You show up at a PTA meeting to discuss homework policy and find there are only five other parents in the audience. Two of them favor a minimum of three hours homework, seven days a week, while the other three strenuously argue that homework should be abolished.

PTA meetings are not the only decision-making processes plagued by low attendance: Aristotle writes of the Athenian assembly of the fifth century B.C. that "absenteeism was common" despite "all sorts of tricks" used to encourage people to attend.

Nor are PTA meetings the only forums in which participants' views can be extreme. For example, in her book Tree Huggers, Kathie Durbin describes timber policy in the Pacific Northwest in the 1970s to 1990s as the outcome of a conflict between extreme environmentalists who would not countenance the felling of a single tree and timber companies who wanted to clearcut every forest in sight.

How can we explain these observations? Do they depend on the specific characteristics of the three situations? Or is participatory democracy inherently flawed? Jeffrey Rosenthal of the Department of Statistics, my colleague Matthew Turner, and I have recently studied an abstract game-theoretic model that encompasses the three situations, as well as many others. (The paper will be published in the American Economic Review this year.)

The model abstracts from the details of any given situation. We take a policy to be simply a number (the number of hours of homework, the tax rate to impose, the percentage of forests to fell). Different people favor different policies. Anyone may participate, if they wish, in the process that selects a policy. (Anyone may attend a PTA meeting or make a submission to a public hearing.) But participation is costly (at least in terms of time). The policy chosen is a compromise among the participants' favorite policies. In a simple case that we study, the compromise is the median of the participants' favorite policies. (When the number of participants is odd, the median is the middle policy when the participants' favorite policies are put in order; when the number of participants is even, the median is the mean of the two middle positions.) This case is of particular interest because the median is an equilibrium outcome in models in which a compromise is reached by a procedure involving majority voting.

Which patterns of participation are stable ("equilibria")? Suppose that you, a potential participant, expect the participants' favorite policies to cover a broad spectrum. Then your participation will have little impact on the participants' median favorite policy. Suppose, for example, that you favor no homework and expect that in your absence the participants' median favorite policy at a PTA meeting will be 30 minutes of homework a day. Suppose also that you expect there to be participants whose favorite policies are 25 minutes a day and 35 minutes a day. Then your presence at the meeting will change the median by at most 2.5 minutes, from 30 to at least 27.5. (The number will be exactly 27.5 if no participant has a favorite policy between 25 and 30.) So unless your cost of participation is very low you won't participate.

What configuration of the participants' favorite positions will draw you away from the good book you'd like to read? Suppose you think that there will be two equally numerous extremist camps at the meeting---five people who favor no homework, five who press for three hours a day. If you don't show up the outcome will be a compromise of an hour and half a day. If you do show up you'll change the balance of the meeting entirely; the median will be your favorite position---no homework at all!

Further, such a configuration, in which there is a large "gap" exactly in the middle of the set of the participants' positions, is the only type of configuration that will provide you---or anyone else---with a strong incentive to attend. That is, the exclusive participation of extremists is a characteristic of any equilibrium in which attendance is positive.

Rosenthal, Turner, and I show also that low attendance is a characteristic of any equilibrium. The reason is subtle. Suppose that there is a small chance that anyone who intends to participate is prevented from doing so (their car gets a flat tire on the way to the meeting, for example). For any given meeting some people who plan to attend may not be able to do so: the set of people who manage to participate is random, depending, for example, on whose car gets a flat tire. Now suppose that the number of people who intend to participate is large. How likely is it that the set of favorite positions of the people who actually manage to participate (those who do not get a flat tire on the way to the meeting) consists of two subsets of exactly the same size, separated by a large gap? A central limit theorem from statistics tells us that such a configuration is unlikely. Even if the set of favorite positions of the people who intend to participate takes this form, the numbers of people on each side who are prevented from participating are unlikely to be exactly the same. (If you toss 1000 pennies the chance that you get exactly 500 heads and 500 tails is very small.) Thus each participant is extremely unlikely to be a swing vote that shifts the outcome from one extreme to the other when the number of participants is large.

But as I have argued above, every configuration for which anyone has an incentive to attend has a large gap exactly in the middle of the set of the participants' positions. Thus we conclude that in the presence of a little randomness a large gap is likely to appear, and participation to pay, only if the number of participants is small! Hence, in any equilibrium few people participate.

These results make us see that Aristotle's complaints, the pleas of the PTA chair that you come to the meeting, and the character of the input into timber policy in the Pacific Northwest have a common denominator. In a wide range of situations in which a collective decision is the outcome of a procedure in which participation is voluntary and costly, the participants are likely to be a small number of extremists.

For a more in-depth analysis of these issues, see M.J. Osborne, J.S. Rosenthal, and M.A. Turner, Meetings with costly participation. (American Economic Review 90:927-943, 2000.)