When institutions matter: electoral systems and intraparty fractionalization in Western Europe

The comparative study of intraparty divisions and their determinants has been a long-debated matter, but some issues remain unresolved. First, the problem of the empirical identification of intraparty groups. Second, the lack of comparative perspective and large-N cross-country and cross-time analyses, given intraparty divisions have been studied mostly through theoretical or, at best, small-N analyses. Third, the underestimation of the potential role exerted by contextual factors because of the emphasis put almost exclusively on intraparty dynamics. To fill these gaps, the article employs a measure of visible intraparty fractionalization allowing for large-N cross-time and cross-country comparison. Moreover, by focusing on about 700 party cases in 11 Western European countries since 1965, the article shows that institutional factors, particularly electoral systems’ characteristics, impact intraparty fractionalization. Specifically, intraparty fractionalization is higher in more disproportional systems and where there are stronger incentives for personal vote.


Introduction
This article provides scholars with an innovative perspective to study intraparty divisions. Naturally, this is not a newly-studied matter: already Voltaire, in 1778, remarked that 'a head of a party is always a head of a faction' (quoted in Sartori 1976Sartori [2005: 1). Such relevance has also been underlined by Sartori, who advanced the idea that political parties are 'an aggregate of individuals forming constellations of rival groups' (Sartori 1976(Sartori [2005: 64).
Focusing on parties' internal divisions might be especially relevant today, given the problems encountered by political parties. Indeed, the number of rebel MPs has weakened party discipline (Sieberer 2006;Carey 2007;Kam 2009), the number of fissions and fusions has been increasing (Ceron 2015), parties' capacity to coherently and efficiently represent and govern has undergone profound challenges (Flinders 2012;Mair 2013), and electoral volatility has grown all over Western Europe (Chiaramonte and Emanuele 2022). All in all, party-centered politics has undergone dire straits, and scholars have increasingly questioned the assumption that parties shall be treated as unitary actors (Scarrow and Webb 2017;Martocchia Diodati 2019).
Therefore, although the relevance of intraparty groups was evident also in the era of mass parties, it is precisely when the established practices and power of political parties have experienced profound shocks that intraparty groups might become the protagonists of the political scene. If, in the past, the role of intraparty groups could have been somewhat counterbalanced by the existence of strong and rooted political parties, the decline of these latter could bring about greater importance for the former. The well-known argument by Sartori (1976), according to which to fully understand political parties, it is necessary to take into account the role played by their internal divisions, cannot be left aside anymore.
As we show later, the study of intraparty groups has suffered from some unsettled issues. First, the problem of their empirical identification. Second, the lack of comparative perspective and large-N cross-country and cross-time analyses, given intraparty divisions have been studied mostly through theoretical or, at best, casestudy or small-N analyses. Third, the underestimation of the potential role exerted by contextual factors because of the emphasis put almost exclusively on intraparty dynamics.
To fill these gaps, in this contribution, we critically discuss the existing literature on intraparty divisions, propose a clear definition and operationalization of intraparty fractionalization, and study their determinants by specifically focusing on the role played by institutional constraints. The empirical analysis, which covers 70 parties in 11 Western European countries since 1965, shows that, far from being a mere product of intra-party competition, intraparty fractionalization is shaped by electoral system features, such as the level of disproportionality and the incentives for personal vote.
The article is organized as follows: the next section discusses the existing literature on intraparty groups and raises hypotheses related to the impact of institutional constraints on intraparty fractionalization; the subsequent section introduces a measure to study intraparty fractionalization and shows its variation in Western intra-party groups. In turn, this has created two major issues: conceptual stretching and empirical detection.
To begin with, the term conflates a wide range of types of intraparty groups. Such groups can be internally distinguished in terms of legal status (whether such groups are allowed or even recognized within a party), structure (the degree to which intraparty groups are organized), attitude (whether groups have a pragmatic or an ideological raison d'être), motivation (whether they have office-seeking or policy-seeking goals), and, finally, stability (whether they are durable or more volatile groups). 1 This clearly raises a conceptual stretching issue (Sartori 1970) because the same label has been used to identify different objects with different levels of abstraction.
The second issue-strictly related to the first one-has to do with detecting factions as an empirical object of study. In this regard, the literature is by far scarce. All the classical contributions on party factions either relied on purely theoretical accounts or, at best, put forward guidelines for an operationalization that remained on paper. 2 The first to provide an empirical account is the cross-national International Comparative Political Parties (ICPP) project (Janda 1980). Here, parties' internal divisions were coded according to six variables: legislative cohesion, ideology, issues, leadership, strategy, and purges. Since then, most of the subsequent literature has focused on one of these aspects, namely, the intraparty competition for leadership. Indeed, they have identified factions by relying on motion lists to select the party leader (Giannetti and Laver 2009;Boucek 2012;Ceron 2012Ceron , 2015Martocchia Diodati 2019). Following this literature, the number of factions in a party corresponds to the number of motions presented during a party leadership race. This is undoubtedly a significant step towards a proper empirical analysis of intraparty divisions, as 'it defines in an exhaustive and mutually exclusive way different subparty groups' (Martocchia Diodati 2019: 594).
An early clue on this kind of operationalization can already be found in Sartori (1976Sartori ( [2005: 98) concerning identifying Italian DC intraparty groups through the motions presented in the party congress. When referring to these groups, Sartori wittily used the looser term 'fraction' instead of the more specific 'faction'. This distinction is not just a terminological one but is crucial for understanding that we have to be cautious about the effective ability of this kind of operationalization to really capture the actual divisions within a party.
Indeed, we acknowledge this method is far from perfect, in that it is likely to underestimate the 'real' number of factions. This is because factions might avoid showing themselves by explicitly supporting a motion, or they can form a coalition with other factions to support a single candidate, or because the party finds a compromise for behaving as a unitary actor by presenting just one candidate for the leadership (see Allern and Karlsen 2014;Ennser-Jedenastik and Müller 2014). 3 What this operationalization really does is give us an account of the visible level of fractionalization within a given party, thus allowing us to put forward a comparative empirical research about intraparty divisions and their determinants. At the same time, for the above considerations, we prefer to use the more generic Sartorian concept of fraction instead of the more specific but also less empirically graspable faction.

