the range of 3,300,000-3,399,999 rather than 3,400,000-3,499,999. Hopefully the mods will unresolve and correct this for me per my comments on the questions.)
This exercise wasn’t too useful as there weren’t enough other people participating in the markets to significantly move the prices from my initial beliefs. But I suppose that’s evidence that they didn’t think I was significantly wrong.
Thanks for writing up this post, @Eric Neyman . I’m just finding it now, but want to share some of my thoughts while they’re still fresh in my mind before next election season.
My independent estimate from the week before the election was that Harris getting one extra vote in PA would increase her chance of winning the presidential election by about 1 in 874,000.
My methodology was to forecast the number of votes that Harris and Trump would each receive in PA, calculate the probability of a tie in PA given my probability distributions for the number of votes they would each get, then multiply the probability of the PA tie by the probability that PA is decisive (conditional on a tie).
I used normal distributions to model the expected number of votes Harris and Trump would each get for simplicity so I could easily model the outcomes in Google Sheets (even though my credence/PDF did not perfectly match a normal distribution). These were my parameters:
Simulating 10,000 elections in Google Sheets with these normal distributions found that about 654 elections per 10,000 were within 20,000 votes, which translates to a 1 in ~306,000 chance of PA being tied. I then multiplied this by a ~35%[1] chance that PA would be decisive (conditional on it being tied), to get a 1 in ~874,000 chance of an extra vote for Harris in PA changing who won overall.
99% of the votes in PA are in right now, with the totals currently at: 3,400,854 for Harris and 3,530,234 for Trump.
This means that the vote totals for Harris and Trump are both within 1 standard deviation of my mean expectation. Harris is about half an SD low and Trump was about half an SD high.
From this, it’s not clear that my SDs of 80,000 votes for Harris and 90,000 votes for Trump were too narrow, as your (or Nate’s) model expected.
So I think my 1 in ~874,000 of an extra vote for Harris determining the president might have been more reasonable than your 1 in ~3.4 million.
[1] Note: I mistakenly privately thought models and prediction markets might be wrong about the chance of PA being decisive, and thought that maybe it was closer to ~50% rather than ~25-35%, but the reason I thought this was bad and I didn’t realize until after the election: I made a simple pattern matching mistake by assuming “the election is a toss-up” meant “it will be a close election”. I failed to consider other possibilities like “the election will not be close, but we just don’t know which side will win by a lot.” (In retrospect, this was a very silly mistake for me to make, especially since I had seen that as of some late-October date The Economist said that the two most likely outcomes of the 128 swing-state combinations was 20% that Trump swings all seven and 7% that Harris swings all 7.)
I haven’t looked at your math, but I actually agree, in the sense that I also got about 1 in 1 million when doing the estimate again a week before the election!
I think my 1 in 3 million estimate was about right at the time that I made it. The information that we gained between then and 1 week before the election was that the election remained close, and that Pennsylvania remained the top candidate for the tipping point state.
I’m curious if by “remained close” you meant “remained close to 50/50″?
(The two are distinct, and I was guilty of pattern-matching “~50/50” to “close” even though ~50/50 could have meant that either Trump or Harris was likely to win by a lot (e.g. swing all 7 swing states) and we just had no idea which was more likely.)
Note that I also made five Manifold Markets questions to also help evaluate my PA election model (Harris and Trump means and SDs) and the claim that PA is ~35% likely to be decisive.
Will Pennsylvania be decisive in the 2024 Presidential Election?
How many votes will Donald Trump receive in Pennsylvania? (Set)
How many votes will Donald Trump receive in Pennsylvania? (Multiple Choice)
How many votes will Kamala Harris receive in Pennsylvania? (Set)
How many votes will Kamala Harris receive in Pennsylvania? (Multiple Choice)
(Note: I accidentally resolved my Harris questions (#4 & #5) to the range of 3,300,000-3,399,999 rather than 3,400,000-3,499,999. Hopefully the mods will unresolve and correct this for me per my comments on the questions.)
This exercise wasn’t too useful as there weren’t enough other people participating in the markets to significantly move the prices from my initial beliefs. But I suppose that’s evidence that they didn’t think I was significantly wrong.