Rational voting

by Chris Bertram on September 27, 2004

Jim Holt in the New York Times raises the old question of whether it is rational to vote . The issue is this (for those who don’t know): the rational voter decides what to do by weighing the expected cost and benefits of actions. Suppose I value the victory of Party X at $1000. In working out the expected benefit of voting, I also have to take account of the probability of my vote making a difference, a probability which is extremely low (say 1/100,000). Assigning, therefore an expected benefit to voting of 1c, I see that going to the polling booth involves an expenditure of time, shoe leather and incurs the opportunity cost of missing a few minutes of my favourite soap opera. Since these costs will certainly by incurred if I vote, and dwarf the expected benefit of voting, the expected net benefit is always negative, and so, rationally, I shouldn’t vote.

What’s wrong with this argument? Well, one thought, which I remember hearing first from my friend Alan Carling, is this: the argument involves inconsistent assumptions about rationality. The assignment of a low probability to my vote making a difference assumes what the conclusion of the argument denies, namely, that rational persons would vote. But the argument says they wouldn’t. Well if they wouldn’t then I would be the only voter (a dictator, in effect). In which case I would certainly be rational to vote since I can count the full expected benefit of $1000 in favour of doing so. But if that’s the case, and I should vote, then so should everyone else … in which case I shouldn’t … in which case nor should they … in which case I should ….



lago 09.27.04 at 9:31 am

And what about the rational ignorance that results from the cost of becoming informed about the best candidate outweighing the expected benefit to you if that candidate wins?


John Quiggin 09.27.04 at 9:35 am

I don’t think the argument you have here works the way you want. It would imply that there’s an equilibrium where 100,000 people vote.

A much easier explanation is that even very mild altruism gives good grounds for voting. Suppose I think that Party X would make the average person better off by $100. If n is the size of the community, my chance of being decisive is (of the order of) 1/n and the aggregate benefit of Party X is $100n so the expected community benefit of my vote is $100.

So, as long as I’m willing to put in 1c of my own effort to make the community better off by $100, I’ll vote.


NBarnes 09.27.04 at 9:36 am

Also, I would put the value of a Kerry win instead of a Bush win as being worth at least an order of magnitude more than $1,000 to me (as a young male of draft age). Possibly two orders.


cdm 09.27.04 at 9:39 am

I think the point is simply that it is (apparently) irrational to vote taking as given the action of others (as in a Nash equilibrium). This requires no presumption of whether others are acting rationally; it is simply a question of what a rational individual should do confronted with the (unexplained) actions of others. From this perspective, the fact that people do vote in such numbers implies either that people are not rational, are altruistic, or that there are other benefits not captured in the story you tell.


John Quiggin 09.27.04 at 9:44 am

A slight correction. I read the 1c as the cost of voting in the post – make this, say, $5 and the argument isn’t changed fundamentally.

With more realistic numbers the altruistic case for voting is not a no-brainer. If you don’t see a big difference between the parties or don’t care much about the good of the community, then you won’t vote.


Chris Bertram 09.27.04 at 9:49 am

Just a comment on what cdm said ….

The story is meant to bring out the tension between the two sides of rational choice theory, the normative and the explanatory. Taken just as a theory that tells me what I ought to do, then there’s no problem with just taking the actions of others as given (as you suggest). But RC theorists of politics also seek to explain political life on the assumption that people are rational (or some close “as if” assumption). If you take _that_ line, then I think the paradox comes off.


Jack 09.27.04 at 11:13 am

Who says what goes into the decision? The normative rational choice model as presented seems radically vunerable to miss-specification.

For example the assignment of probablility in the original argument is flawed by, as Chris suggsets, not taking into account similar decisions on the part of other voters. According to the calculation noone would vote in a safe seat. Is it an important part of the RC theory that the rationala ctor should be self contained?

Similarly it is evident that there is more to be gained in an election than just the result. The Monster Raving Loony Party gain something. Overall vote shares count. The fear of regret at not casting the decisive vote. Loss of rhetrical advantage by non-voters. If the voters for these people are rational actors then clearly both probabilities and values are not what the simple model presented suggest.


abb1 09.27.04 at 11:32 am

Assesing monetary benefit of ‘victory of Party X’ is difficult. For this and other reasons voting for a party is mostly an emotional, not rational decision.

