Tuesday, November 6, 2012

Should You Vote (For President)?

It's been a long time since this blog has been touched, and old readers shouldn't get too excited about any future posts, but for today I have something.  I've spent a good chunk of election day trying to find an answer to a simple question:  should you vote?

There's a whole bunch of ways to answer this question.  You could say yes because it's fun, you could say no because you're too busy, you could just ask your facebook friends.  Each of these ways could give you a different answer, and each may be a valid decision process.  I decided instead to break out the old economics textbook and answer the question through a different method:  examining the value of an individual vote for president.

The Value Of A Vote
How do you calculate the value of a vote?  There's probably multiple ways to do this also, but I chose ( you'll never guess this) to do some math.  Specifically, I broke it down in to three steps:
1.  Find the chance your vote swings your state's outcome.
2.  Find the chance your state swings the national election's outcome.
3.  Find the value of choosing the president of the United States.

To do all of these calculations I needed to make some basic assumptions and approximations.  I'll outline some as we go, but here are a few of the basic ones:
I pulled odds and expected percentages from Nate Silver's 538.  It's a site with a very mathematical background, and is similar to what I would've done if I had more infinite time/money/patience etc.  Some say it leans towards Obama in this particular election, but that should have little impact on my results here.
I assumed voter turnout would be the same as in 2008, and got those numbers from this website.
I assumed states would not have a tie in popular vote (by assuming vote total odds were continuous).  No ties made it easier for me, and shouldn't affect results.
I ignored the special conditions for Nebraska/Maine.  Both of these states break some of their electoral votes up by district.  This assumption will make a difference for these two states, but saved me significant time and will have limited impact on other states.

Odds Your Vote Swings Your State
One of the most overwhelming hurdles, to me at least, while voting is the realization that your vote is just one out of millions.  What are the chances that an election, even in a state, are decided by just one vote?  It seems small, and it is, but unless you vote in Washington D.C. it's higher than you'd think.

I'll save the full values till the end, but here are some samplings:
Iowa is decided by one vote in 1 out of every 200,000 elections.
In Florida it's 1 out of every 600,000 elections.
In California it's 1 out of every 70 trillion elections.
In Washington D.C. it's 1 out of every 2*10^123 elections.  Sorry D.C.

The message to take away here is that if you're in a close state your one vote can make a difference.  For states that are more firmly entrenched for a particular candidate your odds of changing the outcome with your vote quickly get very small.

For the mathematically inclined, these numbers were done by taking the one voter wide section in a normal distribution where the mean is 0, and the section has a z score determined by the difference in popular votes from Silver's numbers, and the margin of error he gives on his predictions.

Odds You State Swings The National Election
 In a small state your vote is more likely to change to result.  A bigger state has much higher odds of swinging the election as a whole however.  Any time an individual state votes for the winner and their electoral vote total is larger than the electoral college's margin of victory you can say that state "swung the election".  This means states with large electoral vote totals are more often "swing states".  This is a little counter-intuitive, but if you can imagine Romney winning California or Obama winning Texas the election results become pretty clear. 

I wrote a quick computer program to simulate these numbers based off of 538's inputs.  Again, 538 leans more towards Obama than other sites like Intrade do.  If the election is closer than this the numbers here will be slightly too small. 

Here's a sample of the results I got:
Iowa swings the election just .6% of the time.
Florida swings the election about 5.9% of the time.
California swings the election an impressive 23.3% of the time.
D.C. swing the election only .5% of the time.

Again, if you like to follow the math I did a monte carlo simulation to get these percentages.  Be careful though, you have to include a lot of covariance between the states, otherwise your distribution becomes much too narrow.

Odds Your Vote Swings The National Election
The two numbers above don't directly depend on each other.  We can multiply them however to get an easy to digest statistic:  the odds that your individual vote determines the nation's president. 

Here's a sample for our guinea pig states:
If you vote in Iowa you swing the election 1 in 30 million elections.
In Florida it's 1 in 10 million.
For California it's 1 in 300 trillion.
And in D.C. it's 1 in 4*10^165.  Don't even bother.

The Value Of Picking The President
Your odds of swinging the national election are extraordinarily small.  The case for voting is not without hope though, because the value of picking a president is extraordinarily large.  Exactly how large would be the topic of an advanced economics paper, but there's a few ways to make simple back of the envelope calculations.

First, 2.8 billion has been spent on presidential advertising this year.  A very naive estimate could use this number as the value of picking a president, but because of diminishing marginal return it is probably too small.

Second, about 200 million has been spent on advertising in Florida and Ohio, two states where advertising will likely make a difference that swing the election roughly 1 in 20 times.  That implies a value of roughly 4 billion dollars for the whole election.  This should be more accurate, but is also probably an underestimate due to diminishing marginal return.

Thankfully, I'm okay with educated guesses, so based off the two numbers above we'll place the value of picking president at 5 billion dollars.  That's right to an order of magnitude, and it's large.

