# How much is one vote worth?

Just 537 votes in Florida would have been enough to change the outcome of the 2000 election from George Bush to Al Gore – a margin of 0.009% (recount pictured above). And that wasn’t even the closest-won state that year: in New Mexico the margin was a mere 366 votes.1

People say it’s your civic duty to vote, but it also seems like it’s very unlikely your vote will make a difference.

Who is right? Is voting really valuable, or a waste of time?

We looked into the research on this, especially regarding the US Presidential election. The answer, surprisingly, is that the single hour you spend voting for the President and Congress can be the most important thing you do with an hour each four years – and we expect similar numbers for other kinds of elections outside the USA. It also looks like there are effective techniques you can use to ‘get out the vote’, if you want to do more than just vote yourself.

The impact of your vote largely depends on 2 things, which we’ll investigate in turn:

• The chances of your vote changing the election outcome.
• How much better for the world as a whole one candidate is, compared to another.

At first blush it might seem that the chances of your single vote changing the election outcome are zero. But while the chances are low, they could be around 1 in 10 million if you live in a swing state.

And that small chance matters, because who runs the US is kind of a big deal.

## How likely is your vote to change the election?

Given how infrequent real elections occur, we can’t just look at the historical record and observe how frequently an election is won by one vote – we’ll never get a big enough sample. We need a different way to precisely measure the likelihood of one vote changing the outcome: statistical modelling.

Statisticians can model tens of billions of elections – filling them with polling data to simulate voters – and then calculate exactly how often the entire result would be changed by a single vote.

Famed statistician Andrew Gelman, of Columbia University, has done this for US Presidential elections and published several papers outlining the results. He found that if you’re in a ‘safe state’ like California, the odds of your vote changing the outcome really are effectively zero.

But if you’re in a swing state, like Florida, Nevada or New Hampshire, the odds are often around 1 in 10 million that your vote will change the outcome, and can be even better. In the closest state in an election with perfectly even polling, the odds could be as high as 1 in 1 million. (Gelman’s article Vote for Charity’s Sake offers a nice overview of his work, and we’ve stuck some details in this footnote2.)

There are two reasons these votes have a bigger effect than you might first expect. Firstly, US elections are often quite competitive rather than blow-outs. Secondly, almost all the ‘voter power’ in the US electoral system is concentrated in a small number of ‘swing’ states. Effectively, a fifth of voters are given votes five times more powerful than average.

Precisely how likely a vote is to change the election outcome varies depending on:

• how close the polling is in that state;
• how close the election is across the country as a whole;
• how closely the state matches polling across the country as a whole; and
• the population of the state.

But because political polling isn’t good enough to precisely predict election outcomes, we can’t accurately distinguish those elections in which your vote is essential, from those in which it can’t change anything. And this means the 1 in 10 million figure is surprisingly stable across different elections and states. As of writing this article, the 2016 election looks ‘moderately close’, but that only pushes the odds to something like ‘1 in 20 million’.

These chances may sound bad to you, but they’re still 18x better than your chances of winning the US Lottery. And if you think a 1 in 10 million chance of changing an election is small, you need to meet the US Federal Government.

## How much does it matter who wins?

The US Federal Government is big. Really big.

Over each four year term of a President, the US Federal Government will spend \$14.8 trillion.3 Written out as a number that’s \$14,800,000,000,000.

It’s \$46,000 for each American. It’s \$117,000 for each vote cast in 2012.

If you multiply all that spending through a 1 in 10 million chance of changing the outcome, it comes to \$1.5 million. That’s the fraction of the budget you can ‘expect’ to influence by voting in a swing state, in a statistical sense.

Of course, much of the Federal Budget is quite stable, and mostly determined by Congress (though you can vote for Congress too). But it doesn’t have to be that flexible to be important. For example, if one party wants to spend 0.5% of GDP on foreign aid, and the other wants to spend 0.3%, one vote alone could shift – in expectation – \$14,000 into foreign aid. If that were spent on bed nets – which unfortunately it usually isn’t – it would be enough to save 4 lives from malaria.

Spending money is just one thing the government does which happens to be easy to quantify. Its impact through regulations is likely similarly large.

How much could the choice of President affect the economy as a whole? One recent paper published by Brookings Institute economists Justin Wolfers and Eric Zitzewitz, looked at how election ‘prediction markets’ and the stock market moved during sudden shocks in the Presidential race. The results suggested that US stock markets ‘expect’ themselves to be 12% lower if Trump wins the election than if Clinton does.

If this were true, as US stock markets are worth about \$25 trillion, that would represent an anticipated \$3 trillion decline in global wealth.

By this measure then, a single vote with a 1 in 10 million chance of changing the election outcome would be worth \$300,000 to US shareholders as a whole. This also ignores effects on economies overseas, which in some cases seemed to be almost as large – or domestic impacts like unemployment.

(ADDED NOTE: This forecast turned out to be quite wrong, with the stock market rising around 1% on the day after Trump’s election. The author of the paper linked above tries to explain how their prediction was so wrong here.)

On top of this, each four years the US President makes decisions affecting the immigration status of millions of people, and choices about wars that in some cases have resulted in 100,000s of deaths.4

Speaking of war – imagine you think over four years with one candidate there’s a 0.1% chance of nuclear war, and 0.2% with the other. How valuable is it to vote for the safer one?