Here in Switzerland they had a bunch of referenda yesterday. Among other things they voted on a 14 week maternity leave. It passed. Now, how much does something like this is worth to you if you’re a young family? Much more than $1000 for sure.


abb1 09.27.04 at 11:35 am

Ooops. That’s “how much is something like this worth”.


Andrew Boucher 09.27.04 at 12:54 pm

OK voting is irrational. People do irrational things. And so…?


Jason Kuznicki 09.27.04 at 1:09 pm

One should not value one’s participation in the system based merely on the probability that the preferred candidate will win. There are other benefits too: Win or lose, one is supposed to gain peace of mind by participating in the process, by reminding oneself of the freedom to vote, and by participating in a ritual that ties us all together, no matter what the outcome.

This is why citizens of the United States always conduct their elections with such remarkable decency, fairness, and politeness.


KCinDC 09.27.04 at 2:06 pm

John Quiggin, you seem to be assuming that a $100 benefit to someone else is just as good for you as a $100 benefit to you. Few people are that altruistic.


digamma 09.27.04 at 3:41 pm

John Quiggin, you seem to be assuming that a $100 benefit to someone else is just as good for you as a $100 benefit to you. Few people are that altruistic.

You can eliminate that assumption and still arrive at Quiggin’s conclusion. Let’s say a $100 benefit to someone else is worth $1 to me. Multiplying that by every citizen still yields a whole lot of benefit.


digamma 09.27.04 at 3:42 pm

John Quiggin, you seem to be assuming that a $100 benefit to someone else is just as good for you as a $100 benefit to you. Few people are that altruistic.

You can eliminate that assumption and still arrive at Quiggin’s conclusion. Let’s say a $100 benefit to someone else is worth $1 to me. Multiplying that by every citizen still yields a whole lot of benefit.


Anthony 09.27.04 at 4:07 pm

There exist more races on the ballot than just the presidential election. Given other races that it might be worth voting for, it is a trivial addition to also vote for president.

It is a non-trivial matter to determine which, if any, race is worth voting for. It would be a waste of time and effort to figure that out before every election. And this difficult determination will be uncertain, anyways.

The solutoin: Decide if there will be at least one race (or a collection of races) that will make it worth voting over your lifetime. If yes, then vote habitually. If no, then don’t bother.


Guy 09.27.04 at 4:28 pm

If you allow for people to use mixed strategies (people vote with a certain probability), then this paradox disappears. A voter chooses this probability given their likelihood of being pivotal, the cost of voting and the benefit of getting their favorite politician elected. Therefore, while no voter is 100% likely to vote, a non-zero fraction of the population will turn out to vote.

This kind of model tends to underpredict voter turnout, and this underprediction gets worse with population size.


Guy 09.27.04 at 4:31 pm

If you allow for people to use mixed strategies (people vote with a certain probability), then this paradox disappears. A voter chooses this probability given their likelihood of being pivotal, the cost of voting and the benefit of getting their favorite politician elected. Therefore, while no voter is 100% likely to vote, a non-zero fraction of the population will turn out to vote.

This kind of model tends to underpredict voter turnout, and this underprediction gets worse with population size. Once you start adding private psychological benefits to voting (e.g. John Q’s suggestion) turnout in these models obviously increases.


Richard Bellamy 09.27.04 at 4:34 pm

Last time I voted (Dem primaries), I took my 3 year old daughter into the booth with me to show her how to vote. Once we got in, she insisted that we right in her imaginary friend, instead of voting for my preferred candidate (John Edwards).

I insisted that we vote for the real person, and not my daughters imaginary friend. She threw a huge fit.

Thinking back, I’m fairly certain that I made the wrong choice, and had I thought about it rationally at the time, there would have been a positive gain, and no real loss to voting for the imaginary friend.


Zizka 09.27.04 at 5:09 pm

Few of the discussions I read about rationality make me conclude that rationality is really a very good thing atall, either analytically or normatively.

