Could one vote — your vote — swing an entire election? Most of us abandoned this seeming fantasy not too long after we learned how elections work.

But the chances are higher than you might think. If you’re in a competitive district in a competitive election, the odds that your vote will flip a national election often fall between 1 in 1 million and 1 in 10 million.

That’s a very small probability, but it’s big compared to your chances of winning the lottery, and it’s big relative to the enormous impact governments can have on the world.

Each four years the US federal government allocates \$17,500,000,000,000, so a 1 in 10 million chance of changing the outcome of a US national election gives you some degree of influence over \$1.75 million, on average.

That means the expected impact of voting — the probability of changing an election’s result multiplied by the impact if you do — might, depending on your personal circumstances, be very high.

This could, in itself, be a good argument for voting.

Fortunately there is a significant amount of academic research on the importance of elections and how likely one vote is to change the outcome, so I’ve pulled it together to estimate the average value of one vote for the right person.

The answer, as you might expect, depends a great deal on the circumstances of any given election, and indeed most votes predictably have no impact.

But there are common situations in which the expected value of casting a vote will be far higher than anything else you could hope to do in the same amount of time.

Why exactly? Let me explain.

What’s coming up

In this article, we’ll demonstrate that, for many people, voting matters, but not (or at least not only) because of the normal arguments about it being your civic duty.

Your vote could actually change the world for the better, and if you’re in a competitive race the chances are high enough that you should think hard about hitting the voting booth.

First I’ll investigate the two key things that determine the impact of your vote:

• The chances of your vote changing an election’s outcome in a range of different situations
• How much better some candidates are for the world as a whole, compared to others

Then I’ll discuss what I think are the best arguments against voting in important elections:

• If an election is competitive, that means other people disagree about which option is better, and you’re at some risk of voting for the worse candidate by mistake.
• While voting itself doesn’t take long, knowing enough to accurately pick which candidate is better for the world actually does take substantial effort — effort that could be better allocated elsewhere.

Finally we’ll look into the impact of donating to campaigns or working to ‘get out the vote’, which can be effective ways to generate additional votes for your preferred candidate.

We’ll use figures for US presidential elections, because they have an unusually large impact on our priority problems, more of our readers live in the US than any other single country, and more work has been done to model them than other kinds of elections. However, similar reasoning can be applied to elections in other countries.

The probability of one vote changing an election

Given how infrequently national elections are won by one vote, we can’t just look at the historical record and observe the fraction for which that’s true. While we do have examples of large tied elections, there’ll never be enough real-life elections to accurately determine their frequency empirically.

We need a different approach: statistical modelling.

To see how the method works, we can start small. Imagine that you’re on a small committee making a decision. The odds that you’ll change the outcome of a vote like that — assuming 2 options and 4 other voters, each 50% likely to vote for either option — is about 19%. We could confirm that empirically if we liked.

We can then work upwards to the size of national elections: with 8 voters it’s 14%, with 16 voters it’s about 10%, with 32 voters about 7%, and so on. In fact, the likelihood you’ll change the outcome ends up being roughly proportional to one over the square root of the number of voters.

Statisticians who specialise in politics add real polling data to the mix, and compare it to actual election results to figure out how accurately polling predicts how people will vote. This gives them a ‘probability distribution’ for the likelihood that each elector will choose to vote for each candidate.

With all of this information in hand, we can go ahead and model tens of billions of elections to estimate how often the entire result will be changed by a single vote.

The famous statistician Andrew Gelman of Columbia University has done just this for US presidential elections, which are broken down into states, and has published several papers outlining the results.1 2

He found that if you’re in a ‘safe state’ like California, the odds of your vote changing the outcome of a presidential election really is effectively zero (the model spits out 1 in 100 trillion, but it’s very hard to assign meaningful probabilities to something so unlikely). Something similar would be true for voters in ‘very safe seats’ in the UK or Australia.

By contrast, in a small US state polling around 50/50 in a close election nationally — for instance New Mexico, Iowa, or New Hampshire in the 2000 elections — the probability could get as high as 1 in 3 million. (The article Vote for Charity’s Sake offers a nice overview of this research, and we’ve stuck some details in this footnote.)

