What’s your vote really worth? Your ballot paper probably won't be the one that changes the course of an election

Every time there’s an election — and there’s a big one on at the moment — a certain kind of person always goes around telling people how important it is that they vote. I can’t help but ask — are they right? How important is it? How likely is it that any one vote will make a difference to the outcome? And how big an impact will that outcome have?

I was particularly inspired by a recent post on the subject from Robert Wiblin of the effective altruism charity 80,000 Hours, who helped me understand a few things.

First, it’s unlikely that your vote will be the one that, on its own, changes the course of an election. It will probably not turn a win into a tie or a tie into a win. That’s because millions of votes are cast. 

If there were only 10 people voting as well as you, then there would be 1,024 possible different ways the 10 of them could vote, of which about 250 would be a 5-5 tie that you could break. So – assuming all the other voters were equally likely to vote A or B – there would be about a 25% chance that your vote would make the critical difference. If you go up to 100 other voters, it becomes about an 8% chance.

Of course, you can’t be the actual tiebreaking vote in the US election, because it’s not decided on the popular vote. But you could break the tie in a single state which then goes on to break the tie in the electoral college.

But how likely is that? On the face of it, not very. Even the smallest US state, Wyoming, has over 200,000 registered voters; most have millions. The odds of breaking the tie in one state is small: even in Wyoming (and even assuming other voters were equally likely to vote for either party, which they are definitely not; it is one of the safest Republican states, voting 67% Trump in 2016), then it would be less than one in 500. 

And even if you are the one who breaks the tie in that state, that state then has to be the one to break a tie in the electoral college for your vote to affect the overall outcome, and since there are 50 states and the smaller ones have less influence, the odds of your Wyoming vote being the tiebreaker are commensurately tinier.

So that might sound like your vote is essentially meaningless. But it’s a bit more complicated than that, for two reasons. One, not all states are equally likely to be tied, or to be tiebreakers; and two, the value of a vote is determined not just by how likely it is to have an effect, but by how big that effect will be.

The statistician Andrew Gelman’s model suggests that there is about a one in 7.7 million chance that a voter in Pennsylvania could swing the election, because it is a reasonably big state that is likely to be close. A voter in Wisconsin has about the same chance; a voter in Nevada, about one in 12 million. Meanwhile, very safe states – such as the aforementioned Wyoming, or California – are very unlikely: one chance in many billions. So if you live in a swing state, your chances of deciding the election are thousands of times higher than in safe ones. 

Wiblin raises an interesting aside, by the way. You might be thinking “Hang on, there’s no way that a state is really decided by a single vote. If it’s that close, there’ll be court battles about recounts and so on.” And that’s true.

If vote-counting were absolutely perfect, and we knew down to the exact one how many had voted A or B, then candidate A would have 0% chance of winning if they had one vote less than their opponent, and a 100% chance of winning if they had one vote more. But given that counting isn’t perfect, then it’s not quite as clear as that.

If the count finds that one candidate in a swing state has won by a single vote, it’s almost certainly going to be challenged in the courts, and that challenge might be successful. If that candidate is found to have won by 100 votes, the challenge is that much less likely to work. If it’s 1,000 votes, it’ll be less likely still. If it’s 10,000, there might not be a challenge at all. A vote for a candidate in that situation doesn’t instantly tip a result from win to loss, but it might make some small difference to the chance that the result will be challenged and overturned in the courts.

So instead of having a one in 7.7 million chance of being the tipping vote, a Pennsylvania voter might have a somewhat larger chance of helping to avoid a recount or to make their side more likely to win that recount, which then translates into the same one in 7.7 million chance of affecting the election overall.

Still, though, one in 7.7 million doesn’t sound very much. But that’s where we have to introduce our second point, how much the outcome matters. What matters is not just the likelihood, but the likelihood times the outcome. Here, I need to bring in the idea of expected value. 

Expected value is a concept from probability theory. Imagine we were playing a game of dice. We roll a die: on a score of one to five, you give me a pound; on a score of six, I give you £10. Should you take that bet? You’re probably going to lose, after all. But if we played six times, on average, you would lose £5 and win £10, for a total gain of £5. My expected value for each roll, therefore, is £5 divided by six, or 83p. I can’t actually win 83p on any single roll, but if I did it a million times, that’s what I’d expect on average. So the bet is a good one.

