## Why vote?

There’s an article by Stephen Dubner and Steven Levitt in the New York Times today on why it’s rational to vote. They correctly point out that the probability of casting a decisive vote is very small. Unfortunately they don’t seem to be aware of the social-benefit motivation for voting, which is why voting is, in fact, rational behavior. See here for our paper on the topic. For convenience, I’ll repeat our earlier blog entry below. The short version is that you can be rational without being selfish.

The chance that your vote will be decisive in the Presidential election is, at best, about 1 in 10 million. So why vote?

Schematic cost-benefit analysis

To express formally the decision of whether to vote:

U = p*B – C, where

U = the relative utility of going and casting a vote
p = probability that, by voting, you will change the election outcome
B = the benefit you would feel from your candidate winning (compared to the other candidate winning)
C = the net cost of voting

The trouble is, if p is 1 in 10 million, then for any reasonable value of B, the product p*B is essentially zero (for example, even if B is as high as \$10000, p*B is 1/10 of one cent), and this gives no reason to vote.

The usual explanation

Actually, though, about half the people vote. The simplest utility-theory explanation is that the net cost C is negative for these people–that is, the joy of voting (or the satisfying feeling of performing a civic duty) outweighs the cost in time of going out of your way to cast a vote.

The “civic duty” rationale for voting fails to explain why voter turnout is higher in close elections and in important elections, and it fails to explain why citizens give small-dollar campaign contributions to national candidates. If you give Bush or Kerry \$25, it’s not because you’re expecting a favor in return, it’s because you want to increase your guy’s chance of winning the election. Similarly, the argument of “it’s important to vote, because your vote might make a difference” ultimately comes down to that number p, the probability that your vote will, in fact, be decisive.

Our preferred explanation

We understand voting as a rational act, given that a voter is voting to benefit not just himself or herself, but also the country (or the world) at large. (This “social” motivation is in fact consistent with opinion polls, which find, for example, that voting decisions are better predicted by views on the economy as a whole than by personal financial situations.)

In the equation above, B represents my gain in utility by having my preferred candidate win. If I think that Bush (or Kerry) will benefit the country as a whole, then my view of the total benefit from that candidate winning is some huge number, proportional to the population of the U.S. To put it (crudely) in monetary terms, if my candidate’s winning is equivalent to an average \$100 for each person (not so unreasonable given the stakes in the election), then B is about \$30 billion. Even if I discount that by a factor of 100 (on the theory that I care less about others than myself), we’re still talking \$300 million, which when multiplied by p=1/(10 million) is a reasonable \$30.

Some empirical evidence

As noted above, voter turnout is higher in close elections and important elections. These findings are consistent with the idea that it makes more sense to vote when your vote is more likely to make a difference, and when the outcome is more important.

As we go from local, to state, to national elections, the size of the electorate increases, and thus the probability decreases of your vote being decisive, but voter turnout does not decrease. This makes sense in our explanation because national elections affect more people, thus the potential benefit B is multiplied by a larger number, canceling out the corresponding decrease in the probability p.

People often vote strategically when they can (in multicandidate races, not wanting to “waste” their votes on candidates who don’t seem to have a chance of winning). Not everyone votes strategically, but the fact that many people do is evidence that they are voting to make a difference, not just to scratch an itch or satisfy a civic duty.

As noted above, people actually say they are voting for social reasons. For example, in the 2001 British Election Study, only 25% of respondents thought of political activity as a good way to get “benefits for me and my family” whereas 66% thought it a good way to obtain “benefits for groups that people care about like pensioners and the disabled.”

Implications for voting

First, it can be rational to vote with the goal of making a difference in the election outcome (not simply because you enjoy the act of voting or would feel bad if you didn’t vote). If you choose not to vote, you are giving up this small but nonzero chance to make a huge difference.

Second, if you do vote, it is rational to prefer the candidate who will help the country as a whole. Rationality, in this case, is distinct from selfishness.

1. Stephen says:

So if people believe that their preferred candidate winning will give society as a whole a huge benefit, then what does that translate into for themselves?

It strikes me that you still need to put a value on how much they want this societal benefit. I presume that you're not saying that they value the candidate winning as the total of all the expected benefits for everyone in society, right?

