# Why it’s rational to vote (rerun)

In anticipation of tomorrow’s election, I’d like to repost this entry from 2004 explaining why it’s rational to vote. I was talking there about the Presidential election but the argument is relevant for Congress also (that is, if you, unlike me, would have the chance to vote in any closely contested elections):

The chance that your vote will be nationally decisive 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 the Republicans or Democrats \$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 the Republicans (or the Democrats) 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.

## 11 thoughts on “Why it’s rational to vote (rerun)”

1. In many states the odds that you will cast the deciding vote in a presidential election are far less than 1 in 10 million. If you live in, say, Idaho, there is really no chance that you will cast the deciding vote: if the vote in Idaho is close to 50-50, then the Democratic candidate will have won almost every other state in a landslide. And yet, people in Idaho still vote.

I think some additional factors are (1) local elections. Your vote may not be decisive in the national or state races, but it's much more likely to be decisive in, say, your City Council race. If you're going to the polls to vote for you City Councilperson anyway, you may as well vote for President while you're there. (2) Voters want to "send a message." If you're voting for the candidate who is going to win, you want him/her to have more of a mandate; if your candidate is going to lose, at least you can decrease the mandate of the other guy. Also, you might chose to vote for the Green Party candidate if you are sure the Democrat is going to win anyway, as a sign that you are left of the Democrats, or you might vote for the Libertarian as a sign that you're right of the Republicans.

I'm not saying that either of those is "the reason" that people vote, just that I believe them both to be non-negligible factors, perhaps larger than the ones that you consider.

–Phil

2. John,

Greg's comments are interesting, but I don't think they address the positive motivation for voting in a large election.

Phil,

Yes, I wrote that the probability is "at best" 1 in 10 million. Regarding Idaho: as far as the model is concerned, different people should be more or less motivated to vote, depending on their individual values of p, B, and C. The point is that, if the election is expected to be closer (so that your vote is more likely to be decisive), then it will be worth more people's while to vote.

Regarding your additional factors 1 and 2: yes, these are there, and we discuss such issues a bit in our paper. Just to be clear: we are not claiming to explain everyone's voting behavior, we're just explaining why it can be rational to vote in an instrumental sense (that is, without requiring one to rely on "civic duty" or the "consumption value" of the act of voting).

3. > The "civic duty" rationale for voting fails to
> explain why voter turnout is higher in close
> elections and in important elections.

Well I think this claim of yours is incorrect. If an election is close, parties will put in much more effeort to mobilize people, media coverage will be different, etc. etc. etc. (add you favourite confounder HERE) all of which affect the cost of voting.

A quick look in an intro cog. scie. textbook reaveals that rational choiche as a desciptive theory makes no sense. Utility max. behaviour just does not happen with humans in real world siutations. An so when we focus on the predicition alone, we can find plent of other, poentially more compelling stories about why turnout is higher in closer elections.

Tom

4. But if your rational reasoning is right, people should hold out for the aggregate benefit if assured that their vote will actually make the difference. But surely if you were offered, say, \$1 million by a genie who guaranteed the election of the guy you don't like, most people would jump at the chance. Even when you include their altruistic impulses….

In fact, the history of voting in the nineteenth and early twentieth century is replete with people selling their votes for meals and whiskey. That's what a ward heeler used to do. Now of course many of those votes were no doubt in the preferred direction of the voter, but surely not all.

5. Tom,

By saying "it is rational to vote," I don't mean to imply that rational calculation is the reason that people vote. I agree with you about psychological motivations, however I don't think this is inconsistent with a rational-choice analysis. We discuss this in Section 5.2 of our paper (see link at bottom of blog post). To quote myself:

From the perspective of the rational model based on perceived social benefits, we recognize that all human actions, including those that are rational, need some psychological motivation, and it makes perfect sense that a beneficial action will feel pleasant also; higher perceived salience corresponds to greater social benefit from voting. Conversely, the psychological explanation does not stand alone—voter turnout (unlike Academy Award voting) has direct political effects, and it is reasonable and appropriate to study the benefits from voting, even if from a psychological perspective they are perceived only indirectly.

I think this is consistent with what you're saying.

6. Jonathan,

If you offered me the two options:

A: \$1 million, and my preferred candidate loses
B: \$0, and my preferred candidate wins,

I'd take option B. Of course, I'm perfectly willing to admit that my preferred candidate is actually worse (who really knows, ahead of time). But yeah, considering what's at stake, I'd give up the million.

The bit about meals and whiskey is not at all contrary to my argument. Here, the meals and whiskey is being offered for a single vote (not for determining the election outcome).

7. Andrew,

Where did 1/30,000,000 come from? Your chances of casting the decisive vote for Charlie Rangel or Hillary Clinton is quite a bit smaller than 1/GDP so I think the expected value must be bounded above by \$1.

Doug

8. Doug,

Yes, in my district, there's essentially no chance my vote will make any difference. Well, maybe the election for local judge or something like that. The 1/(10 million) came from an average over the 50 states based on presidential election forecasts. The calculation could be refined, but I think it's the right order of magnitude.

9. … a critical & unstated 'assumption' in your vote-rationality model is — an accurate & honest 'voting system', that correctly collects, tabulates, and applies voter-choices to the formal government system.

I submit that such an Accurate & Honest voting system is not in evidence in the U.S. — thus, automatically invalidating your model conclusions.

Even a casual internet search of vote-fraud, lost ballots, miscounted totals, recounts, ballot/voting machine errors, ballot-access rules, etc. — will reveal a very significant amount of system-errors, unreliability, and outright corruption in American 'voting'.

The Bush-Gore presidential vote-fiasco alone is proof positive that the inherent margin-of-error (~1-3%) in our current national voting 'system' exceeds a typical spread of actual voter preference between two candidates.
To this day we do not 'know' the actual popular vote for Bush/Gore… because the closeness of the real vote was within the large margin-of-error of the shabby voting-system in use.

Why tolerate any margin-of-error in voting ?

The current American voting-system is at least inaccurate … and corrupt in many areas.

Would you 'rationally' sit down and play high-stakes poker in a game you knew was rigged against you (??)

___________

"The people who vote decide nothing. The people who count the vote decide everything."

{– Josef Stalin}