In a wider range of ‘tipping point states’ in reasonably close elections, the probability is lower, and closer to 1 in 10 million.

(Note that what matters isn’t the state in which polling is closest, but rather the states that might put someone over the edge of winning the election as a whole — the ‘tipping point state’. If one candidate is ahead nationally then they’ll probably be ahead in the ‘tipping point state’ too.)

As of October 14 2020, Joe Biden’s substantial lead in public opinion polls means Gelman’s modelling indicates that there are only four states where the odds of one vote changing the outcome is greater than 1 in 10 million: New Hampshire, Pennsylvania, Wisconsin, and Michigan.3

However, a perhaps unexpected finding is that even when an election doesn’t look that close, the probability of one vote changing the outcome in a potential ‘tipping point state’ rarely falls to less than half of what it would be in a close-seeming election. This is shown visually in the figure below from FiveThirtyEight’s election modelling. The underlying reason is that opinion polls are often off by a large margin, so when an election is close we can’t rule out that it will be a blow-out for one side — and, similarly, even when a candidate seems to be substantially ahead, we can’t confidently rule out the election being close.

In the UK or Australia, an equivalent analysis would look at the likelihood that a party gains a majority in parliament by one seat, and that that seat is won by one vote.

The factors that push up the leverage of each voter are:

• An election being close to 50/50 nationally
• An election being close to 50/50 in a given ‘tipping point’ seat or voting region
• Being able to accurately determine which elections are closest
• Being able to accurately identify which seats or regions are closest (in which case expected influence becomes concentrated in those places)
• Fewer total voters

Australia has a tenth as many voters as the US, and the UK has a fifth — which, all else equal, would make each vote 2-3x more likely to flip the outcome of a close election.4 Polling is similarly precise in all of these different countries. And the likely ‘tipping point’ seats in US, UK and Australian elections all contain a similar fraction of the population — 10 to 20% — so power is concentrated in a similarly-sized subset of voters.

So from the above we can anticipate that in a similarly tight election, in a ‘tipping point seat’, the odds of a vote changing the outcome would be a few times higher in those countries than in the US.

A similar analysis can be applied to any sort of election.

A common objection to this line of reasoning is that if an election is as close as one vote, it will be re-run or decided by the courts anyway, and so a single vote can never actually make a difference.

To see how this is mistaken, you need to conceptualise the vote margin in large elections as shifting the probability of each candidate winning. If you’re ahead, each extra vote makes you more likely to win without a court battle or a re-run. And if you’re narrowly behind, each extra vote increases your chances of successfully disputing the result. So long as we’re unsure what the vote margin will be, the expected impact of each extra vote remains the same as it would be if all its impact were entirely concentrated on a perfectly tied election.

Finally, there’s another quite different way one can model the impact that each vote has, but it won’t much change our conclusion, so for simplicity I’ll leave it in this footnote.5

Alright, now that we have a sense of the likelihood of swinging an election, we need to know how valuable it would be to do so.

How much does it matter who wins?

Compared to the likelihood of a vote changing the outcome of an election, how much it matters who wins i) is harder to quantify, ii) depends more on your values, and iii) varies widely depending on the candidates running for office. But a quick scan of the numbers and issues at stake suggests that the impact will often be substantial.

In most rich countries, governments tax and spend 25-55% of a country’s GDP. As a rule of thumb, you can roughly think of them as directing a third of a country’s income.

That’s enough money per person, and per vote, that positively influencing how it’s spent can be important enough to offset the low chances of any given vote swinging an election.

Again using US to illustrate, over the next four years6 the US federal government will spend about \$17.5 trillion.

Written out as a number it looks like \$17,500,000,000,000. That’s \$53,000 for each American, or \$129,000 for each vote cast in 2016.

If you multiply all that spending through a 1 in 10 million chance of changing the outcome, in a swing state like New Hampshire, it comes to \$1.75 million. That’s the fraction of the budget you might ‘expect’ to influence by voting in a swing state, in the statistical sense of expectation.