Hypotheses
This section raises hypotheses about the impact of a country's institutional framework on intraparty fractionalization. More specifically, by innovating the related literature, we argue that the electoral system plays an essential role in shaping the number of fractions in Western European party systems. Indeed, although intraparty fractionalization has been widely studied, a central aspect has been overlooked: fractions might be shaped not merely by party-level characteristics but also by systemiclevel constraints.
In this regard, political scientists have always been aware that 'context matters'. Starting from the classical 'holistic perspective' of social sciences and the more recent neo-institutionalist approach (Marsh and Olsen 1989), studying how contextual effects influence or moderate actors' behavior has become a central asset of political science. The study of contextual effects addresses the question of how 'environmental properties determine variation in a given behavior of interest' (Sprague 1982: 100). In particular, contextual effects have been shown to have a powerful impact in many fields of research, such as electoral participation (Lane and Ersson 1990;Franklin 2004) and electoral behavior (Thomassen et al. 2005;Dalton and Anderson 2011), the study of new parties (Birch 2003;Tavits 2006), or government formation (Müller and Strøm 2000). 4 Contextual effects may play an important role in shaping party fractions. Indeed, if it is true that parties are a 'constellation' of fractions, it is therefore also true that contextual features may affect not only parties but also their internal fractionalization. In this regard, some hints about the role played by institutional factors, and particularly electoral systems' features, have been already discussed by scholars (Zariski 1960(Zariski , 1965Cox et al. 1999Cox et al. , 2000Boucek 2009Boucek , 2012. However, all these studies have either only a theoretical perspective or mainly focus on two prominent cases, such as the Italian Christian Democracy (DC) and the Japanese Liberal-Democratic Party (LDP). Therefore, what still lacks is a truly comparative analysis exploring the effect of contextual factors on party factions in a broad number of countries and parties over time.
We argue the characteristics of an electoral system provide political actors with incentives and constraints concerning the decision for a fraction to split (or not) from its party, for a party to contest (or not) a given election, to form (or not) an electoral alliance, and so forth. More specifically, we consider two elements of each country's electoral rules: the disproportionality of an electoral system (Gallagher 1991) and the incentives 'to cultivate personal vote' (Carey and Shugart 1995).
Following the literature on the effects of electoral systems on the number of parties (Duverger 1954;Lijphart 1994;Sartori 1994;Cox 1997), the argument behind the first element is straightforward: the higher the permissiveness of an electoral system, the higher the number of parties, and vice versa. So, with a permissive electoral law, a political entrepreneur has greater incentives either to split her/his fraction from an existing party or create a new political formation from scratch. This is because the representation cost would be smaller than a less permissive system. Conversely, as disproportionality increases, the incentives for party splits or the emergence of new parties become smaller, as obtaining representation will be costlier in terms of votes to get (e.g., see Boucek 2012: 208;Ceron 2015). In such a scenario, and this is crucial for our argument, it would be more convenient for fractions-to say it with Hirschman (1970)-not to play the 'exit' card but rather to use the 'voice' option within their party (Close and Gherghina 2019). As a result, in highly disproportional contexts, the party system's format will be limited. However, we expect a higher fractionalization at the intra-party level: while there will be fewer incentives for inter-party fractionalization, there will also be more chances for within-party fractionalization. More formally, Hp1 states that:

