the voting paradox: solved!

I’ll compose a response to the orgheads later, but today’s post is inspired by Andrew Perrin, who mocked us over at Scatterplot. The issue of the voting paradox came up: rational choice theory predicts no one would vote because most of the time voting doesn’t change the outcome. I expressed my dismay at the issue since I think it’s a lame “paradox.” Andrew responded that it was something that had confused decades of researchers.

Well, I’ve thought it through and just because generations of scientists work on a topic doesn’t mean it’s not lame.  To see why, let’s work through this carefully. Let’s state the basic research question: why do people vote rather than stay home? The rational choice approach suggests that you should look at the costs and benefits. In an overwhelming number of elections, one vote has nearly no chance of determining the outcome. So the benefit is zero. End of story, right? Not so fast. Unfortunately, a lot of folks skip the next part: the cost of voting. Voting takes about 15 minutes, once every year or so. Mathematically, the voting cost is bigger than the benefit. But for practical purposes, vote cost = vote benefit = 0.

What does basic econ say when costs and benefits are both zero? Answer: not much.  When costs and benefits are zero, any behavior is consistent with the model. That’s why there’s no “paradox” at all. There’s no constraint on voting, so cost-benefit analysis is almost useless.

Now, this doesn’t mean the rational choice approach is without merit. In nations that impose fines for non-voting, voting turnout is very high. That’s enough proof, for me at least, that if you add costs, people respond.

How should we analyze free voting (i.e., no constraints through fines, fees, etc.)? Turns out that we have enough evidence for a good rough theory. First, there’s always a chunk of people who vote and they have predictable characteristics. They are more high status, have more education, more cognitive ability, etc. They also tend to be civically engaged. Voting correlates with a lot of other associational life (e.g., clubs) and early life club participation predicts voting. Basically, for whatever reason, there’s a slice of the population that are the joiners and participators. Call them the reseve army of voters. Maybe, from the perspective of evolution, it’s useful to have people around who care about social things.

Second, it seems that the symbolic importance of issues and elections predicts participation on the margin. People are more likely to vote in national elections, they are more likely to vote if there’s an issue that appeals to their identies or interests, etc. Some voters are also likely to vote through political recruitment.

Andrew cited some very clever experiments that explore pivotal (close) elections. While I admire the  effort, I find it beside the point. Pivotal elections are infrequent in the real world. We only care about them because of faulty reasoning about voting. What matters is psychology of voting, whether it be basic disposition, symbolism, or networks. Thankfully, there’s a lot of good work on these issues and it’ll yield a lot more useful stuff than the n+1 study of weird voting situations.


Written by fabiorojas

July 30, 2008 at 1:09 am

18 Responses

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  1. This seems rather disingenuous to me. The benefit asymptotically approaches zero, the cost is, well, kinda low, so we can decide it will be zero and move on to use other explanations? Puh-leez. Has anybody around here tried to get time off from a factory or service job to vote? How about to interpret a California voter’s guide?

    This is complicated, too, by the fact that (a) people vote more when their vote is less likely to be pivotal; and (b) the people who tend to vote more are people whose time is worth more on the market. Does this make voting an inferior good?

    Or, to be more direct: can you cite other instances of behaviors that can a priori be removed from utility maximization (that is, are lame), or is this just a post hoc justification for the fact that a rather important behavior doesn’t fit?



    July 30, 2008 at 1:36 am

  2. I find this voting stuff one thing where the discounted-contingent-claims approach to utility fails – people don’t just care about ends (whether or not my candidate won, whether I consumed enough goodness of voting minus the badness of voting to be net positive) but about means – whether or not I got to participate in the election. Whether or not my voice was heard. Just because price mechanisms have force on the margins doesn’t mean it is a economic-based phenomenon.

    And indeed, it is tough to get off work to vote. One can go further down this trail – why would a white person risk their life to register black voters in the 60s, as this (dangers aside) dillutes their votes?



