The first time I heard the argument against voting based on economic self interest was from a friend in 2004. He was a newly-enfranchised young fellow who gave it to me straight: "my vote matters so little that it's not worth my time to bother casting it." Flustered, I sputtered some typical fluff about Civic Duty and What If Everyone Did That and But So and So's Policies are Really Problematic; none of which had much effect. Later I convinced myself that I was obviously correct because of (mumble), where (mumble) was an delicious blend of political enthusiasm and motivated thinking seasoned with a baseline distrust for simple economic arguments.

Fast forward to a few weeks ago. In a post by incomprehensibly prolific blogger Scott Alexander, among many, many other musings he followed the great rationalist tradition and actually bothered to do the math, as follows:

If the Presidency is worth about $300 billion (in terms of influencing money toward worthwhile projects) and the average American has a 1 in 60 million chance to decide the election, then

Of course, you could still argue that you don't want to vote because neither of the candidates deserves this money; but one of them is going to get it anyway, so if you have any preference at all between the things they're likely to do in office, your vote seriously counts.

I guess that's the main point - if you value five thousand dollars, then your vote is totally worth your time. So please make a plan to vote on Nov 8th and invest $5000 in the future.

But in fact, I think your vote is even more valuable than this rather simplistic argument suggests. So let's math it up. I've come up with a model that I think more rigorously captures both the details of our electoral system (e.g. the electoral college), and includes a better interpretation of what you might mean by "the value" of a vote. Note that I don't know that much about federal budgeting and expenditures, so I'll take the $300 billion dollar numerator at face value. (If you have a better idea for how much the presidency is actually worth, that should just scale the final answers up or down.) Instead I'll focus on the denominator: how the value of the presidency is divided among ballots.

The 1 in 60 million chance of deciding the election came from an analysis of the 2008 election by Gelman, Silver and Edlin, and is calculated as the chance that

So how should we value votes? As Nate Silver is fond of saying, elections are won on the margins - the votes that win the election are the ones that you get over and above your opponents. But votes are essentially indistinguishable. No matter what order you count them in, you should get the same result - you can't single out Bob Anderson of Plainville, PA's ballot as the one that put you over the top. So the value of all votes in a state should be the same, but that value should be set by how close the election is. This leads us to our model:

Here, the value of the presidency is divided among the winning electoral college votes. The value in each state is then divided up among the winning vote margin. By adding up the value of the marginal votes in each state as a fraction of its share of marginal votes in the electoral college, you get the value of the presidency. (This captures the naive idea that if one vote breaks a tie in one state, which breaks a tie in the electoral college, that vote is worth the entire presidency.)

Here's a graph showing the value of votes in the 2012 presidential election (voting data source) using this model:

Because we have focused the value on decisive votes, you can see that the value per vote is way higher than in the previous naive estimates: your vote is now worth between 10,000 and 1 million dollars! Also, the swing states pop right out. Florida leads the pack with a whopping $930k per vote, followed by (in order) NC, OH, NH, NV, VA, AK, CO, IA and PA. (I was kind of surprised Arkansas is on that list, but the margin there was only 42,000 votes.) On the other side, the safest states are Nebraska, DC, Utah, MA, MD, NY, MA, OK, and CA; but even there, the minimum vote value is over $20,000.

One unfortunate thing about this model is that there's a lot of wasted value: the sum value of all votes cast in this model is way, way, way higher than the actual value of the presidency. In fact, for 2012 it is precisely

We don't have the hard numbers for the current presidential election, but based on the best information available about the most likely vote margin in each state and the most likely electoral college margin (collected 11/4/16 just prior to posting this), we can predict your vote's value:

Note that I had to add another factor of 20 to the y-axis in order to capture New Hampshire ($2.0 million/vote), Florida ($7.8 million/vote) and North Carolina ($11.4 million/vote). This still isn't enough to capture Nevada, where polling is within a hair of 50:50 and where this model currently estimates that each vote is worth $50 million at the least. Given the closeness of the race, even the least valuable vote (MD) clocks in at just under $100,000.

