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Blais (2006) and Blais and Aarts (2006) in their review essays on voter turnout call attention to a striking puzzle about the link between electoral systems and turnout, namely that, ceteris paribus, proportional representation (PR) systems with many parties appear to have higher national-level turnout than single-member district (SMD) plurality systems with few parties, yet turnout does not increase with the (effective) number of parties (ENP) at the national level. To address this puzzle we turn to district-specific within-nation panel data from Switzerland and Spain. Our country-specific findings allow us to explain the national-level puzzle as essentially an ecological artefact, in that the multi-member districts found in proportional systems, on average, do exhibit higher turnout than SMDs, but turnout does not rise with district magnitude, m, once we move beyond the contrast between m = 1 and m > 1. Using a more sophisticated approach to measuring political competition that does not treat all PR systems as generating identical turnout incentives (Grofman and Selb, 2009), we seek to explain this puzzle by showing both empirically and theoretically that (1) proportionality does not necessarily increase with district magnitude, and (2) competition does not necessarily increase with district magnitude. [Reprinted by permission of Sage Publications Ltd., copyright holder.]

Blais (2006) and Blais and Aarts (2006) in their review essays on voter turnout call attention to a striking puzzle about the link between electoral systems and turnout, namely that, ceteris paribus, proportional representation (PR) systems with many parties appear to have higher national-level turnout than single-member district (SMD) plurality systems with few parties, yet turnout does not increase with the (effective) number of parties (ENP) at the national level. To address this puzzle we turn to district-specific within-nation panel data from Switzerland and Spain. Our country-specific findings allow us to explain the national-level puzzle as essentially an ecological artefact, in that the multi-member districts found in proportional systems, on average, do exhibit higher turnout than SMDs, but turnout does not rise with district magnitude, m, once we move beyond the contrast between m = 1 and m > 1. Using a more sophisticated approach to measuring political competition that does not treat all PR systems as generating identical turnout incentives (Grofman and Selb, 2009), we seek to explain this puzzle by showing both empirically and theoretically that (1) proportionality does not necessarily increase with district magnitude, and (2) competition does not necessarily increase with district magnitude.

Despite its conceptual centrality to research in comparative politics and the fact that a single measure—the Laakso-Taagepera index (LT)—is nearly universally employed in empirical research, the question of what is the best way to "count" parties is still an open one. Among other alleged shortcomings, LT has been criticized for over-weighting small parties, especially in the case of a one-party majority. Using seat-shares data from over 300 elections, I have calculated LT as well as an alternative measure (BZ) which employs normalized Banzhaf scores rather than simple party seat shares, as weights. The Banzhaf index is a voting power index which calculates a party's voting power as a function of its coalitional potential. Though the two measures are highly correlated, I identify three particular party constellations in which the differences between LT and BZ are systematic and statistically significant. In all of these cases, and especially in the case of a one-party majority, I argue that BZ is a more accurate representation of the actual party system, after any given election, while LT is perhaps better interpreted as a measure of the shape of the party system more generally. These findings have many implications, including with respect to the categorization of party systems and the empirical validity of Duverger's Law.

Despite its conceptual centrality to research in comparative politics and the fact that a single measure - the Laakso-Taagepera index (N[sub]s) - is widely employed in empirical research, the question of what is the best way to "count" parties is still an open one. Among other alleged shortcomings, N[sub]s has been criticized for over-weighting small parties, especially in the case of a one-party majority. Using seat shares data from over 300 elections, I have - for the first time - calculated N[sub]s as well as an alternative measure (N[sub]Bz) which employs Banzhaf scores, rather than seat shares, as weights. The Banzhaf index is a voting power index which calculates a party's voting power as a function of its coalitional potential. Though the two measures are highly correlated, I identify three particular party constellations in which the differences between N[sub]s and N[sub]Bz are significant and systematic. [Copyright Elsevier Ltd.]

Forming an idea of the number of parties competing at elections or winning seats in legislatures is fundamental to disaggregated approaches to mapping party systems. We set out a method for systematically relating the behaviour of any `number of parties' index to the size of the largest party's vote and the numbers of parties in competition. This approach shows that the `effective number of parties' ( N2) can confuse real changes in party competition with mathematical quirks in the way that the index is calculated. We also demonstrate that N2 (and its main rival the Molinar index) behaves in hard to predict and anomalous ways under some configurations of party support. We conclude that the Molinar index should not be further used, and that the N2 score's behaviour can create problems in quantitative applications. Even in less formal historical or comparative analyses N2 always needs to be carefully interpreted. There is no `perfect' measure of the weighted number of parties, but averaging N2 scores with a simple measure of largest party predominance (1/ V1) produces a highly correlated measure ( Nb), but one with lower maximum scores, less quirky patterning and a readier interpretation. A more radical solution is to `spatialize' N2 (or Nb) scores, which allows analyses to take more account of variations in the party competition conditions lying behind any given index number.

I propose an alternative index to operationalize the variable number and size of parties in a party system. To support it, I present a critical overview of the two most common indices, Laakso and Taagepera's effective number of parties and Kesselman and Wildgen's hyperfractionalization, showing the causes of their weakness. Then I explain the computational logic of my alternative, "number of parties," and compare it with the other two, using hypothetical cases. After that, I contrast the Laakso-Taagepera index with mine, using data from actual elections between 1945 and 1981. I conclude that my index outperforms the other two as an operationalization of the variable number and size of parties.

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. Adapted from the source document.

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. © 2013 © 2013 Elections, Public Opinion & Parties.

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. © 2013 © 2013 Elections, Public Opinion & Parties.