Let’s say, then, you’re in Pennsylvania, and you’ve got your one in 7.7 million chance of being the magic vote. How can we work out the expected value of your vote?

The obvious thing to look at would be the overall spend of the United States government. In the 2019 financial year, the US government spent $4.45 trillion. Over four years, that’s nearly $18 trillion, or about $130,000 for each American who voted in 2016. If it was as simple as “If the candidate you want gets in, he will spend that $18 trillion on good things; if the other one does, he will spend it on bad things,” then the expected value of a one in 7.7 million chance of directing $18 trillion is pretty huge. It would be the equivalent of being in charge of a nearly $2,400,000 budget. Even if you lived in California and had only a one in 9.4 billion chance of being the deciding vote, that is an expected value of nearly $2,000.

Of course, that’s not what actually happens. The bulk of the spending is already locked in, on social security, Medicare and Medicaid, road maintenance, interest on the public debt, things like that. 

And under the US system, presidents have surprisingly little power. Wiblin points out that last year, in the UK, the hung parliament meant that Theresa May’s government struggled to pass its agenda, and that this was really weird and unusual and the situation was rectified with an election. But in the US, it’s often true that the House of Representatives, the Senate, and the Presidency are split between parties, they stymie each other, and nothing very much gets done.

Still, the Trump administration, although in general (as I understand it) a legislative and policy failure, did manage to get a tax cut passed in 2017 which the Congressional Budget Office estimates lowered federal revenues by about $300 billion in 2018. That’s about 8% of the total revenue that year. More generally, presidents have huge power over foreign policy – they can decide whether to go to war, for a start, and the war in Iraq cost trillions of dollars. I think it’s pretty reasonable to think that a choice of president could affect a single-digits percentage of total US government spending. If we say 5%, and that you’re a Pennsylvania voter, then your vote probably has an expected value of about $120,000. (If you’re a California voter, on the other hand, it’s probably only about $100.)

There are harder-to-quantify non-budgetary values as well, of course. Donald Trump, I think it’s fair to say, says a lot of untrue things. If you think that his constant lies and misinformation are corrosive to public debate, you might feel that has some value that is difficult to assess in dollar terms. Similarly, he is a destabilising influence on the world order, threatening to remove the USA from international bodies like the WHO and Nato and undermining international cooperation; whether you think that’s a good or a bad thing depends on how you value the world order, but either way, again, it’s a difficult thing to price. 

Also, while – I’m surprised to learn – fewer immigrants have been deported per year under his presidency than under Obama’s, the administration has upped the rhetoric on immigration, and introduced a blanket policy of the separation of migrant children from their parents (separation did happen under Obama/Biden, but in rare cases, not as a general policy). Whether you approve or disapprove – the latter seems bafflingly cruel to me – you’ll want to assign some value to voting for it.

It’s not that you can’t put a dollar value on these things – you absolutely can, just as people can put a value on a year of human life (between $50,000 and $150,000 in the US, apparently). But it’s much harder, and would involve a much longer piece than this. I’d be happy saying they add significantly to the value of a vote; whatever value you give them, the same calculations about the likelihood of that vote having an impact apply.

But these hard-to-quantify values bring us to the third point: how confident are you – or should you be – in your decision? For instance, I would have probably bet that there were more deportations under the Trump administration than under the Obama one, but (as I have just illustrated) I would have been wrong. If I were a single-issue voter who cared solely about the number of deportations, I would have voted exactly the wrong way, if I assumed that a Biden administration would be more like Obama’s than Trump’s. If you are completely uninformed about some policy or other, then your voting will be no more likely than chance to improve the situation.

In general, whichever way you vote, about half of the population will vote the other way. In most situations, if you believe something and half of the world believes the exact opposite, that’s a good reason to pause and wonder how confident you should be in your belief.

