Vote!

Or, less pithily, under certain assumptions that are fairly likely for some of you, voting has a significant expected difference!

[Note: this post has been edited on 10/28/24 with more accurate numbers]

Does one vote make a difference?

It’s that old question again. A lot of people seem to think that one vote makes no difference, and that the only reason to vote would be to feel good about participating in civic life.

Alas: One vote probably won’t make a difference. But, contrary to this common attitude, the chance that your vote will make a difference is large enough that, in many cases, it is really important to vote.

The Potomac River, with VA on left, MD on right. (Algonkian Regional Park can be seen on the MD side.) DC and the White House are 25 miles further ahead. Photo by me, September, 2014, Creative Commons License CC BY-NC4.0. Yes, the picture shows two strands of the river merging back together, rather than going apart, so the metaphor with divergent futures doesn't actually work. But give me a break.

Let’s start with the typical way act-consequentialists analyze these situations. Take the chance that your vote in fact makes the difference in the election, and multiply that tiny number by the very large amount at stake. Consequentialists typically don’t think in terms of acts making or causing a difference, but in terms of a act’s having a certain expected value. I will argue here that voting (in some cases) has a very large expected value. And I should say that I think you should be interested in the result of the argument here even if you are not an act-consequentialist.

Debates about the efficacy of individual actions also happen in questions of greenhouse gas emissions (do the emissions from a single drive have any expected difference in harms from climate change?) and veganism (does my purchase of an animal product have any expected difference in how many animals are harmed at farms?). (See, e.g, a couple papers I’ve written.) One thing I will note is that in these debates, it is mostly (though not entirely) non-consequentialists who argue in favor of the individual inefficacy claim, and consequentialists like me who argue that the expected difference of individual acts is in many domains (like those above) are significant. I have some thoughts as to why the disagreement over what is primarily an empirical question is divided along these ideological lines, but I’ll save that and some other similar considerations for a follow-up post.

One final preliminary point. If you really do think that one vote makes no difference, then why do you think that your vote contributes anything to civic life? Why should you feel good about voting? Now, if you show up at the polling place, and enjoy having some interaction with the other civic-minded folks there (who, perhaps together, can make a difference), then that probably will feel good, and I think that does give a reason to for you to go and vote. But if you are just putting your vote into an envelope and sending it off knowing that it goes directly into an abyss of uselessness, then it seems to me to be more fitting to feel sad about voting. So if you are not convinced by the arguments I give here, I’d recommend to you to vote in person! (Or, at least, if you vote by mail, go and talk about it a lot with other people!)

Why some specific votes have a very large expected difference

Let’s say you live in Pennsylvania. In the last Presidential election, Pennsylvania had 8 million votes. Let’s say this one will have 10 million votes (as that’s a rounder number and, being larger, works against the point I’m trying to make). Let’s say you are the 10M+1st voter. The polls today are saying that it’s a dead heat. (Below I’ll talk about cases that aren’t dead heats.)

I should be explicit that if you live in states that are much less of a toss-up as Pennsylvania, as most states are, and which are much less likely to be deciding states in the election (which those same states are), then indeed one vote is likely to not have a significant expected difference. However, I did some back of the envelope math, and it seems to me that the argument I’m about to give does apply to all seven+ of the swing states in the 2024 election (PA, AZ, WI, MI, NV, NC, GA, plus the Nebraska 2^nd district). I can give more details later if you really want me to.

In an excellent article that is the basis of this post, titled “Why You Should Vote to Change the Outcome”, Zach Barnett shows that the expected value of voting in many cases is not insignificant. I’ll simplify Barnett’s argument and apply it to Pennsylvania. Let’s follow the best most recent assessment and say that Pennsylvania has a 2% chance that it will be the single tipping point state.

Next, history suggests that if a state, by itself, tips the Electoral College scales, the vote margin between the winning and losing candidates in that state will be less than 2.5% of the total vote – and probably much lower. That means that it is most likely that the winning candidate, one or the other, will get between 5,000,000 and 5,125,000 votes (setting yours aside). And that means that there are 250,000 (and one) possible scenarios of how the vote (of all the non-you voters) will turn out. What are the chances that it will be a tie? Well, that very specific middle (5M-5M) possibility is at least as likely as each of the other individual possibilities is, and is probably more likely than the others. Thus there is at least a one in a 250,001 chance that if you vote, you, personally, will cast a deciding vote in Pennsylvania. And probably much greater.

If we multiply the chance that your vote will make a difference in Pennsylvania by the chance that the state on its own will tip the election, the chance that your vote will be the difference in the Presidential election for the entire United States is greater than one in 5 million. It is nearly sixty times more likely than a ticket winning the PowerBall lottery.