The Value Of Your Vote
How much is your vote worth then?  It's just a case of expected value that involves multiplying the three numbers we've calculated together.

Let's look at our favorite four:
Value of a vote in Iowa:  about $155
In Florida:  about $518
In California:  about .0018 cents
In D.C:  about 10^-156 dollars

First, notice that these numbers are fantastically uneven.  This is a side note from the true question of this post, but it's crazy the impact that the electoral college has on things.  For comparison, if the election was determined by national popular vote each individual's vote value is about $366 dollars.  Surprisingly, nobody really benefits from the electoral college.

Anyways, we can now answer our question:

Should You Vote?
If you're in a swing state like Iowa or Florida the value of your presidential vote can be as high as a couple hundred dollars.  That value makes it definitely worth voting, even if you have to stand in line for a couple hours.  You should go vote for president.

If you're not in a swing state however, your presidential vote has almost no value.  That's not to say blindly you shouldn't vote.  If you care about other elections, if you feel it's your civic duty, if you just enjoy voting it's still worth going to vote.  If the only value in voting for you comes from choosing the president though, it's not worth voting.

There you go.  In a swing state, go vote, it's like getting a couple hundred dollars for just going to the polls.  If your state is decided though, think twice about your reasoning because your vote has almost no value in terms of determining the president.

Unsure about where your state fits?  This chart below has everything!

State O Win% Step 1 Step 2 Value($)
Alabama 0.0% 2.85E-16 2.1% 3.06E-08
Alaska 0.0% 4.21E-08 0.5% 1.10
Arizona 1.8% 5.92E-07 2.4% 71.58
Arkansas 0.0% 1.68E-12 0.6% 5.07E-05
California 100.0% 1.54E-14 23.3% 1.80E-05
Colorado 79.8% 3.91E-06 2.1% 417.97
Connecticut 100.0% 7.97E-10 2.4% 9.39E-02
Delaware 100.0% 2.70E-08 0.5% 7.31E-01
D.C. 100.0% 4.34E-164 0.5% 1.18E-156
Florida 50.0% 1.76E-06 5.9% 518.34
Georgia 0.1% 2.10E-08 3.5% 3.69
Hawaii 100.0% 1.10E-21 0.5% 2.99E-14
Idaho 0.0% 4.60E-18 0.5% 1.20E-10
Illinois 100.0% 3.44E-16 6.0% 1.03E-07
Indiana 0.2% 7.21E-08 2.4% 8.72
Iowa 84.1% 4.92E-06 0.6% 155.41
Kansas 0.0% 3.82E-09 0.6% 1.15E-01
Kentucky 0.0% 1.04E-09 1.6% 8.38E-02
Louisiana 0.0% 1.75E-13 1.6% 1.41E-05
Maine 100.0% 2.47E-08 0.5% 6.68E-01
Maryland 100.0% 1.45E-18 2.5% 1.82E-10
Massachusetts 100.0% 1.11E-11 2.9% 1.64E-03
Michigan 99.6% 9.31E-08 5.0% 23.34
Minnesota 99.8% 5.82E-08 2.5% 7.33
Mississippi 0.0% 2.00E-09 0.6% 6.00E-02
Missouri 0.3% 9.51E-08 2.1% 10.21
Montana 2.0% 2.54E-06 0.5% 66.14
Nebraska 0.0% 2.26E-12 0.6% 6.80E-05
Nevada 93.3% 4.46E-06 1.0% 228.16
New Hampshire 84.1% 9.72E-06 0.8% 399.85
New Jersey 100.0% 4.20E-09 4.1% 8.62E-01
New Mexico 99.5% 4.42E-07 1.3% 27.68
New York 100.0% 3.96E-23 9.4% 1.86E-14
North Carolina 25.7% 2.87E-06 3.6% 517.46
North Dakota 0.0% 2.66E-08 0.5% 6.93E-01
Ohio 84.1% 1.18E-06 5.5% 324.73
Oklahoma 0.0% 2.30E-21 1.6% 1.85E-13
Oregon 99.5% 2.18E-07 2.4% 26.07
Pennsylvania 98.8% 1.94E-07 6.0% 58.29
Rhode Island 100.0% 5.21E-13 0.5% 1.41E-05
South Carolina 0.2% 7.58E-08 2.1% 8.14
South Dakota 0.1% 1.22E-07 0.5% 3.17
Tennessee 0.0% 5.69E-10 2.4% 6.88E-02
Texas 0.0% 4.70E-13 6.7% 1.58E-04
Utah 0.0% 2.22E-29 0.6% 6.68E-22
Vermont 100.0% 6.77E-16 0.5% 1.83E-08
Virginia 78.8% 3.11E-06 3.6% 555.59
Washington 100.0% 1.98E-09 2.9% 2.91E-01
West Virginia 0.0% 3.92E-08 0.6% 1.18
Wisconsin 97.1% 7.63E-07 2.6% 99.91
Wyoming 0.0% 1.77E-13 0.5% 4.62E-06