¹ It may be objected that this is not rational choice since it rests on psychological, not material, incentives. Although we do not explore it in this article, we are indebted to Hugh Ward for the argument that many material benefits (e.g. employment contacts) flow from approval within informal social networks.

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⁷ As Palfrey and Rosenthal, ‘Voter Participation and Strategic Uncertainty’, showed, and Feddersen, ‘Rational Choice Theory and the Paradox of Not Voting’, p. 103, reiterates.

⁸ A recent article, by Levine, David K. and Palfrey, Thomas R., ‘The Paradox of Voter Participation: A Laboratory Study’, American Political Science Review, 101 (2007), 143–158CrossRefGoogle Scholar, uses quantal response equilibria (QRE), as opposed to Nash equilibria, inter alia to derive sensible turnout rates in large-N elections. But this is not rational choice: in the QRE equilibrium in a large-N election, the pivotal probability for the individual player is virtually zero, just as in a Nash equilibrium and as Levine and Palfrey themselves say ( Levine, and Palfrey, , ‘The Paradox of Voter Participation’, p. 155Google Scholar). A quantal response only ‘explains’ turnout in this case because it assumes that players make mistakes. Therefore, the standard rational choice paradox remains.

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¹³ As shown in Morton, ‘A Group Majority Voting Model of Public Good Provision’; Morton, ‘Groups in Rational Turnout Models’; Uhlaner, ‘Rational Turnout’; Aldrich, ‘Rational Choice and Turnout’; and Shachar and Nalebuff, ‘Follow the Leader’.

¹⁴ Morton, ‘A Group Majority Voting Model of Public Good Provision’; Morton, ‘Groups in Rational Turnout Models’; Uhlaner, ‘Rational Turnout’.

¹⁵ This is the Coate assumption. The Feddersen assumption is that they behave as simple rule-utilitarians, but this distinction does not affect our argument.

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⁴⁸ A constant term is included so that x_i ¹ = 1 for all i.

⁴⁹ There may be literally hundreds of small unrelated factors that affect the cost of voting for any individual and c_i is the sum of these factors for i. The central limit theorem states (loosely) that the distribution of the sum of a set of N independently distributed random variables converges to the normal as N goes to infinity.

⁵⁰ Of course, there will be negative drawings from the c_i distribution as well as positive ones, reflecting the benefits from voting. The model allows for only a small probability of c_i < 0 since we can think of its distribution as c_i∼N(α ₀, σ ²) with the α ₀ > 0 mean absorbed in the constant term.

⁵¹ Since the σ transformation does not affect significance tests, the tilde over the coefficients is dropped in what follows.

⁵² This assumption is discussed below.

⁵³ To see this, add and subtract inside the bracket on the right-hand side. Then each equation takes the form , with , so each equation is identical apart from the . Hence, any solution holds for all i. If a solution exists this implies . With , so and must intersect at least once since Φ is continuous and increases monotonically; and it is easy to see there are an odd number of equilibria.

⁵⁴ In other words, a player deserves punishment if he behaved badly in the previous period and/or if he deserved to be punished in the previous period but was not. He behaves badly if he did not vote when he meant to or if he did not punish someone who was guilty. We have neglected the case of a player punishing a non-G player. And we have also assumed that a G player at the start of m has no move in m.

⁵⁵ This assumption can readily be changed.

⁵⁶ We assume for convenience that the actual number of non-voters in equilibrium is equal to the expected number, and that the number is continuous.

⁵⁷ As with non-voting, the punishment for not punishing occurs a period later.

⁵⁸ , while . For simplicity, we assume: (1) that N is continuous; (2) if A ⩾ A*.

⁵⁹ The cut-off rule here is: vote so long as c_i ≤ c_A, where c_A = β(A + D), which implies that .

⁶⁰ This is not the only way of dealing with renegotiation-proofness. If my status in the group depends in part on never having been disapproved of by the group while others have, and if this gain in my status outweighs the cost of my having to disapprove of others, it will not be in my interest to renegotiate.

⁶¹ If the other side has a majority of 2, then an increase of turnout of 1 on one's own side would lead one to vote – but rationally there should never be a majority of 2.

⁶² In fairness, the standard model does imply very low levels of turnout, because at very low levels the probability of an individual affecting the outcome of the election is sufficient to outweigh the costs. Beyond that level, PB can be assumed to be 0, as we do assume.

⁶³ These in turn vary with the structure of national political institutions, although our evidence is focused on the United States and the individual level.

⁶⁴ Morton, ‘A Group Majority Voting Model of Public Good Provision’; Morton, ‘Groups in Rational Turnout Models’; and Uhlaner, ‘Rational Turnout’.

⁶⁵ Queries about our module of questions on the YouGov survey should be addressed to Samuel Abrams.

⁶⁶ The actual turnout rate is a subject of debate and disagreement with rates ranging anywhere between 50 and 56 per cent participation. See McDonald, Michael P., ‘Up, Up and Away! Voter Participation in the 2004 Presidential Election’, The Forum, 2 (2004)Google Scholar), issue 4, article 4, for a more thorough discussion.

⁶⁷ Since too few observations are available for some states, we lose information as a result of including this variable, but the substantive results are not much affected. To make the results comparable across models, all regressions are restricted to the set of observations for which we have party contact data.

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⁷⁶ We estimate this by simply omitting political knowledge from the model and then summing the estimated effects of group turnout, discussion, tenure and disapproval.

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