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 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.