    July 30, 2008 at 1:58 am

  3. I’m with Fabio in that I find the whole thing a bit frustrating. Can I posit an alternate utility function and thus escape some of this paradox? Namely, why do we assume the only benefit from voting is that your preferred candidate has a slightly higher chance of winning?

    I don’t mean to add in any ‘psychic’ benefits (not that I think that’s a bad way to go, it just won’t win you an argument with many economists). But what if we move past a simplistic winner-takes-all model? Yes, the plurality vote getter does get the office, but doesn’t mandate matter? In particular, elected officials that win by big margins are (I would wager) more likely to receive big campaign donations the next time around and less likely to face serious challenges, in addition to any benefits they may have (especially mayors, governors, presidents, and the like) in actually achieving their agenda. Didn’t FDR get more done in part because he was perceived as having a mandate?

    So suppose a voter cares about this. Now his or her vote always matters, albeit only a fractional amount. But, as Fabio pointed out, the cost of not voting is also fractionally small. But instead of ~0 = ~0, I would argue ε1>ε2. I don’t fully buy this solution – I agree with other commenters that voters care about means, and we use social norms to enforce the norm of voting, and etc. But this is a solution more or less internal to the framework.

    Is there something lame I’m missing about this solution?


    Dan Hirschman

    July 30, 2008 at 2:43 am

  4. Hi, Andrew.

    Why do you think it’s disingenuous? It’s a straight forward issue: rational choice theory has nothing to say about zero cost-zero benefit actions. I thought you might appreciate this point because, at the very least, it shows a serious limit in the theory. The theory is literally silent on a central point of political research.

    I do agree with your point that some people have jobs that increase the cost of voting. However, that puts you more into an RC camp. All you are saying is that many people can’t afford an hour or so of lost income, which is very true at the low end of the income spectrum.

    At the same time, I would point out that in the recent Indiana primary – where I live – we had mail in voting, early voting, weekend voting, AND extended voting hours. Basically, you literally had a couple of weeks to vote. AND it was a “hot” election. Result? 36% turnout rate. Check out turnout rates of other state primaries here:

    This suggests that even when the costs are drastically reduced, you still only have a voluntary turnout rate of about 40% in the US. As barriers for voting decrease (motor voter registration, email voting, extended hours, mail in, etc), I bet you’ll see the same trend: 40% of the people do most of the voting.

    Personally, I’ve only read the voter’s guide once, but I have never met anyone who has been prevented from voting because of the voter’s guide (though I can easily see that some folks might vote in mistaken ways because of the obfuscation in referenda wording). Unless you can show me some systematic evidence, I bet there aren’t armies of eager voters who have been confused by voter pamphlets.

    I do agree that people are more likely to vote if it’s pivotal. But why should I care? As the Mulligan/Hunter paper shows, pivotal elections are absurdly rare occurences. The pivotal issue is more likely to impact me in a faculty meeting than it ever will in conventional politics.

    I also agree that wealthier people are more likely to vote. But that doesn’t “complicate” things at all – it just shows that people at higher income levels are more likely to be in professions where they have control over their schedule. As you have pointed out, factory workers may not get time off, while a doctor may shuffle an appointment around. You are making the cost-benefit difference non-zero.

    Inferior goods: If you mean that the demand of voting decreases as income rises, I don’t think so. Isn’t it the case that wealthier folks vote more often? That’s normal, you have more money, you buy more. Or am I missing your point?

    At the end, you ask about situations that are zero-cost, zero-benefit. I admit, it is weird, but we can think about more cases. In modern times, I think a lot of cultural tastes are essentially zero-cost, zero-benefit. Few people care what I watch on TV and there are no benefits, except for whatever psychic benefit I get.

    But when you think about it – isn’t voting a weird action? It’s something that takes a few minutes a year, with, at best, a few indirect consequences. We should expect few other activies to be like it.