I should note that a 2010 paper co-authored by statistician Andrew Gelman, Nate Silver, and economist Aaron Edlin argues that, given the polls for the 2008, voters in some swing states had a 1 in 10 million chance of swaying that election. The number I’ve derived using Barnett’s method is in that same ballpark. And Barnett’s argument is beautiful in its simplicity - it sets a low bar for your chances within a state, without complicated modeling or calculations. Hopefully you were able to follow along.

These still may seem like small odds. But the stakes are enormous: the federal budget is about $7 trillion per year. Among other things, the President is responsible for non-military discretionary spending, which is approximately one eighth of the budget. Thus, the President controls $3.5 trillion over four years. Divide that number by the chance that your vote will decide the entire election, and you’ll find that the expected value of your vote is $700,000. Similar calculations can be done for voters in the other swing states. I can’t speak for you, but the time it takes me to vote is worth a lot less than that.

We should be cautious in interpreting these numbers. Although Trump says that he will completely overhaul the federal workforce, he likely won’t spend the entire discretionary budget differently than Harris would. But discretionary spending is just one easily-quantifiable piece of a bigger picture: US Presidents affect the country and the world in ways that go well beyond government spending. These other consequences are enormously important but are impossible to put a number on. The upshot is that a vote in Pennsylvania has an expected $700,000 difference in what the next President can choose to spend, plus one vote’s proportion of the innumerable additional effects of a Trump or Harris Presidency.

Objections and Replies

There are a bunch of ways to object to these valuations. I don’t think any are successful, but I’ll discuss two here.

First, what if it is not a dead heat? What if the polls are showing one candidate ahead by 51 to 49? Here I’ll have to refer you to Barnett’s paper for the details. Barnett shows that under certain conditions, as long as each of the two candidates has at least a 10% chance of winning, then one vote will have a chance of making a difference at least as high as one divided by the number of voters. So this kind of argument applies in a large swath of cases. And if we give each candidate at least a 25% chance, then the likelihood of one vote making a difference is way higher than that.

It should be noted that if the polls are showing 51-49, it doesn’t mean that the leading candidate has a 51% chance of winning. For instance, if the polls in the past have always been exactly right, then based on the evidence of the 51-49 poll, that leading candidate has a 100% of winning. So the determination of chances of winning has to be made on the basis of the margins for error of the polls, and also the past performances of polls. Typically, however, with the way polls have worked, margins for error in polls are 3% or greater, and the polls in the swing states are mostly within that. And polls in the past have been mistaken by quite a bit.

So there are factors that push both in the direction that the chance that one vote in PA will be the deciding vote is greater is more likely that what I say, and in the direction that it is less likely than what I say. However, under the very plausible assumptions that (a) conditional on PA being the single tipping point state, it is most likely that the tally will end up somewhere between 5.125M/4.875M and even, and (b) the likelier individual outcomes are grouped closer towards 50/50, then the expected value of one vote in PA will be greater than the value I gave above.

Second, one might say that if the initial election tally in PA is such that it is decided by one vote, then there would be a recount, and the courts would get involved since both sides would sue, etc. I am quite sure that this is true. But it, most likely, doesn’t undermine the argument. That’s because, after all the lawsuits, etc., it will most likely still come down to some vote tally. Maybe a recount will flip it considerably in the other direction from the initial vote. But this doesn’t show any problem with the argument: the odds are just the same that the recounted number that the courts end up using will be decided by one vote.

And it is also the case that one vote may make the difference in whether a recount is done in the first place (in PA, it happens automatically when the voting margin is within .5%), or there is a point at which lawsuits will not happen, or the courts will dismiss suits from one side or the other. Again, one vote can make that difference. Recounts and such just shift the point of difference-making from the precise majority in the initial vote tally to something different.

However, there is a nearby scenario where your one vote will indeed not make a difference. That’s if (1) it is almost certain that the tally will be, say, between 50.1% and 49.9%; (2) there is high likelihood that if the difference between the candidates is less than, say, .5% the case will go to the courts; and (3) the way the courts decide has little to no connection with the original tally (because, say, once it’s in the courts, it’s like a coin flip, or the judges are biased so that one side is nearly guaranteed to win). Under those three conditions, indeed your vote will have no expected difference. My sense is that the system is not that broken. But this does put a qualification on the Barnett-based argument that I have given. We’d have to put odds on the conjunction of the vote margin being less than the court-triggering margin, and odds of the tally itself having little to no role in the outcome once it’s in the courts. I’d be curious to hear your thoughts about those odds, especially for PA.

But it should be noted that even if those odds are 50%, the expected value of your PA vote would still, most likely, be at least $350,000, and probably greater.

In Conclusion

Vote! (Hedged appropriately, in the ways discussed above.)

I hope that these results encourage you to get out and vote, and to help others vote. They also illustrate a bit of the wonder of being human: while an individual occupies only a small place in the vast world, the difference that one can make in it is surprisingly significant.

Also, if you have friends in swing states who will vote for the better candidate, tell them how important it is for them to vote. (And please share this with them!)