How much is a vote worth? How about a dollar?

This is the first in a series of posts where I'll examine the relative value of a vote for a candidate, a dollar donated to a candidate, and later a dollar spent on other goods--charitable or self-serving.

First, in order to address the election-centered questions, it's important to know what the likelihood that an individual voter will swing an election (and by extension, the number of voters per election swing), a version of Nate Silver's return on investment index.  He probably has more thoroughly calculated numbers for presidential elections that I do, but I'm going to start off looking at congressional elections.  Also, everything here is going to be an estimate; I'm more interested in getting the right order of magnitude than the exact number.

So, what are the odds that a voter, or a percent of voters, will swing an election?  I looked at Nate Silver's projections for the 2010 midterm elections, and recorded the projected and actual margin of victories for all races projected to be within 3 points*, on the theory that if one were to give money to a campaign, it would be hard to do better than to just pick the races Silver says are the closest.  Plotting the distribution, it seems like there is an error of roughly +- 5% on the margin.  Again, this is just an estimate, but for an order of magnitude calculation it'll do.  However, if you're looking not that the margin of the two votes but the absolute Democratic (or Republican) share of the two-party vote, this is a margin of +- 2.5%.  So, if you take a given election that Silver projects to be close (within 3% margin), the odds of a 1% boost in the Democratic (or Republican) vote deciding the election are roughly 20% (+=2.5% means a total range of 5%).  I'll use that number from here on out.

So, what to do with this number?  Well, first, the average congressional district is going to have turnout of about 41%; let's say that a competitive district is closer to 50%.  Given that there are 435 voting congressmen and about 217,000,000 eligible voters, there are about 250,000 votes cast in an average swing district.  So buying 1% of the vote gets 20% of a congressman; voting yourself (or getting someone else to) in a swing district buys about 8*10^(-5) of a congressman.

So, how much do you have to donate to give a candidate an extra 1% of the vote?  It seems that the jury's still out on this one, and different papers I've looked at have given different answers.  One paper that looks cool, by Steven D. Levitt, looks at repeat match-ups between congressional candidates, and claims that $100,000 of 1992 money (or about 160,000 current dollars), donated to a challenger (i.e. non-incumbent), gives the challenger an extra ~0.3% of the vote.  If this number is taken at face value, then that would imply that it takes roughly $2,600,000 to buy a congressional election.

Below I've started a table which summarizes this.  The Equivalent Cost (dollars) tab adds the cost in dollars with $25*(hours), assuming time is worth $25/hour.  I'm also assuming it takes one hour to vote; costs are scaled up to the size of the effect (in this case, swinging one election).  I put a question mark next to the campaign donations row to indicate the large error associated with the money-to-votes calculation (done using Levitt's paper).  I'll add to this table later as I expand on these calculations.

Action	Cost (hours)	Cost (dollars)	Effective Cost (dollars)	Effect	Notes (?=sketchy)
Voting	12,500	0	$312,500	Swings congressional election	Relies on Silver's 2010 projections
Donating to non-incumbent congressional campaign	0	$2,600,000	$2,6000,000	Swings congressional election	? (Relies on Levitt)

*Differences between projections and actual votes: [7.0, 0.29999999999999999, 0.5, 6.0, -4.0, 1.5, 8.0, -1.0, 0.0, -10.800000000000001, -6.5, 13.0, -1.0, -1.0, 5.0, 8.0, 8.0, 3.0, 5.0, 2.0, 4.0, 4.4000000000000004, 1.8, -3.5, -3.0, 2.0, -3.0, 6.4000000000000004, 2.6000000000000001, 9.9000000000000004, 0.5, -4.0, -6.0, -4.0, -5.0].  Positive means actual vote was more Democratic than projection.