    July 30, 2008 at 3:20 am

  5. I agree with Andrew Perrin that treating voting as zero-cost is a bit implausible. But Andrew Gelman proposes another approach to this “problem”:

    His solution is to treat the benefit side of the cost-benefit calculation in terms of collective rather than individual benefits: “If you think your preferred candidate could bring the equivalent of a $50 improvement in the quality of life to the average American–not an implausible hope, given the size of the Federal budget and the impact of decisions in foreign policy, health, the courts, and other areas–you’re now buying a $1.5 billion lottery ticket. With this payoff, a 1 in 10 million chance of being decisive isn’t bad odds.”

    Does this answer Perrin’s original challenge? I’m not sure. And I’m certainly not convinced that a rational choice model of voting is even appropriate here. But this one does offer an explanation for why people are more likely to vote in national elections.



    July 30, 2008 at 12:47 pm

  6. Fabio: I think we’re in agreement on the specifics; I agree that voting is essentially a cultural and psychological act (in fact that’s part of what I say in my upcoming piece in Contexts on voting). But I don’t agree that coming from a strong-RC position one could decide a priori that voting would be expected to be different. The utility-maximization axiom is simply that whenever we make decisions, we seek to maximize utility. It’s clearly a decision whether or not to vote, and there are clearly calculable and finite costs and benefits, but people appear not to behave in ways that maximize their observable utility.

    One of these is the fact that people for whom time is worth more also tend to vote more — that’s why I asked if voting is an inferior good, that is, that “demand” for it appears to decrease as the time-based resources necessary for it increase. I actually don’t think it’s true, but I think it’s another problem with the RC-based formula for considering voting.

    So my point, to summarize, is that if utility maximization is a generic model for understanding human decision making, voting ought to be inside the scope conditions. But when we examine actual voting behavior, it appears that there are quite a number of ways in which it doesn’t (easily) fit the model.

    Peter: Gelman’s approach is an interesting one, thanks for passing it along. But I think it suffers from the same logical flaw as the “civic duty term” others have used. It’s only a $1.5 billion lottery ticket if we posit ahead of time that an individual voter derives utility from the satisfaction that the nation as a whole is better off. That’s no better, really, than deriving utility from the satisfaction of having voted.

    I think the real reason people vote more in presidential elections than local ones is that presidential ones are more exciting. Danielle S. Allen has a terrific chapter on the ritual of voting in her (generally fabulous) Talking to Strangers that provides some of this ammunition. Essentially, people participate because it’s a way of reaffirming membership in a political community. That political community is more exciting to be part of when it’s bigger and more important.

    Note that, if we are willing to introduce what Duffy and Tavits call the “D-term,” or the sense of civic duty, as a form of utility, then Allen’s position helps describe why people derive utility from the D-term. But my contention then is just that if we expand the concept of utility to be that broad we’ve essentially robbed it of any useful meaning and therefore made the utility-maximization thesis trivial.



    July 30, 2008 at 1:56 pm

  7. Andrew: Here’s the point that we disagree on. “Utility maximization” is a vague and, in some ways, meaningless phrase until you get specific. A very plausible definition of utility maximization is that an action 1 is taken over action 2 if b1 – c1 > b2-c2, where b and c refer to costs and benefits. It is a simple deduction from this equation that there is no prediction if b1 – c1 = b2 – c2 = 0. From this point of view, the choice to vote in any relatively large group is not predicted by RCT. This is a normal property of many theories – when you write them down, you often get strange conditions where the theory makes no predictions or bizarre predictions. It is not ad hoc, just basic math.

    You and other commenters point to the costs of voting for some people. All that does is just push you back into RCT land (i.e., c1>0 but c2, b1, & b2=0). That’s an empirical point that the assumptions of the model is wrong.

    Ultimately, what’s at stake is that the “voting paradox” is used by RCT critics to undermine all of RCT. This was pushed by Green and Shapiro. At first, I bought it, but when you actually write down what RTC predicts, voting is a really bad example. There are much more fruitful criticisms of RCT, but focusing on low cost, low pay off situation is not useful.