Amarya Sen (in “Rationality and Freedom”?)recently replaced three common definitions of rationality with his own. I liked his definition fine, but it had lost all the supposed clarity, definiteness and power that made the other, wrong-headed definitions interesting.


AVoter 09.27.04 at 6:05 pm

Naive question: Why do we get to determine arbitrary values for voting based on direct material gain while leaving the costs wide open. Ten dollars is a ridiculous cost for my vote. I can walk to the polls and be done in half an hour. Using electricity to watch a soap opera with zero material value would be much more expensive.


Guy 09.27.04 at 6:15 pm

If you’re a consultant that earns $200/hr, then $10 is indeed a ridiculous(ly low) cost for your vote. :)

More seriously, the model is robust to the choice of voting costs. As long as the benefits of having “my choice” win are sufficiently high relative to those costs, elections will have a non-zero turnout.

I think this result should hold up even if we introduce cost heterogeneity into the population.


John Kelsey 09.27.04 at 6:16 pm

It seems like most of the benefit of voting is outside the realm of having a tiny fraction of a say in deciding who wins. From feeling good about your citizenship to getting the little “I Voted” sticker they give you around here for voting, you get some external benefit.

And of course, that’s necessary, because based on your chances of affecting the election’s ultimate outcome, there’s really almost no benefit to voting. (I can look at the fact that there is substantial turnout to the elections, and thus decide that my vote will have a negligible effect.)



Sebastian Holsclaw 09.27.04 at 6:51 pm

This is one of the paradigm cases of what David Post called “The Reverse Tinkerbell Effect” over at The Volokh Conspiracy (scroll down to the “Tinkerbell Effect” posts.

It describes phenonmena where “The more people who believe it, the less true it is.”

Like many such phenomena it works both ways. The more people who believe voting is crucial, the less likely it is that your vote is crucial. As more and more people believe that their vote isn’t likely to make a difference, it becomes more likely that an individual vote makes a difference.


avoter 09.27.04 at 6:53 pm

If you’re a consultant that earns $200/hr

I guess the rational thing would be to work 30 minutes, vote, then bill the client for an hour. I’ll make $200.01.


Jay 09.27.04 at 7:19 pm

I think there is an intrinsic value to human beings in expressing themselves. This blog, to wit, and my post.

Voting is a form of self-expression, and thus has value, in and of itself.

Of course, not everyone values self-expression, or the particular form of self-expression that is voting, the same. And the number of individuals for which it exceeds the cost of voting is going to vary with the issues on the table. Thus Karl Rove gets the right all fired up about gay marriage and bible banning. Most of these people know they aren’t going to stop gay marriage ultimately, they aren’t going to stop abortion, they aren’t going to establish Christianity as the state religion. But they want to stand up and make themselves heard, and voting is a way for them to do that.

This also holds for me, living in CA. My own vote for John Kerry will not make the slightest bit of difference. CA is a foregone conclusion for Kerry. If I want to change my life, buying a ticket for the CA lottery, which is over $35 million at the moment is more rational than voting. Except I want to be counted. I want to be heard. I want to express myself.

So any model that ignores this expressive value of a vote will underpredict the number of voters.


Jack 09.27.04 at 9:16 pm

Would a student presenting the calculation above in a term paper in Rational Choice Theory would they be expected to adjust both the probability of making a difference and the size and strucure of other rewards for voting?

It seems (as perhaps intended) to me that it would be easy to treat this calculation as being normative and draw a very wrong conclusion.

If I understand Chris’s suggestion properly taking the explanatory view it becomes clear that the calculus presented is incomplete because it suggests actions that people do not take.


Jason 09.28.04 at 2:53 am

2 things,

First, people differ. Different types will generate an equilibrium where some people vote and others don’t. That said, enough people vote irrationally that the rational agent model isn’t a good one.

Second, I don’t see any reason why I should expect my chance of affecting the outcome to be on the order of 1/n. The probability of a binomial distribution being a significant distance from the mean is exponentially small, not harmonically small. I live in CA (although as a non-resident). If I had to estimate the probability of a single voter turning CA from one party to the other, It would be in the trillions to one (actually, it would be higher than this, but largely because of some rare events that might cause most of the population of CA to die, eg. nuclear war, huge earthquake, etc.).