If that number sounds unexpectedly large, remember that we’re shifting around roughly a third of the economy’s output, over several years, and we’re concentrating on the impact a voter can have if they’re among the privileged 20% of the country that lives in a state which can plausibly determine the election outcome.

In the US’s state-based system, 80% of individual voters can’t hope to change the outcome — but that leaves the remaining 20% with 5 times the leverage they’d have otherwise.

Of course, much of the US federal budget is quite stable, but keep in mind that stickiness in how money gets spent cuts both ways: it makes it harder to shift the budget, but if you do, it means those changes will probably stay around for longer.

It’s common for parties to want to shift how several percent of GDP gets spent. But the budget doesn’t even have to be that flexible for your impact to matter.

For example, if one party will spend 0.5% of GDP on foreign aid, and the other will spend 0.3%, a vote with a 1 in 10 million chance of changing the outcome would shift — in expectation — \$17,800 into foreign aid.

There are other kinds of government spending that can have huge impacts as well: R&D into new clean energy technologies is probably one of the most cost-effective ways to limit climate change, and think about the enormous return the world is getting from countries like the UK that decided, years ago, to fund preliminary research into coronavirus vaccines.

But choosing which taxes to impose and how to spend the money raised is just one thing the government does, one which happens to be easy to quantify in dollar terms.

There are major non-budgetary impacts as well, which include:

• Foreign policy: Elected governments decide things such as how much to trade with foreigners (which can affect their wellbeing too), how much to raise tensions with other countries in pursuit of foreign policy goals, and ultimately whether to go to war. Foreign policy is often determined without a lot of input from legislatures, which means a few officials have substantial discretion — and that’s especially important for countries with large militaries or nuclear weapons.
• Stabilising the business cycle: Governments work to raise total spending during recessions and decrease total spending when inflation is too high, in order to limit excessive ups and downs in the economy.
• Regulations: Elected governments make decisions about all sorts of regulations, for instance on consumer products, workplace conditions, environmental standards, and so on.
• Immigration: Elected governments decide how many foreigners can come live in a country and on what basis, ranging from skilled migrants, to economic migrants, to political refugees.
• Social freedoms: Elected governments can influence whether LGBTQ+ people can be public about their sexual orientation and whether they can get married, which recreational drugs people are free to use, how police go about enforcing laws, whether voluntary euthanasia is permitted, and so on.
• Political freedom: Elected governments can try to entrench themselves, or reduce the ability of the public to reflect on political questions, by harassing political opponents, being generally misleading, shutting down hostile media outlets, or making it harder for people to vote.

Measuring the social impact of the different approaches governments might take to these issues is difficult. But it could easily be more important than the shifts in spending that result from a change in government.

To illustrate, imagine that you think the chance of a nuclear war over four years under one presidential candidate is 1 in 1,000, and the chance with the other is 1 in 500. While highly uncertain, these probabilities are both figures nuclear security specialists might give if you asked them about the likelihood of nuclear war. How valuable would it be to vote for the safer leader?

To answer this, we can think about 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 total nuclear war would kill around 80% of the US population.7 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 also affect people overseas, as well as untold future generations.

The policies which are most impactful are not always the most salient. George W. Bush’s famous choice to pursue the Iraq War resulted in the removal of Saddam Hussein, though at the cost of hundreds of thousands of civilian lives and trillions of dollars in spending.8 But President Bush also dramatically raised US spending on antiviral drugs for impoverished victims of HIV in Africa. This ‘PEPFAR’ program probably would not have been pursued in his absence, and likely prevented several million deaths.

Though the above is not a systematic survey, and some examples are atypical, to me they suggest that the outcome of elections will often have significant consequences.

Of course, not every election is that important. Sometimes all the candidates likely to win an election are similarly good overall, or if one of them is better it’s hard to figure out which it is.

In particular, within some electoral systems — for instance those with compulsory voting and electoral candidates chosen by politicians or party professionals — the tendency for parties to strategically bunch together in the middle of the political spectrum is strong.

