## How much is your vote worth?

Tyler Cowen writes:

If it were legal, and you tried to sell your vote and your vote alone, you might not get much more than 0.3 cents.

It depends where you live.

If you’re not voting in any close elections, then the value of your vote is indeed close to zero. For example, I am a resident of New York. Suppose someone could pay me \$X to switch my vote (or, equivalently, pay me \$X/2 to not vote, or, equivalently, pay a nonvoter \$X/2 to vote in a desired direction) in the general election for president. Who’d want to do that? There’s not much reason at all, except possibly for a winning candidate who’d like the public relations value of winning by an even larger margin, or for a losing candidate who’d like to lose by a bit less, to look like a more credible candidate next time, or maybe for some organization that would like to see voter turnout reach some symbolic threshold such as 50% or 60%.

If you’re living in a district with a close election, the story is quite different, as Edlin, Kaplan, and I discussed in our paper. In some recent presidential elections, we’ve estimated the ex ante probability of your vote being decisive in the national election (that is, decisive in your state, and, conditional on that, your state being decisive in the electoral college) as being approximately 1 in a million in swing states.

Suppose you live in one of those states? Then, how much would someone pay for your vote, if it were legal and moral to do so? I’m pretty sure there are people out there who would pay a lot more than 0.3 cents. If a political party or organization would drop, say, \$100M to determine the outcome of the election, then it would be worth \$10 to switch one person’s vote in one of those swing states.

We can also talk about this empirically. Campaigns do spend money to flip people’s votes and to get voters to turn out. They spend a lot more than 0.3 cents per voter. Now, sure, not all this is for the immediate goal of winning the election right now: for example, some of it is to get people to become regular voters, in anticipation of the time when their vote will make a difference. There’s a difference between encouraging people to turn out and vote (which is about establishing an attitude and a regular behavior) and paying for a single vote with no expectation of future loyalty. That said, even a one-time single vote should be worth a lot more than \$0.03 to a campaign in a swing state.

tl;dr. Voting matters. Your vote is, in expectation, worth something real.

1. pwyll says:

Given how toxic and divisive an effect the act of voting has on social community, I’d argue that *not* voting is more of an imperative if you live in a battleground state.

• Andrew says:

Pwyll:

Political conflict is real, with serious issues in dispute. Choosing not to vote won’t make that conflict go away.

• Terry says:

Does it make a difference if you could buy and sell blocks of votes?

What if middlemen could buy up thousands and resell them?

What if there were exchanges where millions could potentially be bought?

I can convince myself either way: that it makes a difference and that it doesn’t make a difference.

• Terry says:

Oops. Above post shouldn’t have been a reply. Should have been it’s own post.

• Andrew says:

Terry:

I think if you could sell your vote, politics would be different. But I haven’t thought much about this; I’m sure some other people have.

• pwyll says:

I believe allowing the buying and selling of votes would go a long way towards fixing the pathologies of democracy. I’d be very curious to see what would happen were one of the states be allowed to run the experiment.

There’s an important distinction however: most discussion of “vote-buying” involves political candidates making promises to compensate voters for voting for them. This is extremely corrosive because it isn’t permanent: the same “bribes” must be promised every election cycle. But if a vote were like a share of stock in a corporation, over time the incentives of the ruling elite should become better aligned with the fortunes of the country as a whole. (Mencius Moldbug has written extensively on this topic.)

• Blake says:

… correct. Voting or Not-Voting by average individual citizens has no effect on that general conflict nor election results.

political campaigns and elections are primarily political theater to persuade people that their ruling government is legitimate

• Andrew says:

Blake:

It is not true that voting by individual citizens has no effect. Any given vote is unlikely to have an effect on the national election, but the probability of an effect is not zero. By voting you have a small probability of making a big difference.

• Blake says:

…come’on now — nobody rationally acts upon tiny microscopic probabilities.

You have a dramatically higher probability of being badly injured in a car crash this year versus changing the outcome of a NYC election– yet you routinely ride in automobiles without a second thought

• Andrew says:

Blake:

Sure, people act on the risks of riding in a car all the time, for example by paying for air bags and other safety improvements.

• pwyll says:

I agree 100% that political conflict is real… you can model an election as a (regularly scheduled) civil war where both sides agree that whoever amasses the largest army is the winner. And to me, scheduling regular civil wars sounds like a great way to dramatically increase social tension and division. It’s a real testament to average Americans that the phrase “election-related violence” isn’t yet strongly associated with our country, but I’m not sure how much longer we can continue to rely on their goodwill.

2. denny says:

“not get much more than 0.3 cents”

… and equally as worthless in affecting election outcomes (with very rare exceptions

3. Z says:

“We can also talk about this empirically. Campaigns do spend money to flip people’s votes and to get voters to turn out. They spend a lot more than 0.3 cents per voter.”

