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The effective number of parties, N = 1/Σp(i)2 (where p(i) is the fractional share of the ith party), usually suffices to describe adequately a constellation of parties of different strengths. Difficulties arise when disparity in party sizes is such that the largest share (p1) surpasses 0.50 (meaning absolute dominance), while N still indicates a multi-party constellation. In such cases N(∞) = 1/p1 is proposed as a supplementary indicator: a value less than 2 indicates absolute dominance. An 'NP' index proposed earlier is a combination of N and N(∞); its values are close of those of N(∞), but NP sometimes falls below 2 even when many parties are relevant for coalition formation. Appendix A offers an alternative approach based on indices of deviation from a norm, but it proves cumbersome.

In the presence of a large lumped category of 'Other' parties the effective number of parties cannot be known exactly. Some approaches used produce large discrepancies. This note shows how the effective number still can be estimated with fair accuracy. The same issue arises with the measures of disproportionality between seats and votes. © 1997 Elsevier Science Ltd.

This paper proposes a model of a legislature, formed by several parties, which has to vote for or against a certain bill in the presence of a lobbyist interested in a certain vote outcome. We show that the ease with which the lobbyist can manipulate a legislature decision increases with the number of elected parties, and, consequently, decreases with an electoral threshold. On the other hand, a lower electoral threshold increases the representativeness of a legislature. We combine these two effects in a notion of fairness. We show the existence of an electoral threshold that optimizes the fairness of a political system, which is close to 1–5%. Namely, the optimal threshold (in our sense) is close to thresholds that exist in most parliamentary democracies.

This paper proposes a model of a legislature, formed by several parties, which has to vote for or against a certain bill in the presence of a lobbyist interested in a certain vote outcome. We show that the ease with which the lobbyist can manipulate a legislature decision increases with the number of elected parties, and, consequently, decreases with an electoral threshold. On the other hand, a lower electoral threshold increases the representativeness of a legislature. We combine these two effects in a notion of fairness. We show the existence of an electoral threshold that optimizes the fairness of a political system, which is close to 1-5%. Namely, the optimal threshold (in our sense) is close to thresholds that exist in most parliamentary democracies.

Altres ajuts: COST Action IC1205 on Computational Social Choice ; This paper proposes a model of a legislature, formed by several parties, which has to vote for or against a certain bill in the presence of a lobbyist interested in a certain vote outcome. We show that the ease with which the lobbyist can manipulate a legislature decision increases with the number of elected parties, and, consequently, decreases with an electoral threshold. On the other hand, a lower electoral threshold increases the representativeness of a legislature. We combine these two effects in a notion of fairness. We show the existence of an electoral threshold that optimizes the fairness of a political system, which is close to 1-5%. Namely, the optimal threshold (in our sense) is close to thresholds that exist in most parliamentary democracies.

This note helps to explain how cabinet-level concentration of power is constrained by party level concentration of seats. Arend Lijphart's Patterns of Democracy (1999) measures concentration of executive power by the frequency of minimal winning and one-party cabinets (MW/OP), and party concentration by the effective number (N) of legislative parties. In his factor analysis, these highly correlated indices are the central features that distinguish consensual from majoritarian systems. The present study establishes a quantitative logical relationship leading from N to the major component of MW/OP, so as to explain the reasons behind Lijphart's empirical observation. The note analyses separately the frequency of various types of cabinet coalitions that Lijphart's book has lumped together. It also offers a new way to visualize the effective number of legislative parties, as twice the minimal number of parties needed to form a minimal winning coalition. © 2002, Sage Publications. All rights reserved.

Political science knows a fair amount about the variables explaining party systems fragmentation or the constant in the equation of the number of parties. But we know much less about changes in the number of parties over time within countries or the slope in the equation of the number of parties. This paper fills this gap explaining the entry of new viable competitors in party systems after the founding election. Using empirical evidence from Spain, we show that when there is an electoral market failure and the higher the number of perfectly elastic voters, the higher the probability of new viable entrants. ; La ciencia política tiene un conocimiento bastante extenso de las variables que explican el sistema de fragmentación de partidos o de la constante en la ecuación del número de partidos. Pero, en cambio, se sabe mucho menos sobre los cambios en el número de partidos en el tiempo en los países o la pendiente de la ecuación en el número de los partidos. Este trabajo viene a cubrir esta carencia al explicar la entrada de los nuevos competidores viables en el sistema de partidos después de la elección fundacional. Utilizando la evidencia empírica de España, mostramos que cuando hay un fallo del mercado electoral y cuanto mayor es el número de votantes perfectamente elásticos, mayor es la probabilidad de nuevas entradas viables.