More stark differences tend to arise in places with low turnout, few checks on executive power, plurality voting along with more than 2 viable candidates, and party primaries in which only the most motivated voters participate. In those elections the differences between candidates tend to be larger, meaning it’s more often important for the right group to win, and it’s easier to tell which group that is.

What if you’re wrong?

So far I’ve argued that voting can represent a great opportunity for social impact if:

• You’re in a close district in a close election
• There is a noticeable difference in the desirability of different candidates winning

But there’s a sophisticated argument against this view:

You can only swing an election if roughly as many people are voting for the outcome you prefer as the outcome you oppose. But if the public as a whole is roughly split down the middle, why should you trust your own judgement on the matter? Sure, you’ve looked into it and think that your view is right. But so have many other voters and about half of them still disagree with you. So because there’s no principled reason to trust your judgement over that of others, even after doing your political research you should still think you’re only about 50% likely to be voting the right way.

This is an application of the case for epistemic modesty, and it has some bite. If you think half of your fellow voters are getting things wrong, why should you think you’re getting it right?

This uncertainty about whether you’re truly voting the right way reduces the expected value of voting. If you had no confidence at all in your judgement — in other words, if you thought you were as likely to be wrong as you were to be right — the expected value would fall all the way to zero.

However, to go as far as that this case for intellectual modesty requires that other voters be your ‘epistemic peers’ — basically that they be as smart, informed, honest, and motivated as you. And there are a number of reasons you might think you can cast a ballot more wisely and altruistically than average.

First, the level of information most voters have about politics and policy is quite low. Some typical examples in the US, taken from Ilya Somin’s 2013 book Democracy and Political Ignorance include:

• “A survey before the 2014 election … found that only 38 percent of Americans knew that the Republicans controlled the House of Representatives at the time, and the same number knew that the Democrats had a majority in the Senate. Not knowing which party controls these institutions makes it difficult for voters to assign credit or blame for their performance.”

• “For years, there has been an ongoing debate over the future of federal spending… Yet a 2014 survey found that only 20 percent of Americans realize that the federal government spends more money on Social Security than on foreign aid, transportation, and interest on the government debt. Some 33 percent believe that foreign aid is the biggest item on this list, even though it is actually the smallest, amounting to about one percent of the federal budget, compared with 17 percent for Social Security.”

• “In 1964, in the midst of the Cold War, only 38 percent were aware that the Soviet Union was not a member of the U.S-led NATO alliance.”

This should not be surprising and in my view is no reason to think poorly of your fellow citizens. People have jobs to do, family members to take care of, and personal projects to pursue. For most folks, following the ins and outs of policy debates is neither easy nor rewarding, and because they don’t live in close districts it’s not the best way for them to improve the world, either. On top of that, following the news can be bad for people’s focus and mental health.

While the polling above appears dismal, there is an active academic debate about how problematic it really is for voters to lack the basic knowledge they would seemingly need to vote wisely. The damage is partly reduced by uninformed voters making different random errors that cancel out, people using heuristics like ‘am I better off than I was four years ago’, and politicians paying attention to things voters are more likely to know (e.g. ‘I want better healthcare’) while ignoring their views on things they won’t (e.g. how best to organise a healthcare system).

Nonetheless, for our purposes the fact remains that simply looking up basic background information — like who is in government, where different parties or people stand on the issues, what experts say about those issues when surveyed, and so on — will give you a big edge over others when it comes to determining which candidate will produce better outcomes.

If you’re trying to figure out how best to treat a disease you have, it’s one thing to think you can do better than your doctor, and quite another to think you can do better than a random stranger.

Secondly, if you’ve read this article to this point, you’re likely unusually interested in figuring out which election outcome is best for the world as a whole.

But not all voters focus on that question. Some always vote for the same party as a matter of habit, without giving much thought to the expected impact on the world. Others care about which outcome is best for them and their family, or the country in which they live. Others vote to express their ideals, or their loyalty to a group, or just for fun.