Hp1
The higher the disproportionality of an electoral system, the higher the intraparty fractionalization.
Focusing on the second element of each country's electoral rules, we expect that the personalization of the electoral system (Carey and Shugart, 1995;Renwick and Pilet, 2016) could play a role in our story. For example, let us imagine a country A with a closed-list PR system, thus not providing voters with any stimuli for personal vote. Here, the voter will be allowed to cast a vote for a party without any indication related to within-party competition or individual candidates. On the other hand, let us imagine a country B, with an open-list PR system, where voters do have the possibility to make an intra-party choice. In country A, we expect that parties will more likely act as unitary actors, as intra-party divisions have fewer incentives to emerge by becoming visible in voters' eyes and weighing their electoral strength among each other. In other words, the intra-party competition will be more likely to remain invisible. Conversely, in country B, thanks to preferential voting, it will be easier for fractions to be recognizable by voters and exploit the electoral competition to show their strength vis-à-vis the other fractions. This rationale was already acknowledged by Zariski (1960: 24), who, referring to the Italian DC, argued that 'when proportional representation is coupled with genuine provisions for preferential selection among candidates of the same party, minority factions may flourish'. A similar argument is made by Cox et al. (1999: 35-37) about the role of the Single Non-Transferable Vote system in institutionalizing the factionalism of the Japanese LDP. 5 More formally, Hp2 is as follows:

Hp2
The higher the incentives for cultivating personal vote, the higher the intraparty fractionalization.

The dependent variable
Measuring the level of intraparty fractionalization is not an easy task. Resorting to the discussion in the theoretical framework, and following the abovementioned literature (Giannetti and Laver 2009;Boucek 2012;Ceron 2012;; Martocchia Diodati 2019), we start from party leadership races. We apply the index by Laakso and Taagepera (1979), originally developed to measure the Effective Number of Electoral Parties (ENEP), to visible intraparty fractionalization. Such a measure has already been employed by Boucek (2012: 45-46) to analyze intra-party factions in the Italian Christian Democracy's party congresses and national party councils  and the Japanese Liberal-Democratic party's parliamentary party . 6 Hence, our measure of Visible Intraparty Fractionalization (VIF) in a given party is calculated as the number of candidates in a given party leadership race weighted by their relative electoral strength, that is, the percentage of votes received by each candidate in that race. 7 Formally, we define VIF as follows: where f ij is the proportion of votes that a fraction i have obtained during an intraparty competition in party j.
Let us give a brief example. Let us imagine a party, say, Party A, which, according to its last leadership race, has four fractions, one having received 35% of congress delegates' votes, the second having obtained 30%, the third one 25%, and the fourth one 10%. Then, following our formula above, the number of VIF of party A is equal to: So far, the literature has mainly relied on a case-study or a very small-N approach, focusing on just one or a few political formations (e.g., Boucek 2012; Martocchia where p is the party's national share of votes. The ENEP counts the number of parties in a given party system by weighing them for their relative electoral strength, and it is considered the standard measure of party system fractionalization (Cox 1997;Lijphart 1999;Golosov 2015). 7 For a further use of the Laakso and Taagepera's index to the competitiveness of leadership races, see Kenig (2009a).

