External validation of voter turnout models by concealed parameter recovery

One can also view our concealed parameter recovery approach as an unusual variation on the statistical methodology of testing for overidentifying restrictions. The (known) concealed parameter is, at the same time, formally identified in the models we estimate. Thus a model for which the estimated value of the concealed parameter is significantly different from the known value should be rejected.

For example, Armantier (2002) runs standard tests that are used in the empirical literature to distinguish between private and common value auction environments using bid data generated from a common value laboratory experiment and shows that they cannot reject the null hypothesis that the data was generated in an auction environment with private values. Frechette et al. (2005) and O’Brien and Srivastava (1991) conduct similar analyses in the context of multilateral bargaining environments with different protocols and stock markets with multiple assets, respectively.

Palfrey and Prisbrey (1997) use a weighted altruism model in their estimation of a model of voluntary contributions to public goods. This framework can be easily applied to a voting model and has appeared many places in the experimental and behavioral economics literature.

For models of voter turnout that endogenize the cost of voting see, e.g., Degan and Merlo (2011), Feddersen and Pesendorfer (1996, 1999) and Matsusaka (1995). For a survey of models of voter turnout see, e.g., Merlo (2006).

For an alternative structural estimation of pivotal-voter models using data from local school budget referenda in Oregon, see Hansen et al. (1987).

Note that in the experiment, subjects are not given the choice of voting for the opposing party’s candidate.

In this model, it is still the case that \(d=0\). In the special case of \( \alpha =0\) for all i, the model reduces to the pure model of instrumental voting described in the previous subsection.

A possible alternative model of altruism would be to have individuals value the payoffs to members of the other group as well as payoffs to members of their own group. This would reduce the incentive for minority voters to vote and increase the incentive for majority voters to vote. We did not explore this alternative model for two reasons. First, a group-based model of altruism parallels the ethical voter model, which is based on group utilitarianism. Second, there was no evidence of this kind of asymmetry in the data: i.e., less over-voting (relative to Nash equilibrium) by minority voters compared to majority voters. There is also an existing body of experimental evidence of group-based social preferences and cooperative behavior in other contexts. See for example Chen and Li (2009).

In the context of the experimental environments we study here, this model can be thought of as a statistical version of what Riker and Ordeshook (1968) first put forth formally as the “citizen duty” model.

Citizen duty and preference for expression are only two of the many possible rationales for the inclusion of a d-term in the calculus of voting equation that have been put forth over the years. For example, Tullock (1967) observed that this net cost of voting could just as easily be negative as positive.

In particular, the turnout rates predicted by the model are constant across elections and across groups.

This is also similar to earlier models of turnout and public good provision studied by Morton (1991).

Without this “group utilitarian” modification, the solution would be trivial: One voter from the majority party should vote, while everyone else abstains.

Coate and Conlin also assume that m and M are random variables, which is necessary for them to generate randomness in the electoral outcomes in equilibrium. We do not do this here for two reasons. First, in the experiments that generated the data, m and M are common knowledge. Second, because we have a finite number of voters, the outcomes of elections are random in equilibrium even if m and M are common knowledge.

The Levine and Palfrey data also contain 850 elections with 3 voters (i.e., a treatment with \(N=3\), \(m=1\), and \(M=2\)). We do not include these elections in our analysis since, when there are only three voters, the notion of group, which plays a critical role in the altruistic and the ethical model, is not applicable to the minority party (and is also not very meaningful for the majority party).

For further details about the experimental protocol, design, and results, see Levine and Palfrey (2007).

While we do not have a general uniqueness proof of the LQRE for the four voting models we consider, we do know that the Bayesian equilibrium (without QRE disturbances) in each of these models is unique for the experimental environment considered here. For this environment, our numerical procedure for computing the LQRE equilibria for each of the four models never encountered issues related to multiplicity for all possible parameter values.

This is different from the QRE estimation in Levine and Palfrey (2007), which conditioned estimation of a pivotal-voter model on all of the experimental parameters as well as the realized idiosyncratic costs.

To understand the intuition for why this is the case, note that for each group/party, “producing” turnout is analogous to producing a public good by voluntary contribution. From a selfish instrumental standpoint, each voter in a party has strong incentives to free ride by abstaining. The ethical voter model instead assumes that all voters make a group-efficient turnout decision and completely ignores the free rider problem. The very high turnout predicted by the ethical voter model thus simply reflects the typically large gap between efficient levels of public good production and individually rational levels that are produced by voluntary contribution. The fine details are somewhat more complicated because there are two competing groups, but this is the basic intuition.

Note that the instrumental model is nested in both the altruistic and the expressive models (corresponding to \(\alpha =0\) and \(d=0\), respectively). Neither \(\alpha \) nor d are estimated to be significantly different from 0 at any conventional significance level, and a likelihood ratio test fails to reject the constrained model (instrumental) in favor of either of the two models with an additional parameter.

The reason why both C and d are imprecisely estimated in the expressive model is that separate identification of the two parameters is tenuous.

Equivalently a fine could be imposed on non-voters. Many countries have compulsory voting that is enforced with fines (e.g., Australia).

Note that this is about twice the subsidy based on the naive calculation corresponding to shifting the voting cost distribution down by \(10\%\) of C (\(S=.06\) or .08, depending on the model).

The subsidy is equal to .64 according to the naive calculation.

It is “nearly” rather than “exactly” because all the models have a QRE component, so even voters with negative (net) voting costs abstain with positive probability. The same would be true if we were to shift the cost distribution down by .64 according to the naive calculation.

As long as \(S>.55\), all voters would have negative net voting costs and hence turnout would be nearly \(100\%\).

Obviously, it can also be applied to assess the empirical performance of many other models of voter turnout in addition to the ones considered here.