Does the price you’re willing to pay per vote in a bulk buy of votes translate directly into the price you should be willing to pay for a single vote (if you’re totally rational)? Could it make sense to want to spend \$10 per vote for 10,000 votes but not want to spend more than \$1 for just 1 vote? Maybe this is never the case if you’re strictly rational…

• Andrew says:

Z:

When it comes to campaign spending, I think it should be proportional. Switching 1 vote will give you very close to 0.001 times the difference in win probability that you’d get by switching 1000 votes.

In practice, nobody’s trying to switch some precise number of votes (or get some precise number of people to turn out); rather, campaigns are trying to switch lots of people’s votes, even if in any given case the probability of switching is low.

4. Steve says:

I have a serious question for Andrew. Voting power makes it look like any single vote has very little power because my vote will almost never be the swing vote. Are their other ways to model the importance of the vote? For instance, I am also sending a signal with my vote to others in my voting coalition. My vote may not have much power to ever swing the vote since I am sitting here in New York, but it may signal to those in my coalition in Ohio that their vote is going to matter and they better get to the polls. I don’t know if that makes sense, but I am wondering if their are other ways of mathematically representing the importance of voting other than voting power.

• Another Steve says:

I have a similar question. I’m wondering how you square your TLDR with the evidence you provide. You say, ‘Voting matters. Your vote is, in expectation, worth something real.’ But to me, doing something with a 1 in a million chance of having an effect decidedly does *not* matter. And that’s the optimistic end of the range. In your NBER paper with Nate Silver, you say the average probability your vote will make a difference is 1 in 60 million, yet from reading your comments here and elsewhere, you still seem to think people should vote, or that it is important. I don’t see that conclusion following: to me, your papers are evidence that voting is one of the least useful things you could spend your time doing.

• Andrew says:

Another:

As Aaron, Nate, and I put it, voting in a national election is like buying a lottery ticket where, if you win, there will be potentially huge effects, not so much for you personally but for the country and the world. It is what it is.

Do you perceive voting as less useful than other things you could be doing, like watching a movie or going out to dinner? That’s your call.

• Another Steve says:

Andrew:

Appreciate your response. While I do think voting is less useful than some random leisure activity I might be doing (going to the movies, etc), I also think it’s just not the best way to be politically active. If your goal is to participate in democracy in some meaningful way, it seems to me that there are more efficient ways of doing that: lots of campaigns mine social media data, so commenting online or tweeting will get your ideas registered with politicians; ‘voting with your dollar’ by shopping at certain places and not others seems just as effective as voting to me (i.e., buy meat from what you consider ethical producers, and not from factory farms, if that’s what you care about). With the amount of data mining and research campaigns to, IMO just living your life as you normally would does as much to get your ideas noticed by politicians as voting does (the websites I view paint a picture of what issues I care about, and that’s probably being mined by politicians, or will be in the near future). Even calling a politician is probably more effective than voting, since from what I hear, they pay attention to that.

• Andrew says:

Another:

I agree that there are many ways to be politically active, and voting is just one of the ways. What is more effective will depend on who you are. And of course lots of people who are politically active in other ways, also vote.

• jim says:

An individual vote is valuable because it backs up whatever comes out of the individual’s mouth (or keyboard). That is to say If you speak out you can influence other people, but if you don’t back that up with a vote or people will rightly perceive you as a hypocrite and not listen to you.

• jrkrideau says:

I am writing in a Canadian context so it might not apply as we have a multi-party environment but a politician friend of mine says always vote because it raises the percentage of the vote going to the party versus the other parties even if there is no realistic chance of winning this time

That is, it establishes the legitimacy of the party in the eyes of the public for the next election.

• bxg says:

If “huge potential effects” should enter the calculus of whether to vote, we need the expected value of these effects – considering how likely large effects are to happen, and whether they will be very good or very bad. But in close elections, half of the voting public thinks each way. It seems even a little bit of humility should shrink your estimate of the expected _net_ value to vastly below the value you would get based on “I’m right about everything and half my neighbors are just silly/selfish/short-minded”.

But maybe you are really smart, and in fact tend to be right about these things. Then your “should I vote?” calulation might be special. But here we are offering advice to the general public – on both sides of issues – and to them (us) what’s the relevance of hypothetical huge stakes? Yes big things might happen, but the generic you is manifestly not particularly great at even guessing even the sign of these consequences (because the election is close.)
Should I vote is the chance of making a difference (small but not negligible in close elections) times the expected value of the difference you should make (which, for a generic “you”, should also be small in close elections.)

(My objection is wrong to the extent that if we all have our own value functions and are out for what’s best for us personally – because maybe we can guess the consequences to ourselves reasonably well – but Andrew’s “huge effects” statement was in the context of “not so much for you personally, but for the country and the world”.)