To answer this we need some idea how much society would be willing to pay to avoid a nuclear war. It’s really hard to estimate, but let’s spitball it and say that each US resident would be willing to pay \$1 million to avoid dying in a nuclear war, on average. (For comparison, the US government will spend about \$7 million or so to save a life.) A major nuclear war would quickly kill around 80% of the US population.5 If you do the math, then a vote with a 1 in 10 million chance of changing the election outcome would be worth \$25,000 to your fellow citizens through its effects on the likelihood of a nuclear war alone. And a nuclear war would obviously affect even more people overseas, as well as untold future generations.

So on the basis of the above – let alone anything more difficult to measure we haven’t considered here, such as cultural effects – each vote in these states seems really valuable.

## What does this mean?

Voting is probably the single most high impact hour you’ll spend each four years if:

• you think that one candidate is significantly better than the other for improving the welfare of people around the world, all things considered;
• you live in a competitive US state (this election that means: New Hampshire, Colorado, Florida, North Carolina, Pennsylvania, Wisconsin, Michigan, Nevada, Iowa, Virginia, New Mexico, Minnesota, Maine, Arizona or Ohio).6

There’s nothing too surprising about this. The US Federal Government spends a lot of money and affects people’s lives in major ways. Occasional Presidential and Congressional elections are the main ways people influence its priorities.

When you only spend an hour voting for an organisation that’s spending tens of thousands of dollars on your behalf – and has nuclear weapons – it’s important to vote, and vote wisely!

Of course this relies on you being right about which candidate is best, and others in your country being wrong. But if you do think you have above average judgement – what should you do?

## What you can do now

Firstly, if you’re a US voter in a swing state, find where you can vote and cast an early ballot (or go on Tuesday November 8th):

Given how important each vote is, it could also be very valuable to encourage others to do the same.

One particularly effective way to do this is to contact all your friends and family in swing states, and convince them to vote, and vote the way you think they should. Here’s a list of those states. If you click on the state, it will give you a list of all your Facebook friends that live there:

Alternatively you can use the new site SwingVoter Go (based on Pokemon Go) to find your friends in these states!

After that, you can volunteer to phone-bank or door-knock to get out the vote for the candidate you prefer. When they involve personal contact, these techniques have been demonstrated in experiments to increase the number of people who vote for a given candidate, which is why political campaigns do them! They are especially valuable the day of, and the day before, the election. Here are some cost-effectiveness estimates of different activities from the book Get Out The Vote from the Brookings Institute:

GOTV effortEffectiveness per contactStatistically reliable?\$ cost per vote
Door-to-doorOne vote per 14 contactsYes\$29 a
Volunteer phone bankingOne vote per 38 contactsYes\$38 b
EmailNo detectable effectLarge number of studies show average effect cannot be large-
LeafletingOne vote per 189 voters reachedNot significantly greater than zero-

a At \$16 and 6 contacts per hour
b At \$16 and 16 contacts per hour

According to these studies, one vote costs \$38 through volunteer phone banking – or a few hours of time. If true, this suggests that phone banking could create a significant amount of value. To get started you would want to check out the campaign site of the candidate you want to assist.

Voters in safe states like California can try to make their vote matter as well by using ‘vote swapping’, where they cast a vote for a third party in a safe state, in exchange or someone in a swing state casting a vote for a major party candidate. There are various sites that try to facilitate this.

Finally, if you have more money than time, you can also donate (though there aren’t many days left for anyone to spend money this year!).

Has this impacted how important you think voting it is? Make sure your friends realise it’s important to vote as well:

### Notes and references

1. You can see plenty of other examples of very close election results here.
2. We found three papers estimating the probability of one vote being decisive in one of the biggest elections in the world - that for the US presidency: a. Probability of Events that Have Never Occurred: When Is Your Vote Decisive? b. What is the probability that your vote will make a difference? c. Empirically investigating the electoral college In these papers the authors look at various historical presidential elections from the perspective of someone who has access to polling data a few weeks ahead of the vote. With what likelihood should they have expected that one vote would change the outcome? The first found a probability of 1 in 10 million of a typical single vote being decisive in 1992, with a chance of 1 in 3.5 million for a swing state such as Vermont. The second found a probability of 1 in 60 million for a single vote in a random state, and 1 in 10 million for swing states such as New Mexico or Virginia. That compares to an electorate of around 120 million voters. As they put it:
”A probability of 1 in 10 million is tiny but, as discussed by Edlin, Gelman, and Kaplan (2007), can provide a rational reason for voting; in this perspective, a vote is like a lottery ticket with a 1 in 10 million chance of winning, but the payoff is the chance to change national policy and improve (one hopes) the lives of hundreds of millions, compared to the alternative if the other candidate were to win.”
The third paper also looks at the 2000 Presidential election, the closest in modern history, and (using a rougher methodology) found that the probability of an average vote changing the outcome was around 1 in 6 million in 2000. If we take the same range from an average to swing state proposed in the second paper, that suggests that voters in the key swing states could have a 1 in 2 million chance of swinging the election. The reality is that in most states, including California, New York or Texas, an additional vote has no ability to swing the outcome, because these states are not close themselves. Even if they were close, they couldn’t swing the electoral college from one candidate to another, because a close election in California implies an incredibly unbalanced election in the rest of the country. Almost all of their influence instead becomes concentrated on a handful of swing states.
3. See https://en.wikipedia.org/wiki/United_States_federal_budget
4. The Vietnam War and 2003 Iraq War both resulted in casualties in the 100,000s, including all deaths.
5. https://www.ncbi.nlm.nih.gov/books/NBK219165/
6. See the ‘voter power’ bar on the right hand side of http://election.princeton.edu