Suffice it to say, in this election your vote is extremely valuable, just based on how close the race appears to be. Given the consequences of that vote for our country and the world, this year probably represents the most important your vote will ever be. I sincerely ask that you make a plan and vote, wherever you live! I promise it will be worth your time.

~~~~~~~~~~~~~

* Scott proceeded to discount his calculation based on how likely you think it is that your candidate actually will do a better job than the other guy. This matters for the self-interest argument, insofar as it's not in your interest to buy yourself a punch in the face, whether you expect it or not. This would have a strong impact, as would taking into account unknowable outcomes (e.g. "What would have happened after 9-11 if Gore were president?"); however I think these get further from the intuitive idea of vote value, e.g. how important is my vote in implementing a trillion dollars worth of policy?

Fast forward to a few weeks ago. In a post by incomprehensibly prolific blogger Scott Alexander, among many, many other musings he followed the great rationalist tradition and actually bothered to do the math, as follows:

If the Presidency is worth about $300 billion (in terms of influencing money toward worthwhile projects) and the average American has a 1 in 60 million chance to decide the election, then

**your average American's vote is worth $5000**. That bears repeating: choosing not to vote in this election is the same as saying you don't care how five grand is spent in your country. For over half of Americans who earn less than $60k a year, this calculation makes voting the most economically advantageous thing they do for an entire month.* The only reason this isn't obvious is that multiplying big numbers in your head is hard.Of course, you could still argue that you don't want to vote because neither of the candidates deserves this money; but one of them is going to get it anyway, so if you have any preference at all between the things they're likely to do in office, your vote seriously counts.

I guess that's the main point - if you value five thousand dollars, then your vote is totally worth your time. So please make a plan to vote on Nov 8th and invest $5000 in the future.

But in fact, I think your vote is even more valuable than this rather simplistic argument suggests. So let's math it up. I've come up with a model that I think more rigorously captures both the details of our electoral system (e.g. the electoral college), and includes a better interpretation of what you might mean by "the value" of a vote. Note that I don't know that much about federal budgeting and expenditures, so I'll take the $300 billion dollar numerator at face value. (If you have a better idea for how much the presidency is actually worth, that should just scale the final answers up or down.) Instead I'll focus on the denominator: how the value of the presidency is divided among ballots.

The 1 in 60 million chance of deciding the election came from an analysis of the 2008 election by Gelman, Silver and Edlin, and is calculated as the chance that

- your vote is the deciding vote in the state, and
- your state is the deciding vote in the Electoral College.

The actual probabilities range from about 1 in 10 million (New Mexico) to less than 1 in 10 billion (Oklahoma). While this is a rigorous approach, I don't think it vibes with the idea of vote value per se, since the argument hinges on uncertainty. In their approach, a vote is only valuable if it is the deciding vote, and then it carries the entire value of the presidency; that value is then spread around by the uncertainty of the outcome. However this seems kind of like a lottery to me - where anyone or no one can win, and after the lottery all the non-winning tickets are worthless. If we had perfect information about the electorate - if we knew exactly who was going to vote and for whom, or if we just wait until we get the results - the value of every vote goes to zero in all but one out of 10 million cases. As such, the real value of all votes cast in 2008 would be zero in this model, since we didn't get "lucky" enough to have a one-vote election.

So how should we value votes? As Nate Silver is fond of saying, elections are won on the margins - the votes that win the election are the ones that you get over and above your opponents. But votes are essentially indistinguishable. No matter what order you count them in, you should get the same result - you can't single out Bob Anderson of Plainville, PA's ballot as the one that put you over the top. So the value of all votes in a state should be the same, but that value should be set by how close the election is. This leads us to our model:

Here, the value of the presidency is divided among the winning electoral college votes. The value in each state is then divided up among the winning vote margin. By adding up the value of the marginal votes in each state as a fraction of its share of marginal votes in the electoral college, you get the value of the presidency. (This captures the naive idea that if one vote breaks a tie in one state, which breaks a tie in the electoral college, that vote is worth the entire presidency.)

Here's a graph showing the value of votes in the 2012 presidential election (voting data source) using this model:

Because we have focused the value on decisive votes, you can see that the value per vote is way higher than in the previous naive estimates: your vote is now worth between 10,000 and 1 million dollars! Also, the swing states pop right out. Florida leads the pack with a whopping $930k per vote, followed by (in order) NC, OH, NH, NV, VA, AK, CO, IA and PA. (I was kind of surprised Arkansas is on that list, but the margin there was only 42,000 votes.) On the other side, the safest states are Nebraska, DC, Utah, MA, MD, NY, MA, OK, and CA; but even there, the minimum vote value is over $20,000.

One unfortunate thing about this model is that there's a lot of wasted value: the sum value of all votes cast in this model is way, way, way higher than the actual value of the presidency. In fact, for 2012 it is precisely

*10 quadrillion dollars*. For scale, that's 100x the GDP of the world. Now that's not monetary exchange*value in any sense - it's not like you could collect all those excess votes and buy yourself a solar system. I think it just underscores the inherent inefficiency of a winner-take-all system for actually achieving the real distribution of values in the country.*We don't have the hard numbers for the current presidential election, but based on the best information available about the most likely vote margin in each state and the most likely electoral college margin (collected 11/4/16 just prior to posting this), we can predict your vote's value:

Note that I had to add another factor of 20 to the y-axis in order to capture New Hampshire ($2.0 million/vote), Florida ($7.8 million/vote) and North Carolina ($11.4 million/vote). This still isn't enough to capture Nevada, where polling is within a hair of 50:50 and where this model currently estimates that each vote is worth $50 million at the least. Given the closeness of the race, even the least valuable vote (MD) clocks in at just under $100,000.

Suffice it to say, in this election your vote is extremely valuable, just based on how close the race appears to be. Given the consequences of that vote for our country and the world, this year probably represents the most important your vote will ever be. I sincerely ask that you make a plan and vote, wherever you live! I promise it will be worth your time.

~~~~~~~~~~~~~

*footnotes:** Scott proceeded to discount his calculation based on how likely you think it is that your candidate actually will do a better job than the other guy. This matters for the self-interest argument, insofar as it's not in your interest to buy yourself a punch in the face, whether you expect it or not. This would have a strong impact, as would taking into account unknowable outcomes (e.g. "What would have happened after 9-11 if Gore were president?"); however I think these get further from the intuitive idea of vote value, e.g. how important is my vote in implementing a trillion dollars worth of policy?

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