How Turnout Depends on the Number of Parties: A Logical Model

Abstract

We illustrate the power of “logical models” (Taagepera, 2007) by offering a three-parameter model of the relationship between the effective number of parties and electoral turnout that makes use of the constraints on what parameter values are internally coherent given boundary conditions to specify functional form, and seeks not optimal curve fitting but rather a direct model testing. In our model, one parameter reflects an effect that generally acts to increase turnout as the effective number of parties increases, another an effect that generally acts to decrease turnout as the effective number of parties increases, while a third parameter allows for baseline variation in turnout across countries (or within countries across elections). We fit this model to district-level data from 237 elections held in 17 countries, representing a wide range of electoral system types generating multi-party contests, with over 20,000 district-election observations. The basic intuition, that turnout rises to a peak as the effective number of parties increases and then falls slowly, fits our data pretty well.

Acknowledgements

Grofman's work on this project was supported by the Jack W. Peltason (Bren Foundation) Chair, University of California, Irvine, and by the UCI Center for the Study of Democracy. A previous version of this article was presented at the conference “The Effects of District Magnitude,” Lisbon, May 29–30, 2012. We are grateful to the participants for their helpful comments. We also thank Kathrin Ackermann for her superb research assistance.

Notes

*This article makes reference to supplementary material available on the publisher's website at http://dx.doi.org/10.1080/17457289.2013.858345

1. For example, one of the factors involved in complexity in multi-party settings is attempting to assess what governing coalition will form (Kedar, 2009).

2. The general reasons why the effective number of parties (Laakso & Taagepera, 1979) is used so widely in comparative politics rather than the raw number of parties is that the total number of parties that run in elections depends heavily on the ease of registering parties and independent candidates that net only a few votes, and there is considerable variation in party size. The effective number of parties provides a single meaningful number. Nonetheless, when we have replicated our analyses with the raw number of parties, we get similar results (data omitted for space reasons).

3. Since most modeling in political science still uses linear models, whether they be OLS or otherwise, it is often easy to miss curvilinearity in the data in the absence of theoretical expectations that lead the analyst to look for such nonlinearities. There are, however, a number of recent papers that offer theoretical reasons to expect curvilinearity. For example, in the election context, Ashworth et al. (2006) argue that, rather than high competition levels leading to greater turnout, the relationship between competition and turnout should be curvilinear. The reason they offer is that, while the relationship should be positive for “Downsian” voters, it should, they claim, be negative for “expressive” voters who vote as an expression of identity politics, and the fact that there is a mix of the two types in the population will lead to a curvilinear pattern. Stoll (n.d., cited in Moser and Scheiner, 2012: 186–195; see also Madrid, 2005) argues that, rather than the number of parties increasing with social heterogeneity, as is commonly supposed to be the case, the pattern should instead be curvilinear. As summarized by Moser and Scheiner (2012: 186–187) her argument is that, past some point, “further increases in diversity – when there are larger numbers of (especially small groups – actually reduce the likelihood that any of the groups will play a meaningful governing role. Under these circumstances, political entrepreneurs have no incentive to create parties (or promote) candidates” for small groups, leading to “a consolidation of the party system.” However, while both these papers propose a curvilinear model, neither seeks to model the functional form that governs that curvilinearity.

4. Simple systems include first-past-the-post/single-member plurality (SMP) electoral systems such as the United States or India, single transferable vote (STV) systems like Australia and Ireland, single non-transferable vote (SNTV) systems like (pre-1996) Japan, or systems of so-called “districted proportional representation (DPR)” such as Spain or Switzerland (Monroe & Rose, 2002) in which there is no cross-district compensation for disproportionalities resulting from the vote seat-translation at the district level.

5. The data are freely available from the CLEA website at http://www.electiondataarchive.org/index.html. We use download release 4 (September 14, 2011).

6. For SMD countries, there are also some difficulties in obtaining a sufficiently long time series that reports votes for minor parties at the district level. Countries such as the United States and Great Britain are not in the database we used.

7. Detailed results are available from the authors on request.

8. Curve fitting is a statistical endeavor entirely or mostly oblivious to the nature of the data and the substantive implications of the findings, and (adjusted) R-square is the supreme criterion. In the present case, the curve-fitter might well be satisfied with a linear fit, because, given the huge scatter, no other curve would raise the dismally low R-square to any appreciable extent. A curve fitter is not disturbed by this line projecting to turnouts larger than 100% (if, as expected, the linear fit is upward sloping) since there are no data points where the unrealism of this prediction might be noted.

9. Model fitting involves two stages: (1) developing a theoretically grounded set of expectations and developing a functional form that is appropriate, and (2) evaluating the goodness of result. The first stage is often art as much as standard procedure. For the second, model-fitting is in stark contrast to what is usually referred to as curve-fitting that aims at maximizing R² or related empirical goodness-of-fit measures. For example, if we hypothesize that a given parameter has a certain value or range of values then we would reject the model fit, regardless of R², if that parameter was clearly outside the predicted range rather than testing a null hypothesis of no relationship. In the approach to model testing we use here, for each of the countries, we simply report which parameters violate our theoretical expectations – discovering that our model does quite well. (An alternative approach might constrain a parameter at a specific value and then use a likelihood ratio test to compare the log likelihood of this model to that of the unconstrained model where the parameter is freely estimated.)

10. The chief concern with omitted variables is that their omission will lead to a misspecification of the relationship between the included variables and the dependent variable of interest. Factors that affect T but are unrelated to N are irrelevant. To the degree T and N interact through other processes (and through third factors) not included in our simple model, they would be likely to produce different patterns. Thus they would tend to blur – not reinforce – the relatively simple pattern our simple model expects. Two potential confounders that might affect both N and T are electoral system strength and social heterogeneity. Our implicit assumption is that those factors indirectly affect T via N. Problems would only occur if those factors had an impact on T other than through N, for instance, electoral system → competition → T (Cox, 1999), or social heterogeneity → conflict → T (Huckfeldt, 1979). But as far as we are aware, none of those theories would suggest the particular pattern of nonlinearities between N and T we observe for almost all the countries in our data set.

11. Determination of causality requires attention to issues of model specification. At the suggestion of a reviewer, we have now replicated our analyses with raw numbers of party instead of effective number of parties, N, and the results look quite similar. This reinforces our view that one primary causal process affecting turnout involves number of parties, e.g., some aspect of complexity of choice lowering turnout while more options for choice raising turnout (e.g. Brockington, 2004; Geys & Heyndels, 2006). We deliberately have not made direction of causality central in our article because, in the sciences, many processes involve simultaneous determination, e.g., in Ohm's Law about the relationships between measures of resistance and measures of flow. While we do agree that it is quite likely that T and N interact reciprocally by various other processes, and third factors act on both, thus connecting them indirectly, our various theoretical lines of argument all suggest that T is primarily a function of N rather than N a function of T.