    July 30, 2008 at 4:14 pm

  8. just push you back into RCT land

    Precisely – it shows that there’s no a priori reason to label voting a bad case! If there are measurable costs (and potentially benefits) to voting, it’s not necessarily the case that b1-c1 = b2-c2 = 0. So, yes, I’m pointing out that the assumptions of your model are wrong; that there are, in fact, measurable costs and benefits to the act of voting vis-a-vis other possible activities and that therefore if RCT describes decision making in general the predicted outcome is not random.

    I don’t think the “voting paradox” undermines all of RCT, except insofar as RCT purports to explain all decision-making behavior. I do think it requires RCT to take one of two positions, each of which is somewhat unsavory:
    1.) Define utility so broadly and tautologically as to make the theory trivial; or
    2.) Define certain substantively important decisions as outside the scope of RCT.

    Note that the voting paradox did not arise out of critiques of RCT. Rather, it arose out of the fact that voting is front-and-center in the discipline of political science and it’s pretty hard to explain (not to predict, though).



    July 30, 2008 at 4:30 pm

  9. Dan mentioned the possibility of “psychic benefit,” and I think this is exactly the way to go. Few people believe their vote will turn out to be the marginal vote, but they get a warm fuzzy from “participation in the democratic process.” They wear their “I voted!” sticker proudly throughout the day and use it to shame colleagues who haven’t indulged.

    This is completely consistent with rational-choice theory. Of course, because utility functions aren’t observable, it’s a case where RC doesn’t have much predictive power.

    BTW, Fabio, I think you mischaracterize the RC argument against voting (ignoring the warm fuzzies for the moment). It isn’t that “In an overwhelming number of elections, one vote has nearly no chance of determining the outcome.” It’s that “In essentially all elections, one vote has effectively no chance of determining the outcome.” Let’s call a spade a spade.


    Peter Klein

    July 30, 2008 at 5:10 pm

  10. “…rational choice theory predicts no one would vote because most of the time voting doesn’t change the outcome.”

    This is a huge misrepresentation of rational choice. The more correct statementment is to say: “If a person’s marginal benefit from voting is a function only of the probability that her vote determines the election, and there is an opportunity cost of voting, then because the likelihood of affecting the election outcome is essentially 0, rational choice says the person would not vote.” (Actually, you must add some conditions to ensure finiteness, but you get the idea.)

    But it is just as much rational choice to say the following: “If a person’s marginal benefit of voting is a function of the good feeling felt after voting, and there is an opportunity cost of voting, then rational choice says that only persons who receive a sufficiently strong good feeling will vote.” So there you have it: rational choice predicts voting!

    Long story short, it is incorrect to say that rational choice in and of itself predicts no voters because you can change the assumptions of the model to make pretty much any behavior the optimal one. This is the blessing and curse of rational choice.

    But before we completely tear down rational choice, let’s remember that, technicially speaking, rational choice is ultimately a behavioral methodology whereby the researcher infers the utility function from observed choices. In the case of voting, if voting is observed under some settings but not others for an individual, then the infered utility function would have to assign more utility in the former cases than in the later cases. Hence, the very fact that we observe people vote implies that rational choice, when applied in its purest form, must predict voting in some circumstances!.

    The common misconception that rational choice predicts no voting is based on a different way people apply rational choice: researchers make ad hoc assumptions about utility functions and then deduce optimal behavior. This approach is the most common way people apply rational choice because of (1) the impracticality of observing enough choices to infer a utility function and (2) making assumptions first and then deducing optimal behaving can actually be very insightful. That people used rational choice to predict no voting thus means that they had bad assumptions and, either for good or bad, is not an indictment on rational choice itself. Or put more positively, the earlier work shows that people must not vote solely out of a belief that their vote affects the election’s outcome.

    My own sense is that, in the words of rational choice, the utility derived from voting arises from many different things. Here’s a quickly made, non-exhausitive list, which I put in order of importance:
    – the feeling of participating in democracy, which is surely something socialized into us that affects our preferences.
    – signaling that you are a good person, something we’re indirectly taught.
    – the opportunity cost of voting (is it raining, will you miss too much work, etc.).
    – the perceived significance of the election on one’s one well-being.
    – the probability of being pivotal.
    These all can fit comfortably in a rational choice model.