• Andrew says:

Bxg:

The reason to prefer one candidate or another is not that “I’m right about everything” or “I’m really smart.” The reason is that people have different views. Say, for example, that you have a certain view about government spending, or abortion spending, or some other issue of importance. To feel strongly about these issues does not mean you feel you’re right about everything, nor does it mean that you feel you are really smart. You can perfectly well believe that a certain policy could lead the country in to recession or war and have negative consequences for the country and the world. It’s not about personal taste, it’s about consequences.

• bxg says:

Yikes, I a bit taken aback that I’ve been so inarticulate as to be this badly misunderstood. Can I try again? Prefer the candidate you think is best for your view, yes! But as to the specific question of “should I bother voting”, not only does “what does the chance my vote makes a difference” matter (you basically concede that) but equally so does “what’s the expected improvement vs if my side wins vs if it loses”.

For the latter factor, thinking about the huge effects that might happen depending on who wins is just wrong/insufficient. You explicitly give a lottery argument, as if the “huge effects” were the payoff. No, a priori (i.e. as you vote), your rational _expectation_ of (the value of) these effects is what matters – _that_ is the relevant payoff when considering if the lottery is worthwhile. If you ignore the fact that the populace is split (again: that this is a close election, and we are given advice to a ‘generic’ potential voter) you are going to grossly overestimate the magnitude of the latter.

• Brian says:

It’s even less than that. Your vote won’t be the swing vote. The margin of error in counting is more than a single vote. At *best*, your vote will be the one that makes the margin of victory sufficiently narrow to have the outcome of the election decided by either a court, or a coin flip, depending on the state.

• Andrew says:

Steve:

Yes, people have looked into this. The short answer is that if there is coordination, you can consider voting as a group act rather than as an individual act. If 1000 people decide as a group to act, then that can make a real difference. Lots of politics works in this way.

5. Mathijs Janssen says:

I think it is misleading to interpret the fact that campaigns spend more than .3 cents per voter as proof that the value of an individual vote is higher than .3 cents. What matters for campaign spending is how much you spend relative to the opposition. It’s typically modeled an all pay auction and the equilibrium strategy (it’s a mixed strategy) is proportional to the benefit of winning the election. The probability of any vote being pivotal does not enter into the strategy. So the campaign spending speaks to just the value of winning the election, not that value times the probability of being pivotal, which is the value of a vote (as you know).

• Andrew says:

Mathijs,

I don’t take the fact that campaigns spend more than 0.3 cents per voter as direct proof of anything about voter motivation. From straight statistical analysis, I judge the average value of a vote to be much greater than 0.3 cents per voter. In addition, we can look at what campaigns actually do. Put it all together, and I don’t see any compelling reason to think that Cowen’s quoted statement is at all reasonable.

• Mathijs Janssen says:

Thanks for the reply. I think your phrasing in the last paragraph, “We can also talk about this empirically. Campaigns do spend money to flip people’s votes and to get voters to turn out. They spend a lot more than 0.3 cents per voter”, is a bit misleading. It certainly reads like you are adducing direct evidence. I also think the paragraph is a bit misplaced, since it only gives evidence for the fact that winning an election is valuable and no reasonable person – I have no idea if Mr. Cowen may be counted among them – would dispute this. The interesting discussion is about the probability of pivotality.

The straight statistical analysis, that the probability of pivotality is 1 in a million, is very interesting. I’m neither a political economist, nor empirically equipped to assess the claim, but that is certainly orders of magnitude higher than I would have guessed. So it seems valuable to highlight this very interesting contribution and not muddy the water. Anyway, let me do my part in that and stop with this distraction now. :)

6. Robert Foster says:

For a different empirical study of this, it would be interesting to look at what political machines in American history that did indeed buy votes, such as Tammany hall, were offering to voters. My understanding (as a layperson, so I can easily be incorrect here) is that the exchange was often services for votes rather than cash, but surely those can be equated to a rough estimate of the monetary value of the services.

• Peter Dorman says:

There is also an enormous literature on clientelist politics. From what I’ve read of it (I have a theoretical interest in clientelism more generally), there is less focus on valuing the provision of services and benefits and more on the factors that conduce to such arrangements in the first place. But yes, one could try to monetize the flow from patrons to clients and do analytical work on it. One problem, of course, is that clientelism is based on a different set of motives for political support than what we normally assume animates politics in the US. And in theory the client’s reward is predicated on the observability of their vote. It’s not about whether you are responsible for swinging the outcome.

7. aqsalose says:

The paper from 2007 was quite interesting, especially the appendix with the reasoning why probability of vote being decisive is approx. inversely proportional to the number of voters, 1/n. Then I started wondering that in addition to being uncertain about the vote differential d, the potential voters are also uncertain about n (or equivalently, turnout T). If you entertain a small a priori belief probability that T will be small (and abysmally small possibility that it is abysmally small), it might still influence your calculation of expected benefit of voting?