First of all, I dispute that losing by less than 1-in-100 of the electoral body is a “large margin.” Secondly, I don’t think it’s very plausible that shifting order 1 million votes with $1 billion in additional funding has less than a 2% chance. ($1,000 per vote is well within the statistics I’ve seen on GOTV efforts, and actually seriously on the high end).
III. “I mean presumably even with 10x more money or $6bn, Hillary would still have stood a reasonable chance of losing, implying that the cost of a marginal 1% change in the outcome is something like $500,000,000 - $1,000,000,000 under a reasonable pre-election probability distribution.”
I don’t think this is the right way to model marginal probability, to put it lightly. :)
I don’t think this is the right way to model marginal probability, to put it lightly. :)
Well really you’re trying to look at d/dx P(Hillary Win|spend x), and one way to do that is to model that as a linear function. More realistically it is something like a sigmoid.
So if we assume:
P(Hillary Win|total spend $300M) = 25%
P(Hillary Win|total spend $3Bn) = 75%
Then the average value of d/dx P(Hillary Win|spend x) over that range is going to be 2700M/0.5 = $5.5Bn per unit of probability. Most likely the value of the derivative at the actual value isn’t too far off the average.
This isn’t too far from $1000/vote x 3 million votes = $3Bn.
I think your model has MUCH more plausible numbers after the edit, but on a more technical level, I still think a linear model that far out is not ideal here. We would expect diminishing marginal returns well before we hit an increase in spending by a factor of 10.
Probably much better to estimate based on “cost per vote” (like you did below), and then use something like Silver’s estimates for marginal probability of a vote changing an election.
To be clear, I have nothing against linear models and use them regularly.
Well if we go with $1000 per vote and we need to shift 3 million votes, that’s $3bn. Now let’s map $3bn to, say, a 25% increased probability of winning, under a reasonable pre-election distribution.
Then you can think of the election costing $12bn, for a benefit of 4tn, which is a factor of 400.
I’m really confused by both your conclusion and how you arrived at the conclusion.
I. Your analysis suggest that if Clinton doubles her spending, her chances of winning will increase by less than 2% (!)
This seems unlikely.
II. “Hillary outspent Trump by a factor of 2 and lost by a large margin.” I think this is exaggerating things. Clinton had a 2.1% higher popular vote. 538 suggests (http://fivethirtyeight.com/features/under-a-new-system-clinton-could-have-won-the-popular-vote-by-5-points-and-still-lost/) that Clinton would probably have won if she had a 3% popular vote advantage.
First of all, I dispute that losing by less than 1-in-100 of the electoral body is a “large margin.” Secondly, I don’t think it’s very plausible that shifting order 1 million votes with $1 billion in additional funding has less than a 2% chance. ($1,000 per vote is well within the statistics I’ve seen on GOTV efforts, and actually seriously on the high end).
III. “I mean presumably even with 10x more money or $6bn, Hillary would still have stood a reasonable chance of losing, implying that the cost of a marginal 1% change in the outcome is something like $500,000,000 - $1,000,000,000 under a reasonable pre-election probability distribution.”
I don’t think this is the right way to model marginal probability, to put it lightly. :)
Well really you’re trying to look at d/dx P(Hillary Win|spend x), and one way to do that is to model that as a linear function. More realistically it is something like a sigmoid.
For some numbers, see this
So if we assume: P(Hillary Win|total spend $300M) = 25% P(Hillary Win|total spend $3Bn) = 75%
Then the average value of d/dx P(Hillary Win|spend x) over that range is going to be 2700M/0.5 = $5.5Bn per unit of probability. Most likely the value of the derivative at the actual value isn’t too far off the average.
This isn’t too far from $1000/vote x 3 million votes = $3Bn.
Thanks for the edit! :) I appreciate it.
I think your model has MUCH more plausible numbers after the edit, but on a more technical level, I still think a linear model that far out is not ideal here. We would expect diminishing marginal returns well before we hit an increase in spending by a factor of 10.
Probably much better to estimate based on “cost per vote” (like you did below), and then use something like Silver’s estimates for marginal probability of a vote changing an election.
To be clear, I have nothing against linear models and use them regularly.
Well if we go with $1000 per vote and we need to shift 3 million votes, that’s $3bn. Now let’s map $3bn to, say, a 25% increased probability of winning, under a reasonable pre-election distribution.
Then you can think of the election costing $12bn, for a benefit of 4tn, which is a factor of 400.