    July 30, 2008 at 5:13 pm

  11. These all can fit comfortably in a rational choice model.

    …a rational choice framework, sure, but not one that actually predicts, since the independent variable (utility) is inferred from the dependent variable (voting).



    July 30, 2008 at 5:31 pm

  12. @ap – Well, the model could then be used to make new predictions, such as that increased costs of voting that did not affect the psychic benefits (long line, rainy day) would have an effect on reducing turnout (which they seem to), and similarly increased psychic benefits (election perceived as important, candidate seems awesome) would increase turnout, right? Or is that still a trivial prediction?


    Dan Hirschman

    July 30, 2008 at 7:56 pm

  13. Dan: I see what you’re saying, sure – but the problem is that we don’t actually know what would or would not affect the psychic benefits. For example: if we were to observe (which we don’t) that voting went up with rainy days or long lines, we could construct a post hoc argument that people must have felt particularly virtuous to be braving these hardships. This argument would be nonfalsifiable, and perfectly in line with the RCT model, as would the alternative that we actually do observe.

    So no, the prediction there isn’t trivial, and it’s a useful contribution. But the content of it would have to come from elsewhere, since we have no way of calculating the relative utility of psychic vs. other benefits, or even what kinds of things are likely to be understood as psychic benefits.



    July 30, 2008 at 8:22 pm

  14. Must we assume that the only motivation is individual motivation? I vote not as an individual but as a member of that group of people that supports a candidate.
    Try applying these same individual-rationalist arguments to the question of why people cheer in unison at football games or sing in church?

    What about other forms of voting? if you own stock, you get a proxy card. All you have to do is mark an X and drop it in a mailbox. But I’d guess that the return rate is considerably less than voter turnout in the least significant elections, even though the outcome may have a direct financial impact.

    By contrast, voting on American Idol does have a calculable cost (not much, but more than presidential elections or corporate proxies), yet more people vote at AI than in the presidential election even though their vote has almost no chance of being pivotal, and the outcome will not affect their lives at all.


    Jay Livingston

    July 30, 2008 at 9:15 pm

  15. JL: good points, all. My bottom line is that I think the comparison to singing in church and cheering in football games is very apt — voting is not, essentially, an individual act, but rather something an individual does (in part) to (re)affirm connection to a collectivity. Hence it’s impossible to understand without reference to collective representations.



    August 1, 2008 at 2:56 am

  16. […] real voting paradox Posted in fabio, political science by fabiorojas on August 18th, 2008 A week ago, I argued that the “voting paradox” is a social science red herring: rational choice theories make predictions only when there are differences in utilities/prices. […]


  17. […] orgtheory voting arguments: the voting paradox, the “real” voting paradox, Casey Mulligan on pivotal elections, genetics and voter […]


  18. Lots of reasons to vote that you all seem to miss, and that would add to the benefit side of the fence.

    There’s that “I Voted” sticker, of course, which gets worn proudly on a lapel (perhaps the only proud wearing on a lapel of a flag all year) or stuck to a kid’s bicycle helmet.

    There’s the chance to see your neighbors in a public place, something that not everyone gets to do very often.

    Every once in a while the voting machines change, and the geeks in the crowd get to examine them for their likely flaws. (I’ve been using the “assistive technology” machine just to see how bad the user interface and design is; with a few more elections I might actually be good at it.)

    Every once in a while there’s a contest, especially in local elections. Even more so, if you get to be a certain age, you know people who are in politics by name and face (and they are your neighbors), and you go to vote for them. Hey, you might even pay cash money to put a sign in your yard for them.

    So – cost to vote can’t be seen strictly in some kind of autistic economist cash only point of view. I’d be more likely to compare it with the kind of decision someone makes when deciding to show up at church or at Rotary or at the knitting meetup (the one with the cute girls). It’s not a one time game, it’s a repeated game; there are side effects to being part of the process that are cumulative; the expected costs and benefits of voting are probability functions, not prices.


    Edward Vielmetti

    September 27, 2008 at 6:30 am

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