Political science knows a fair amount about the variables explaining party systems fragmentation or the constant in the equation of the number of parties. But we know much less about changes in the number of parties over time within countries or the slope in the equation of the number of parties. This paper fills this gap explaining the entry of new viable competitors in party systems after the founding election. Using empirical evidence from Spain, we show that when there is an electoral market failure and the higher the number of perfectly elastic voters, the higher the probability of new viable entrants. ; La ciencia política tiene un conocimiento bastante extenso de las variables que explican el sistema de fragmentación de partidos o de la constante en la ecuación del número de partidos. Pero, en cambio, se sabe mucho menos sobre los cambios en el número de partidos en el tiempo en los países o la pendiente de la ecuación en el número de los partidos. Este trabajo viene a cubrir esta carencia al explicar la entrada de los nuevos competidores viables en el sistema de partidos después de la elección fundacional. Utilizando la evidencia empírica de España, mostramos que cuando hay un fallo del mercado electoral y cuanto mayor es el número de votantes perfectamente elásticos, mayor es la probabilidad de nuevas entradas viables.

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.

What determines the number of parties in a national assembly? Previous work has emphasized either sociopolitical heterogeneity or electoral system permissiveness, or brought them together on an empirical basis. Here an equation is developed that satisfies two theoretical boundary conditions and expresses the effective number of assembly parties (N) in terms of both the number of politicized issue dimensions (I) and effective magnitude (M) of electoral system: N= I.6M.15 + 1. Actually, depending on circumstances, any of the three variables could become the dependent one, affected by the two others. Empirical evidence is presented, based on Lijphart's (1984) data on 22 stable regimes.

What determines the number of parties in a national assembly? Previous work has emphasized either sociopolitical heterogeneity or electoral system permissiveness, or brought them together on an empirical basis. Here an equation is developed that satisfies two theoretical boundary conditions and expresses the effective number of assembly parties (N) in terms of both the number of politicized issue dimensions (I) and effective magnitude (M) of electoral system: N= I.6M.15 + 1. Actually, depending on circumstances, any of the three variables could become the dependent one, affected by the two others. Empirical evidence is presented, based on Lijphart's (1984) data on 22 stable regimes.

The mechanical effect of electoral systems, identified by Maurice Duverger, can be estimated by means of a quantitative model. The model predicts the range within which the effective number of parties in a district should fall for a given magnitude (number of seats) of the district. At the national level, a related model predicts the effective number of parties based on the effective magnitude and the number of seats in the national assembly. The institutional variables considered—magnitude and assembly size—define a great portion of the structural constraints within which a given country's politics must take place. The model developed provides a good fit to data in spite of its having been developed from outrageously simple starting assumptions. © 1993, American Political Science Association. All rights reserved.

This article examines the forces shaping changes in the number of parties between consecutive elections. We argue that the transaction costs in electoral coordination depend on the turnout level in the previous election. The greater the number of peripheral voters entering the electorate, the less likely a substantial change in the distribution of partisan support in the subsequent election. The argument is tested using data for 313 parliamentary elections in 63 countries from 1990 to 2011, and two cases studies of countries using compulsory voting (the Netherlands and Australia). ; We acknowledge support from the Ministry of Economy and Competitiveness [Grant number CSO2017-85024-C2-1-P (AEI/FEDER], ICREA under the ICREA Academia programme, and the Australian Research Council Discovery Project [Grant number DP190101978].

This paper proposes a new method to evaluate the number of relevant parties in an assembly. The most widespread indicator of fragmentation used in comparative politics is the "Effective Number of Parties", designed in 1979 by M. Laakso and R.Taagepera. Taking both the number of parties and their relative weights into account, the ENP is arguably a good parsimonious operationalization of the number of "relevant"parties. This index however produces misleading results in single-party majority situations as it still indicates that more than one party is relevant in terms of government formation. We propose to modify the ENP formula by replacing proportions of seats by voting power measures. This improved index behaves more in line with Sartori's definition of relevance, without requiring additional information (such as policy positions) in its construction. We thus advocate for the use of our "Effective Number of Relevant Parties"in future comparative research.