Because what you probably need is the value that someone would place on the hypothetical choice of switching presidents from the preferred to the other. What are you implying that most people will say to the question "Would you rather have \$100K and switch presidents or keep your candidate in office and get zero?" I'm pretty sure that most people would take the money.

So in that case, it's the \$100K (or less probably) that should be discounted to reflect the chances of being pivotal. I agree with you that people's views of benefits others may get from politics may influence their desires to be altruistic. I view it as the same thing that prevents most people from throwing a soda can out the window on a road trip when you can look around and tell that nobody's going to catch you doing it. Whether or not this is "rational," I'm not sure, but it seems to me that people vote because they feel good about voting. Closer races probably get more attention and focus people's interest on what's at stake, which likely feeds into desires to "do the right thing."

In the end, though, it seems that under your setup, the relevant question is how much would someone pay to switch the president from the bad one to the good one, and what does this value become after multiplying it by the probability of being pivotal. Is this worth more than the effort of voting. I suspect that it's usually not.

2. Andrew says:

Stephen,

Thanks for the comments. Here are some replies.

You ask, "So if people believe that their preferred candidate winning will give society as a whole a huge benefit, then what does that translate into for themselves?" It translates into happiness that others are happy, that America is strong, that kids somewhere are healthy instead of sick, etc. It's similar to my motivation to do a better job teaching statistics; to explain political science to the world through books, articles, and blog entries; to donate to charity.

In answer to the question in your second paragraph: Yes, the model (see page 5 of our paper) has a factor "alpha" which represents how much you value others' welfare, compared to yours. We assume that alpha is positive but much less than 1, for most people.

In answer to your third paragraph, I don't know how people would respond to the question, "Would you rather have \$100K and switch presidents or keep your candidate in office and get zero?" It's hard to know, because people don't really get this option. But I do think it's reasonable to put in a small effort to give a small probability of making a really big chance. For some back-of-the-envelope calculations, see the bottom half of page 6 of our paper.

Regarding your fourth paragraph: I agree that people vote because they feel good about voting (or would feel bad if they didn't vote). But I don't think this makes the rational-choice model any less valid. As we discuss in Section 5.2 of the paper, psychological and rational explanations are complimentary, not competing. It can be easier to enjoy an activity if there's a rational reason for doing it.

Finally, regarding your last paragraph: the evidence is that turnout is higher in close elections, that people really do donate small amounts of money to political campaigns (without any direct benefit to themselves), and that people vote strategically (which is consistent with the rational desire to make a difference with one's vote). I'm sure there's also some civic duty, etc.–the point is that, given the large stakes in modern elections, it can be rational to vote.

3. Martyn says:

I remember having an argument on this topic as an undergraduate. I was unable to convince my friend, a philosopher, that voting was rational. So I'm glad to see some good arguments in favour of it.

This is far from my field of expertise, but voting seems more rational to me if I consider it as a means to an end – influencing government – rather than just getting the right candidate in office. Assuming that elected officials want to be re-elected the next time, they should not pursue policies that are radically opposed by the other parties unless they are protected by a large majority. Even if my preferred candidate has very little chance of winning, I should still register my dissent in order to prevent the other candidate from doing too much damage (as I see it) while in office. Conversely, even if my prefered candidate is almost certain to win, I should still vote to protect them from the dissenters in my constituency, so that they pursue the policies I support with more vigour. This argument probably works better for a parliamentary system like the UK than for the US presidential election, but I wonder if is susceptible to the same social motivation argument.

4. Bill Carone says:

You have an equation in your paper

B=Bself + aNBsoc

and you give an example a=0.1 in your paper. I'm trying to assess my own 'a' value.

Say I want to use a=0.1 for my own decisions. Does that imply I must e.g. prefer (a)increasing 6 billion people's wealth by a penny to (b) increasing my own wealth by one million dollars (since B=\$6 million in case (a))?

5. Bill,

That's right. a=.1 for the whole world would correspond to your analysis. If you would rather have the million, then a

6. Bill Carone says:

I see.

Voting is rational (according to your definition), but not consistent with the decisions in the rest of your life. If you want to be a consistent decision-maker, you shouldn't vote. Do I have your position correct?

