No Poll-Box Paradox

Chris Bertram of Crooked Timber raises an objection to the notion of rational non-voting (the idea that people don’t vote because the miniscule expected benefits are swamped by the costs):

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 ….

It sounds like a paradox, but really it’s not. I play on the seeming paradox when I say to my students, “For economists, the question is not why people don’t vote; it’s why they ever do!” But even that question is not terribly difficult to answer.

First, “rational” is not synonymous with “self-interested” or “materialistic” or “concerned only with outcomes.” It simply means acting consistently with one’s own preferences, whatever they might be. There is nothing irrational about voting because (a) one feels a civic or moral obligation to do so, or (b) one derives some kind of direct pleasure from the expressive act. And indeed, that is probably a sizable chunk of why most people vote. Some people entertain the myth that “every vote makes a difference,” but most will admit – at least upon prodding – that their one vote will almost certainly not alter the outcome of the election. They continue to vote because they derive some kind of direct psychic benefit from voting (or perhaps a psychic cost from shirking).

But second, even if we assume people are solely concerned with their votes’ effects on electoral outcomes, no paradox arises. The reason is that the expected benefits of voting decrease as the number of voters rises. If no one else were going to vote, your vote would be decisive. If ten other people planned to vote, it would still make sense to cast a ballot. Even if a hundred or a thousand other people intended to vote, it would not be outrageous to think your vote could stand a chance of making a difference. When the number of other voters numbers in the millions, of course, the expected benefit is vanishingly small. The rational non-voting argument, like most economic arguments, is a claim about the marginal chooser – in this case, the voter right on the cusp between voting and non-voting. Since the marginal benefit of voting declines with the number of voters, there is a turning point at which the marginal benefit sinks below the cost. For the marginal voter, the expected benefit should be approximately equal to the marginal cost.

The two arguments work best in concert. The rationality of voting when few others will vote assures that at least a certain number will vote. Feelings of civic duty and self-expression pump up the marginal benefits further. Together, these factors assure a large enough voting public to make the probability of your vote affecting the outcome tiny relative to the material costs. Only if your psychic benefit of voting is large enough to cover this rather sizable gap will you choose to vote.

Chris’s objection can be rejuvenated by observing that while it’s rational for some people to vote, the model does not determine which people will vote. Suppose, for instance, that the marginal voter’s expected benefit of voting – including all psychic benefits – dips below the cost when the number of other voters reaches (say) 10,000, in an overall population of 100,000. There is still a question of which 10,000 voters will vote. Why these 10,000 people instead of those 10,000 people? The problem has multiple equilibria, and there’s no obvious means of determining which equilibrium will in fact occur (a common feature of economic models that produce multiple equilibria). But the most plausible argument (to me, at least) is that the voting contingent will consist of those voters with the highest psychic benefits, which are independent of the number other voters.

To recapitulate: The rational non-voting theory is a theory of the marginal voter. It is fully consistent with the existence of a situation in which some people vote and others don’t, even if people assess the benefits of voting purely based on its electoral effects. However, recognizing the existence of benefits derived from the act of voting itself allows for a more complete explanation of (a) why as many people vote as do and (b) which people do in fact vote.

UPDATE: I originally attributed the CT post to Henry Farrell, an error I've now corrected.