If you truly aspire to vote for the outcome that is ideal for the whole world, considering everyone’s wellbeing in an impartial way, you are more likely to succeed at that goal than the many other voters who aren’t even trying.

Finally, even if it were individually rational to decide there’s no value in trying to figure out the right way to vote because of ‘epistemic modesty’, the approach would foster collective laziness — leading all voters to be less informed than they otherwise would be, and likely worsening political outcomes. That would make it strange to recommend it to you all as a general policy.

Overall, while the risk of mistakenly voting for the wrong candidate reduces the value of voting, I don’t think it reduces it dramatically — at least not in the most important cases, where the difference between your options is a stark one.

If you think your research can get you to be 75% confident about which candidate is better, that is half as valuable as being 100% confident you’re making the right decision.

Is deciding how to vote too much effort?

While we haven’t been able to place a clear dollar value on a vote in a close district in a close election, we saw that in the US each of those votes influences more than a million dollars worth of government spending, and could have the same or greater impact in other ways.

This suggests that a vote for someone who substantially increases the value of that spending — or otherwise improves government policy — could be worth the equivalent of tens or hundreds of thousands of dollars to your fellow citizens.

If you divide that by the time it takes to vote — minutes in some countries, hours in others — this looks like a great opportunity to do good.

Compare it to earning money to give to the very best charity you can find: even if you assume that the organisation can turn \$1 into something as valuable as giving other people in your country \$100, you’d need to be able to give ~\$1,000 in an hour to make it as valuable as a vote worth \$100,000.

But the true cost of voting is much more than the time it takes to vote. In practice you need to do the research described above to figure out who is best to vote for. This additional effort substantially reduces the good you can do per hour.

Some people will follow politics and policy and form views about who it is best to vote for regardless. For them, figuring out how to vote is not an additional cost beyond what they are doing anyway. They may even find the process fun or energising.

But others don’t like politics and wouldn’t spend any time on it unless they felt it was their responsibility to do so. For them we can think of each hour spent deciding who to vote for as substituting for an hour of work or study that they could have otherwise directed towards improving the world.

How long does it take to decide how to vote? That will depend a lot on the election and how difficult it is to analyse the issues at stake. In some countries one party is clearly far more focused on the wellbeing of the world as a whole, or simply far more competent, than the other. But in other countries it’s legitimately hard to tell what outcome will be best.

Hypothetically, we can imagine someone who doesn’t follow politics at all between elections, and then tunes in to make a decision on who to vote for, and starts reading to try to make an informed choice. If this would require them to do the equivalent of a week’s work, it would increase the effective cost of voting 10-100 fold.

If they’re in a high-impact job already, working to solve a pressing global problem, it would be easy to see how it could be better for them to remain focused on the work in which they’re most specialised, and leave politics to others. Depending on someone’s salary, working for a week and donating the money to an effective charity could also easily be more impactful than doing research and then casting a vote.

If you’re short of time, I can think of two shortcuts you could use to quickly cast a vote that’s more likely to be for the right person than the wrong one.

The first is just to find someone you think is bright, shares your values, and follows politics more than you do, and ask them who to vote for.

The second is to look at opinion polling globally. Even if your country is split down the middle, the world as a whole might very strongly prefer one candidate,9 which is a very important piece of information from an ‘epistemic modesty’ perspective. Foreigners don’t get to vote in other countries’ elections, but they too have preferences about the outcome, are affected by the results, and their outside perspective might even give them insights that locals are missing.

Regardless, one thing to remember is that it will be easiest to tell which candidate is best to vote for in an election in which the difference is large — and these are also the elections in which a vote is of greatest value.

Another is that political participation is open to anyone who, for one reason or another, doesn’t have an especially impactful job at the time.

It’s hard to give general advice here, because in addition to all the variables like election closeness discussed above, individual voter’s opportunity costs vary a great deal. But if I had to give a rule I would say:

• If you already follow politics well enough to vote wisely (and you’d vote in a close election, etc.), it will often make sense to vote.