7. Bill,

I don't think it's possible or even meaningful to be a "consistent decision maker" in all of one's life. I think the most that one can hope for is to be rational within different spheres of activity. For example, in playing chess, I could follow more or less rational strategies, but is it rational for me to play chess at all? Maybe, maybe not, but this shouldn't stop me from considering the rationality of various strategies within a chess game.

To put it another way, utility theory is a means, not an end, and to apply it to all of life would make it a pretty useless tool, I think.

8. Bill Carone says:

I also don't think it is possible to act entirely consistently; we don't have the time to pull out our probability wheels and our u-curve every time we make a decision. We use heuristics in order to get things done in a reasonable amount of time, and these heuristics are different for different spheres of activity. For important decisions, we use more rigorous methods, for unimportant ones, we use heuristics.

However, "'tis a consummation devoutly to be wished". I wish my heuristics were completely consistent with one another and with my rigorous decision-making methods. I often critique and redesign my heuristics to be more consistent with the rest of my life. Decision theory has helped me with that, just like probability theory can help me with my faulty heuristics and biases I use when making inferences. Wouldn't it be great if people's heuristics about data analysis were just as good as the most rigourous Bayesian methods? (Well, maybe not for us statistics teachers, but go with me here).

I'm not sure about your chess analogy; my chess strategies do not impinge on the rest of my life like informed voting does. When they do, my chess strategies change. For example: "Why did you make that move?" "I had a plane to catch, so I threw the game." Or: "Oh man, I just played chess for 8 hours and didn't show up for work. Even though I played the best chess in my life, it was a huge mistake to do so."

I don't know what you mean by utility theory being a means and not an end. It is a means to choose a course of action based on your information and preferences.

I also don't know what you mean by "all of life". I only use it to make decisions about health and wealth. I wouldn't apply it to religion, marriage, ethics (unless I turn into a consequentialist someday), or other "Big D" Decisions, where making distinctions and trade-offs aren't called for (e.g. "for better or for worse" shows explicitly that we are not talking about trade-offs). Voting, I think, clearly does not fall into that category.

9. Brutal Truth says:

What is the point in bothering to vote for candidates in the U.S. anyway? Issues on the ballot I can understand taking the time to vote for or against for example if you live in California there's a reason to vote for Proposition 19 if for nothing else to further expose the Obama administration as reactionary by its response to its passage. But for actual candidates? Total waste of time.

After the Dems swept into power in Congress in the 2006 midterms on a platform of "Vote for us if you want to end the war in Iraq" and then refused to actually do anything to, you know, end the war in Iraq like cutting off funding for it by refusing to vote on supplementals that should have told anybody with common sense that they've been bamboozled. After Obama gets elected as a supposed "change agent" and despite a Dem majority Senate and House gives the American people a health care "reform" bill written by the big insurance companies and Big Pharma; after giving us a Wall Street "reform" bill written by Wall Street that does nothing to prevent their casino-like behavior; after getting a Nobel Peace Prize then immediately escalating the gas pipeline war in Afghanistan; after promising us a government of accountability then adamantly refusing to have his attorney general go after the war criminals Bush, Cheney, Rumsfeld and Rice, as apparantly accountability for torture, for the mainstreaming of police state measures with the Patriot Act and the starting of a war of aggression isn't worthy of examining as he's "moving forward, not looking backward"; as Obama has enshrined these police state measures and expanded upon them; and as Obama's surrounded himself with an administration made up of ruling class pukes from the Council on Foreign Relations, Trilateral Commission, Bilderberg Group and A.I.P.A.C. what can a sensible, informed person conclude but that it is an exercise in futility to go out and vote? Why bother when all the wealthy ruling elite will allow us to vote for are obvious conservatives and thinly-disguised conservatives? That's as undemocratic as Saddam Hussein's elections but with somewhat more sophistication.

Unless private money is taken completely out of political campaigns with each candidate instead getting an equal (but small) amount of public funding with campaigns lasting a couple months instead of a couple years the system will continue to be as artificial as professional wrestling. Candidates will continue to be nothing more than puppets of their wealthy corporate backers, answering to them instead of the average people of this country. Face it America: You don't have a democracy. What you have is a dog & pony show every few years, designed to make you think you have a say in what kind of government governs you. It is painfully obvious that you don't.