# Why it’s rational to vote

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.

See here for the full paper (joint work with Aaron Edlin and Noah Kaplan)

## 10 thoughts on “Why it’s rational to vote”

1. Andrew:
You claim it's rational for you to vote because your vote will benefit the country as a whole. I think there is a mistake in your reasoning. The country is not a passive thing. The country is made mostly of voters. Each one of the voters in the country votes for whoever he thinks is best. For your vote to benefit the other voters, you have to assume implicitly that you are smarter than the other voters and that you know what's good for them better than they know themselves. Clearly this cannot be true for the average voter. Therefore it cannot be rational for the average voter to vote on this ground.
Regards,
Olivier.

2. Oliver,

I think we're mostly in agreement. You write, "Each one of the voters in the country votes for whoever he thinks is best." That's what I'm saying too.

But I disagree with your statement that "you have to assume implicitly that you are smarter than the other voters and that you know what's good for them better than they know themselves." There are a lot of legitimate differences in opinion in politics (as in other areas of life). Just as I might think Roger Clemens has another good year in him and you might think he's washed up, we could disagree on other issues. Person A might support the Iraq War while Person B opposes it. Person A might support a national health insurance plan while person B supports a tax cut. You can disagree with someone without thinking (explicitly or implicitly) that you're smarter than them.

3. You can disagree with someone without thinking (explicitly or implicitly) that you're smarter than them.

Doesn't this mean that you acknowledge that there is an irreconcilable divergence of interests?
To put it more bluntly doesn't this implicitly says to the other party "Smarter or not screw you"?

4. Andrew:

You say that "my candidate's winning is equivalent to an average \$100 for each person".

Now is this \$100 real or imaginary? For it to be real, my candidate winning would have to be really equivalent to \$100 for each person who voted the other way. I'd have to be really smarter than everybody who voted for the other candidate, because it's worth \$100 to them that I override their opinion.

This is why I believe that: "For your vote to [really] benefit the other voters, you have to assume implicitly that you are smarter than the other voters and that you know what's good for them better than they know themselves."

Now if my candidate's winning is equivalent to an *imaginary* \$100 for each person, then you and I can agree on the rest. There are a lot of legitimate differences in opinion in politics (as in other areas of life).

But then we are back to the usual explanation for why people really vote: for the satisfying feeling of performing a civic duty which is completely imaginary.

Olivier.

5. Jean-Luc,

I don't think a disagreement implicitly says "screw you." For example, suppose I think your party is going to wreck the economy (possibly inadvertently). Then I'll want to vote to stop that from happening. It's not "screw you," it's "I disagree with you about economic policy." Or, maybe, "I think your party is controlled by malign interests (big business, big labor, whatever)."

Politics is multidimensional. I might not think you're stupid to vote for party X, I might just think that your priorities are different from mine. Maybe you're voting for party X because of abortion, and I don't care about abortion.

Olivier,

I could be talking about a real \$100 (if I think the opposition party is likely to lead the country into a recession). Again, I don't have to be "really smarter than everybody who voted for the other candidate," I just might have a different opinion about the economy.

Or I could be talking about an imaginary \$100–that is, some outcome whose expected value is equivalent in some sense to approximately that amount. For example, suppose I fear that the opposition party will leave the country vulnerable to an outside attack–if the increased probability of attack is, say, 10%, and the attack has a 1% chance of killing 100,000 people, etc. etc. This is speculative but I wouldn't call it "completely imaginary." I would say that voting is giving me a very very small chance of making a very very big difference

6. Assume that 150 million Americans and you voted for X, while 150 million voted for Y. So you cast the decisive vote for X. Assume you are 1% altruist, i.e. you are willing to exchange \$1 of utility gain (or loss) for yourself, against \$100 utility gain (or loss, respectively) for any other American.
Let's count your personal utility gain from casting the decisive vote. The 150 million Americans who voted X are happy with you because your action allowed them to get their preferred candidate elected. Let's assume having one's preferred candidate elected is worth \$100 to each person. You're up \$150,000,100 so far.
The 150 million Americans who voted Y are unhappy with you because your action prevented them from getting their preferred candidate elected. Each one has a net utility loss of \$100. You're down \$150,000,000 from this side.
In total, your net utility gain is equal to \$150,000,100-\$150,000,000=\$100. A far cry from the \$300,000,000 you computed.

7. Oliver,

Thanks for the comments. I would agree with you if the issues were purely distributional–e.g., Republicans voting for more money for sunbelt states and Democrats voting for more money for the northeast. But there are a lot of issues that are not distributional, where voters can think that their preferred candidate can have a positive impact on the country as a whole, not just on their supporters.

121 million Americans voted in the last presidential election, 62 million for Bush and 59 million for Kerry. If I prefer one candidate because I think he'll be better for the economy, national security, and other issues, I don't think he'll just be better for the 60 million or so people who agree with me. I think he'll be better for almost everyone. In polls, people will tell you which candidate they think is better on different issues–here, "better" does not just mean for the people of their political party.

8. Dear Andrew:
Thanks for your responses. This is interesting.
I think our differences boil down to whether your personal \$1 utility gain from forcing somebody (call her Mrs. Jones) who voted for Y to be ruled by X can be deemed "rational" or not. I gladly acknowledge your utility gain is real for you. I just disagree that it can in any way, shape or form be construed as "rational".
What distinguishes "rationality" from "subjectivity" is a minimum requirement of universality. For one, Mrs. Jones wouldn't agree that she's better off being ruled by the candidate she voted against. So, right there, this is sufficient to undermine your claim at universality and rationality. It becomes a case of: "he said, she said".
But, objectively, as the matter in dispute is whether candidate X or Y is better for Mrs. Jones herself (not in general, not for the country, not intrinsically), most neutral outsiders would opine that you are infinitely less qualified to judge this than Mrs. Jones herself. Therefore your belief that Mrs. Jones will benefit to the tune of \$100 from being ruled by Candidate X is entirely subjective and not rational.
As a result, we go back to the usual explanation, i.e. people vote for the subjective joy of voting. Although in your case it can be made more precise: you vote for the subjective joy of making Mrs. Jones "better off against her will".
Regards,Olivier.

9. Oliver,

Thanks for the comments. Since it's my blog, I think it's only fair for you to have the last word!