• If you wouldn’t follow politics except in order to have a social impact, and you have the opportunity to instead spend the requisite time specialising in a high-impact job working on a pressing problem, or earning to give for effective charities, or something similar, that will often be the better option.

How much does it cost to drive one extra vote?

What if you think the outcome of an election is important enough that you want to do more than just vote yourself?

For most of us, the low hanging fruit is to contact friends and family in competitive districts, encourage them to vote, and make the case for our preferred candidate. Unsurprisingly research shows that personal appeals from friends and family have a big impact, and have 10 times or more than the effect of an appeal from a stranger.

But having exhausted your friends, you might decide you want to give money to a campaign as well. How much do you have to give to get your candidate one extra vote?

With billions of dollars spent on political advocacy in the US each year, this has been the focus of substantial research. Campaigns can randomly target ‘get out the vote’ efforts on some voters and not others, and then see how much more likely those voters are to show up.

This table from the 2015 edition of Get Out The Vote summarises the results of those sort of experiments, with the cost per vote in the final column:

According to these studies, for those methods shown to work — such as door-knocking or phone-banking — persuading one stranger to vote for your preferred candidate costs \$30-100, or a few hours of work as a volunteer.

If, having compared the candidates and the closeness of the election, you think a vote for the right person is in some sense worth thousands of dollars, that sounds pretty good. However, it has to be compared to the best alternative ways to use your money to improve the world, which may also offer a huge return on investment.

On top of that I’ve been advised by researchers I trust, who have investigated the topic in detail, that these figures are underestimates, at least for the big elections you’re most likely to follow.

That’s for multiple reasons. One is that all results in social science tend to look weaker over time as they’re scrutinised and people attempt to replicate them.

Another is that political campaigns, at least in the US, have more money for each voter they’re chasing than they did in the past. New technologies also make them better at targeting the voters most likely to be convinced. As a result, swing voters in swing states are already contacted with campaign messages again and again, reducing the impact of any further prompts.

For instance, a 2020 paper looking at TV ads in recent US presidential elections suggested a cost per vote of \$100-1,000, which is probably now more typical.

However, not all campaigns are as well resourced, and the less funding they have the cheaper it’s likely to be for them to find additional supporters.

The campaigns for Joe Biden and Donald Trump, along with allied groups, are likely to have about \$30 per voter in potential tipping point states. Both have set new fundraising records for presidential campaigns.10

But the Biden campaign had just a tenth as much — \$3 per voter — in the 2020 Democratic primaries through Super Tuesday (after which the primaries began to wind down).

That difference is even starker when you consider that a much larger fraction of voters are open to switching their support in primary elections than in general elections (though keep in mind the differences between candidates within a party are less than the differences between parties).

This level of funding in general elections is somewhat unique to the US. Different campaign finance arrangements mean that parties in the UK and Australia both have closer to \$10 per voter in a marginal seat. 11 12

In these circumstances the experiments suggesting a cost of \$40-100 per vote could even be overestimates, but I haven’t yet investigated the research on the impact of campaign spending outside the US.

The question of when political campaigns are the best use of someone’s charitable giving is also beyond the scope of this article, and seems likely to hinge on how well funded the campaigns are and how large the difference is between candidates.

But if you can encourage someone to vote for <\$100, while you think the social value of an extra vote is >\$10,000, then it should be possible to make a case that it’s competitive with other options. That is something I hope to investigate in more detail in future.

And if voting yourself is worthwhile, contacting friends and family to encourage them to do the same will also usually be above the bar.

Overall, is it altruistic to vote?

The answer is clearly yes, under the following conditions:

1. The election concerns important issues, such as the allocation of large amounts of money, or the foreign policy of a country with a large military
2. One candidate is substantially better than the other, and you’re in a position to know which one that is
3. The election is somewhat competitive, and you’re able to vote in a competitive seat, or district, or state

In a situation like that, the hour you spend voting is likely to be the most impactful one in your entire year, and could on average get you some influence over how hundreds of thousands or millions of dollars are spent. For this reason I vote whenever I get the chance.

