Swing Vote!

Hey everyone. Sam made a post a couple weeks ago analyzing votes (and money) for congressional elections. This is a post about analyzing votes for US Presidential elections. This is the result of a 2am conversation-calculation with Sam and a mutual friend Gary Wang. Also, we base a lot of intermediate steps in the result comes from the website FiveThirtyEight.

The question we're going to try to answer is the following. You are Joe Smith from Iowa (or, insert your favorite state here), who decides to stay home on November 6, election day. What's the probability you wake up the next morning, open up Google News, and feel really stupid? Or more succinctly, what's the probability that a single vote in Iowa will make the difference in November?

Because of the complicated rules of the electoral college, this is a little difficult to estimate, but we do a rough approximation here. What we'll do is think about is 1) the probability that without Iowa, neither Obama or Romney would have enough electoral votes to win the White House, and 2) the probability that Iowa's vote splits exactly evenly, given that 1) is true (so we do a conditional probability). Then we multiply 1) and 2).

1) 538 has run simulations based on lots of polls that estimate the probability distribution of electoral votes Obama will receive. For reference, we estimate this to be a Gaussian centered at 302 with standard deviation 54,

(σ = 54,  x₀ = 302) The probability that, without a state with EV electoral votes and a priori probability of Obama winning pO, neither candidate has enough to win is: 

The first term is from erasing Iowa when it was critical for an Obama victory, and the second term from erasing Iowa when it was critical for a Romney victory. Note that this number basically scales with the number of electoral votes a state is worth. We haven't yet taken account how likely it is the state is to split evenly; just the likelihood it is a state without which one side wouldn't have enough for victory.

2) Now we need to estimate the odds that a state will split its vote evenly, assuming it is critical for the election. 538 has a probability distribution for every state, but we need to modify it slightly -- the one from 538 is a probability distribution for the number of votes Obama will get; we want the probability distribution assuming 1) is true.

Here we make an assumption: Given that the electoral college is so close that one state is decisive, we can assume that the election was very close and the national popular vote was very close to 50/50. We also assume the difference between a state's vote and the national vote is roughly constant (see 538 for more on this). We'll then shift the mean of our Gaussian so as to make the popular vote 50/50. For example, according to 538, currently Obama is expected to win 50.9% of the popular vote nationally, and 51.5% of the vote in Iowa. We'll subtract 0.9% from the Iowa expected vote, and treat the probability distribution instead as a Gaussian centered at 50.6%. 

Now we're interested in the probability of one vote changing the election. So here we should be a bit careful. We approximated our proportion of vote as a continuous variable, but of course it's actually discrete. Really our probability distribution is the sum of a bunch of tiny delta functions with some discrete spacing 1/N where N is the voting population of the state. We guessed the voting population by taking the voting population in 2008 and multiplying by the state's growth over the four years.

Thus, the probability of Joe affecting the election in Iowa is obtained by integrating P(p) around p = 0.50 with a width of 1/N, which we can approximate very well by P(0.50)/N. 

Awesome! Now multiply by 1) and you win!

We did this out using data as of Aug. 8 for a bunch of swing states (and California, for fun) and got the following probabilities: 

Colorado	6 × 10-7
Nevada	6 × 10-7
New Hampshire	5 × 10-7
Virginia	5 × 10-7
Iowa	4 × 10-7
Ohio	4 × 10-7
Pennsylvania	3 × 10-7
New Mexico	2 × 10-7
Florida	1 × 10-7
California	7 × 10-10

Thus this means that, if you are from a swing state, you've got about a one in 10 million chance of affecting the election. So you need to vote in 10 million elections to affect one of them. If each person spends about an hour voting and values his time at about $25 an hour, then about $250 million of time is needed to flip an election1. In Sam's previous post, there was roughly a 10% inefficiency between advertising and voting, so we claim that order of $2 billion would buy a Presidential election. Sam earlier said it was about $2.6 million to buy a cheap House seat (challenger, close election), so to buy control of the House probably costs maybe $100 million or so. 

So according to our numbers, it seems of comparable efficiency to donate money to House campaigns and Presidential campaigns (depending on your value of control of the House and White House and whatnot). 

^{1: Not actually true. Spending $25 has a 1 in 10,000,000 chance of flipping an election is what we actually mean. Thus flipping an election with a probability p is of the order of $250 million in time.}

Continuing the chart from the previous post:

Action	Cost (hours)	Cost (dollars)	Effective Cost (dollars)	Effect	Notes (?=sketchy)
Voting for congressman	12,500	0	$312,500	Swings congressional election	Relies on Silver's 2010 projections
Donating to non-incumbent congressional campaign	0	$2,600,000	$2,600,000	Swings congressional election	? (Relies on Levitt)
Voting for President in Swing State	5,000,000	0	$125,000,000	Swings Presidential election	Relies on analysis done in this post
Donating to Presidential campaign	0	$1,000,000,000	$1,000,000,000	Swings Presidential election	Relies on analysis done here and Levitt