On top of that, does anyone truly believe that if the presidential election came down to 1 vote in 1 state, that it would be that vote that would actually change the outcome rather than an tortuous litigative and bureaucratic procedure?


Jim Birch 09.28.04 at 6:16 am

One major problem with expected utility problems like this is that they assume rationality is identical with self-interest. A selfless act can be totally rational. If you’re truly not at all altruistic and moderately smart then I expect that shooting strangers and removing wallets from the bodies would be a better income stream than voting.


John Quiggin 09.28.04 at 6:37 am

Jason, could you spell out your argument? Are you assuming that the vote is given by a binomial distribution with some fixed p, not equal to 0.5?

If so, don’t you concede the possibility that some shock (say, a scandal about Schwarzenegger) might shift p close to 0.5. Given a reasonable Bayesian prior over p that includes some weight in a neighborhood of 0.5, I don’t think you can get far away from a 1/N chance of a tied vote.

Your second point actually weakens your case. In a close election, no vote is decisive, but every change in the (election-night) margin changes the odds. 1000 extra votes for Gore in Florida would have made a big difference, so every vote cast in Florida made (roughly) 0.001 of that big difference.


JamesW 09.28.04 at 1:45 pm

So it’s just about rational to turn out on polling day but not to invest a lot of quality TV time in finding out whether the candidates are honest and competent. Seems to fit the data.


Jason 09.28.04 at 3:25 pm

Yes, I’m assuming there is a proportion p of the population that will vote one way, and a proportion the other. Personal observations, Polls, Election Markets etc. (pick your poison) offer up something like estimates of p, and these estimates are unlikely to be too far removed from p. If you view polling for example as an (unbiased) sample of how the population *will* vote, then if the estimate of p is say consistently .6, then the probability under say a uniform prior that p is below .5 is ummm, this will take a fair dredging of my knowledge here (my earlier comment was just based on the fact that this was a “rare event” and thus likely to be exponentially small).

Let’s see, uniform prior = beta(1,1). Suppose we sample 3000 people and get 0.6, so 1800 positives, and 3000 total. Thus posterior = beta(1801,1201).

Now, head to beta tables … Bah! Can’t find any tables … OK, let’s look at LR’s and bound the probability … LR is proportional to p^1801 (1-p)^1201. Thus, the ratio of the probability of being 0.6 to that of being 0.5 is (0.6/0.5)^1801 (0.4/0.5)^1201. Taking logs and seeing if my calculator is OK with this, 1801 ln 0.6 + 1201 ln 0.4 – 3002 ln 0.5 = 60 (give or take), so the probability of being smaller than 0.5 in the posterior distribution is certainly smaller than e^-60, or 1 in 10^26.

Also, CA is a democrat state, so a scandal about Arnie is unlikely to change the outcome, unless it was a scandal like he’s going to give all his money out to Californians, but only if they vote Bush. It’s hard to think of a scandal that could swing CA to the knife edge, while keeping the rest of the country on a knife edge too.

I don’t think my second point weakens my case (actually I don’t really think I have a case, my first point is that if everyone were rational there would still be people voting, my second is that your 1/n estimate is way off). Your 1000 people block voting is not the same as 1 person voting. Yes, if I can get 10% of the population to sign a contract pledging to block vote then we could swing almost any election. This is essentially what special interest groups try to do (and use peer pressure to enforce the vote).


Jason 09.28.04 at 3:32 pm

The caret’s disappeared in the formatting. It may make the calculations a little hard to follow. A lot of the numbers squished together are supposed to be “to the power of,” in particular the final answer was 10 to the -26


jr 09.29.04 at 4:18 am

Your argument assumes that people routinely assign a dollar value to their time and effort in ordinary life activities. They don’t. Except for those few of us who bill time at our jobs, does anyone ever even think of valuing a purchase at its cost plus a dollar amount for the time spent shopping? Or the cost of a meal as the check plus some value for the two hours spent consuming it? The few minutes spent voting (not an unpleasant activity) have no calculable dollar cost. If it cost even a trivial amount to vote (say $5) one would expect to see the voting rate drop precipitously.

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