Why do we vote?

I always get in arguments with mathematically-inclined people about whether to vote or not. The mathematically-inclined point out very reasonably that the chances of your vote being decisive are perishingly slim.

(These mathematics are explained clearly in this PBS video by economist Gordon Tullock explaining why he does not vote. Hat-tip: Marginal Revolution)

But there is always a part of me that believes there has to be more to the story. My intuition is that if there really was no practical purpose, people wouldn't do it. The behavioral scientist in me argues that if a behavior has no rewards, people will have no motivation to do it.

Anyway, there are probably many reasons why people vote aside from the likelihood of actually changing the election. There could be psychological factors such as the desire to fit in. There could also be strategic factors like creating a perception of support for your candidate to lure other voters in.

In light of it being election day, I thought I would post two of the more interesting theories I have seen about why rational people should vote.

Eblin et al. argue in The Economist's Voice that rational altruists should vote because of the easy and extensive benefits to your fellow citizens:

Consider the upcoming presidential race. The two candidates have significantly different policies, and it seems plausible that the average benefit to the citizenry is $1000 or more per citizen to have the better candidate win. If one candidate makes global depression, nuclear war, or global climate change less likely, the benefits might be much greater.Multiply that number by the 300 million people in the U.S. or the 6 billion in the world (if the issue is depression, global climate change or nuclear war) and an altruist might rationally vote even if the chances of being pivotal were 1 in 300 million. And, in a close election, the chance will be much greater than that. Certainly, the chance of being pivotal was higher than that in the year 2000, especially for residents of Florida.

Consider this thought experiment. Suppose that an election looks close so that it seems that the percentage tally will be between a 48-52 split and a 52-48 split between two candidates. If there will be 100 voters, and you are the potential hundred-and-first deciding whether to vote, here is you're the situation. Assume for simplicity that each percentage split has equal probability. (Little changes if we assign a more complicated probability distribution.) Then there will be either 48, 49, 50, 51, or 52 voters voting for candidate A, each with probability 1/5. If there are 50 voters, then the election is a tie and your vote will be pivotal.

Suppose there are 300 citizens (the voter turnout is 50% and 1/3 of citizens are underage); then, under the assumption that you believe that candidate A will benefit the population on average by $1000 per person, your vote will confer an expected benefit of 300($1000)(1/5)=$60,000 on your fellow citizens. Even if you are no great altruist, you might be happy to give up a half hour of your time to vote if your vote is expected to confer that much benefit on others. How else could you create that much value?

Now consider the private benefits to voting. The expected private benefits are ($1000)(1/5)=$200, which would probably be worth half an hour of voting. So in this small election, either public or direct private benefits can justify your vote.

How does it look in a larger election? Suppose there are 100 million voters and 300 million people in the country. Then everything scales linearly. The probability of being pivotal falls to the order of 1 in 5 million as we explain in a recent article in the journal Rationality and Society. The expected benefit to others is then 300,000,000($1000)(1/5,000,000)=$60,000 -- exactly as in the small election! The expected direct private benefit, on the other hand, shrinks to become negligible at only ($1000)(1/5,000,000) = .002 cents.

Hence, in a large election, if you vote for the chance of being pivotal, it makes no sense to do so for the direct private benefits you expect if your preferred candidate wins. It makes a world of sense, however, to vote in the hope of conferring benefits on your fellow citizens if you believe that your candidate is better for the typical citizen.

Brian Weatherson at Crooked Timber takes a different tack. He argues that it signals to voters like you that they should also vote -- because you as a group would be more likely to win:

It might be that the probability that others like you will vote in the next election is a more-or-less continuous function of the number of people like you who vote in this election. This doesn't seem too implausible actually. If we imagine Democrats having received tens of millions fewer votes in 2000 and 2004, then it is hard to imagine the level of support and work for Democrats in this cycle that there actually are. If you imagine the prior Democratic vote being a little closer to actuality, but still down, it is easy to imagine support and enthusiasm levels this time being down.

If I were being more careful here, I'd try to work out the marginal efficiency of voting at producing future votes. That is, the marginal difference that an extra vote for your team in this election makes to the expected number of votes your team gets in subsequent elections. I suspect it is well over 1. That's because dropping a party's support by a lot, say 10,000,000 in a US context, feels like it could be so demoralisingly bad that the party struggles to recover. But this would need a lot more work. It is certainly easy to come up with not completely crazy looking mathematical models where the efficiency is over 2. If everyone like you is a little demoralised by your not voting, even if it's just a very little bit, that could easily translate into votes lost next time.

Moreover, that effect need not be localised. It's pretty unlikely that New York will be a swing state any time soon. It's unlikely that it will be close, and conditional on it being close, it's unlikely that the national race will be close. But cross-voter incentives can work across state lines. People pay attention to what the national popular vote number is, and it effects their marginal disposition to vote/campaign for candidates that you like. So a vote in New York now may translate into benefits in swing states (which probably means the Mountain West) in the future.

So I think there's a reason for rational altruists to vote, even when they know they won't be the marginal vote. They are signaling to like-minded voters that they will vote, and that's crucial for sustained success. This isn't an alternative to Edlin et al's model. In fact I think they are right that someone with no other-related preferences shouldn't vote, even given the dynamic effects of voting described here. But I think the dynamic effects (a) make the case for voting considerably stronger, and (b) explain why it is rational to vote even in states that won't be close, and won't be swing states if they are close.

Both views are things to ponder on this election day.

For the record, I voted very, very early this morning in NYC, and even then there was a line. Record turnouts are likely to be the norm for this election. If you still need to vote, I would go early.