When they vote, some of my friends feel very nervous about whether they’re voting for the right person. While there’s a lot they don’t know, surveys how much the public knows about policy issues suggest that they’re a lot more informed than the average voter, and so their input should increase the odds of the better candidate winning. We shouldn’t make the perfect the enemy of the good.

All of that said, I respect people who consciously opt out of following politics, in order to preserve their focus on other important work that improves the world. Following politics and developing informed views can absorb a great deal of time. While spending one hour voting is highly impactful, spending hundreds of hours tracking politics in between elections isn’t — at least if you aren’t regularly taking action based on what you’re learning.

Finally, while persuading other people to vote takes more time or money than simply voting yourself, in elections where you’re confident one candidate is much better for the world than another, joining or donating to a political campaign may also represent a high-impact way of improving the world.

Has this article helped you better estimate how important it is to vote?

As we noted above, appeals from people they know are much more likely to influence people’s behaviour than TV ads or impersonal mail. Some experiments suggest a personal appeal from a friend could increase someone’s likelihood of voting by as much as 10 percentage points. So consider sharing this article with your friends.

Notes and references

1. Gelman and colleagues have three papers estimating the probability of one vote being decisive in one of the biggest elections in the world — those for the US presidency:

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.

2. This piece also builds upon previous articles from the rationality community such as Politics as Charity by Carl Shulman and Voting is like donating thousands of dollars to charity.

3. You can see a current and easy-to-read version of the model here, updated figures here, and the numbers we used from October 14 here.

4. The chance of a tie in an election with perfectly even polling is proportional to 1 over the square root of the number of voters, and note that sqrt(10) = 3.2 and sqrt(5) = 2.2.

5. The alternative model is the following:

Political parties and candidates know they’re in a competitive and strategic race to get the most votes. If they can’t get elected they can’t achieve anything, so they constantly adapt their positions, and add or remove interest groups from their coalition, to ensure they have a decent chance of winning.

If young voters in the US suddenly started voting at the same rate as seniors — 70% rather than 42% — any political party that didn’t adjust its positions to increase its appeal to those voters would quickly become irrelevant.

So rather than thinking of your vote as having a tiny chance of completely swinging an election outcome, you can instead think of it as having a high chance of nudging every party just a little bit in the direction of the political views held by you and people like you. This is one reason there’s still value in voting, even if this year’s election doesn’t happen to be especially close: by indicating you’ll vote in future years you give politicians much more reason to appeal to you.

I’ll also just add that in multi-party systems, such as those involving proportional representation, rather than completely flip an election result your vote is more likely to change which grouping of parties forms a coalition government, and their relative influence within the coalition.

What both of these alternative analyses have in common is that they replace a very small chance of a hugely valuable outcome, with a higher chance of a somewhat less valuable outcome.

While formalising either of these models is going to be more challenging, I expect that these two changes will usually roughly cancel out, leaving the overall expected value about the same.

6. In the US, the federal budget is actually the result of an interplay between three elected actors: the President who is reelected each four years, the House of Representatives which is reelected each two years, and the Senate which is reelected each six years.

As you can vote in all of these elections simultaneously (and often state and local election too!), and on average their terms are four years each, for simplicity I’ll treat them though they were all elected simultaneously each four years.

As you might expect, if you can vote in marginal elections for two of these bodies at once the case for voting will be about twice as strong as if you can only do so for one.

7. See this book on the likely deaths from a nuclear war.

8. From Wikipedia: “Body counts counted at least 110,600 violent deaths as of April 2009 (Associated Press). The Iraq Body Count project documents 185,000 – 208,000 violent civilian deaths through Feb 2020 in their table.”

9. For examples of this kind of polling, see:

10. The most extreme case I could find was the Senate campaign of Al Gross in Alaska who has raised \$57 for each person who voted in Alaska in 2016.

11. In 2017 the Conservative party spent £18.6m contesting about 10% of the actual seats. Across the whole election 32 million votes were cast. This comes to about \$8 USD per voter.