That said, as Wiblin points out, there are reasons to assume – given that you are reading this – that you are better informed than the median US voter. He quotes Ilya Somin’s 2013 book Democracy and Political Ignorance as saying that less than 40% of US voters knew who controlled the Senate or the House ahead of the 2014 elections, or that the government spends more on social security than on foreign aid. (Which it does, by about 20 times.) Keeping informed about politics is time-consuming and difficult and most people don’t enjoy it. So if you are doing some relatively basic things, like reading about the US election, and know basic things like who your congressional representative is, you are probably better informed than most. Still – if you are, say, 75% confident that you know the best way to vote on some issue, then you should downgrade the value of your vote by 25%.

(I will note that I don’t trust people who claim to be 100% confident in anything.)

There are also ways of improving your confidence that your vote will do what you want it to do. Wiblin points out a very simple one – find a clever friend whose judgment you trust and whose political opinions you largely agree with, and ask them who they would vote for. You can also look up surveys of expert opinion on various matters, such as Chicago Booth’s US Economic Experts Panel.

So: is voting worth it? First – of course, there are other reasons to vote than for cold-eyed estimates of your impact on spending. It communicates your preferences; some people see it as a civic duty; some people just enjoy it. And there are other things to vote for than simply the president: votes for local congressional races, or even votes for local district attorneys or dog-catchers or whatever, have much smaller electorates and so it is more likely that your vote will be the deciding one (although on the flip side, the impact of those outcomes will be smaller).

There are a couple of other considerations, as well. Wiblin points out politicians can ignore demographics that don’t usually vote – young people, for instance. If you are in such a demographic, then simply by raising the percentage of that group which voted, then you will make a small contribution to forcing politicians to pay attention to what that group wants. Something similar applies if your voting in a safe state makes it less safe.

Another, more complex, point is that other people think like you, and your behaviour affects other people’s behaviour. So there’s effectively game of the prisoner’s dilemma going on. If there are five of us voting, and I know that two people will vote A, and I want to vote B, then I won’t bother if I know that the other two are going to stay home. But if they know I’m going to stay home, they won’t bother either. By reliably voting, you make it more valuable for other people to vote. 

But to come back to our original question, if you are thinking in pure numbers terms, then, in short, whether voting is worth it depends on where you live. If you live in Pennsylvania, then your vote probably has about a one in 7.7 million chance of affecting perhaps 5% of US government spending over four years – so it has an expected value of about $120,000. Depending on how confident you are in your ability to pick the right candidate, then your vote could be worth as much as that. Even if it takes, say, two hours to vote (which is apparently a thing), then you are doing $60,000’s worth of good per hour. 

Of course, if you have to spend many hours researching your vote beforehand, that reduces the value accordingly, but unless you value your time extraordinarily highly, then it is probably a good use of your time per hour.

If you live in Kentucky or Wyoming or Oklahoma, though, where the odds of your vote being decisive is more like one in 50 billion, then the calculations are very different. Your vote is probably worth something in the region of $20. I could entirely understand not wanting to spend two hours in the rain, or many hours looking up the impacts of different policy measures, for that. That doesn’t mean it’s not worth it if you see it as a civic duty, but I don’t think you could be reasonably judged for deciding it is not worth your time.

If you’re in one of the in-between states – Colorado, or Alaska, or Ohio, places in the one-in-150-million-ish zone – then you have to decide. The sum I’ve been using is $18 trillion (the total US government budget over four years), multiplied by 0.05 (the percentage I estimate that would be affected by a change in president), divided by the chance that you will cast the deciding vote, as given in Gelman’s model here. The expected value of your vote in those in-between states ends up in the mid-four-figures, usually. Then you need to work out how many hours it will take you to do the research if you need to, and to vote. And if you’re not confident that your vote will help – if you think you’ll just be adding noise to the already noisy system – then you may well decide it’s not worth it. 

If, though, like me, you live in north London, haven’t been to the USA for years, and are in no way a US citizen, then you can’t vote at all. Your opinion, as opposed to your ballot, is worthless and you are paying too much attention to the Americans again. But since you will have elections of your own in your own backwoods country, you might dust this piece off in four year’s time.

---