Diodati 2019). Conversely, we innovate the existing literature on the topic by studying the impact of systemic-level factors on intraparty fractionalization in Western
Europe through a large-N cross-country and cross-time comparative analysis. To do so, we apply this measure to cover a wide range of party leadership races, from party councils to open primary elections (see Pilet and Cross 2014). 8 Overall, we have focused on 11 Western European countries since 1965, for about 700 party cases. In particular, we have considered Austria, Belgium, Denmark, Germany, Greece, Ireland, Italy, Norway, Portugal, Spain, and the United Kingdom. The choice of the countries and timespan reflects constraints of data availability: we have included all observations for which we have been able to retrieve reliable data on our dependent variable. 9 At any rate, the selected countries show variation in the political and institutional settings, particularly in our key variables of interest. More specifically, our sample ensures variation in the electoral systems' characteristics both in terms of disproportionality levels (from the highly proportional Austrian and Danish systems to the highly distortive Spanish or UK systems) and incentives for cultivating personal vote (from the Portuguese closed-list system to the Irish single transferable vote system up to the UK plurality system). Moreover, our 50-year timespan allows for detecting cross-temporal variations in the hypothesized association between electoral systems' features and intra-party fractionalization. To the best of our knowledge, this dataset represents the largest data collection on intraparty fractionalization and its determinants in time and space within the context of Western Europe.

The independent variables
Concerning the operationalization of the independent variables, we proceed as follows: to test Hp1, we measure the Disproportionality of electoral systems by relying on Gallagher's (1991) least squares index of disproportionality. 10 Then, to test Hp2, starting from Carey and Shugart's original classification (1995), we follow the measure of personal vote developed by Johnson and Wallack (2012) and take the data from Emanuele (2018). The measure consists of three elements of each country's electoral law. First, 'Ballot', is the level of party control over the party list and candidates' rank on such a list. Second, 'Pool', is the degree to which votes are shared where v i is the share of votes of party i and s i is the share of seats of party i. Data for Disproportionality come from Gallagher (2019). among each party's candidates. Third, 'Vote', measures whether votes are cast for parties or candidates (Carey and Shugart 1995;Johnson and Wallack 2012). 11 The resulting variable is named Personal Vote.
We also control for other systemic-and party-level factors. Starting from the former, we control for the level of fractionalization in the party system. Indeed, as mentioned above, our hypotheses about the impact of electoral system features on intraparty fractionalization are raised by thinking about the incentives for party fractions to either make an exit or voice within their party. So, it is important to control for the number of existing competitors, as this latter might influence the decisions of intra-party actors. We operationalize party system fractionalization via the Effective Number of Electoral Parties (ENEP), developed by Laakso and Taagepera (1979, see also above). 12 A second systemic-level control variable is related to party system stability. Over the last years, this factor has become increasingly crucial for understanding party system change, as empirical evidence has widely shown that Western European party systems have become less stable than before (Chiaramonte and Emanuele, 2017). The more the system becomes electorally unstable, the larger the number of voters who are available to potentially change their previous voting choice. Consequently, intraparty fractions may have higher incentives to split from their party and face electoral competition. We employ the standard measure of electoral instability, namely, the Pedersen's index of Electoral Volatility (1979), whose data come from Emanuele (2015).
Moving to party-level variables, we control for two party-organization-related factors: party membership and leader selection rules. Starting from the former, a larger membership might increase the likelihood of forming divergent fractions within parties. This reasoning was put initially forward by Sartori (1976Sartori ( [2005: 79), who stated that 'party subdivisions are the inevitable consequence of the mass membership of mass parties'. Therefore, we control for the ratio between party members and a country's electorate (M/E). For each party in a given election, we divide the total number of party members in the electoral year by the country's electorate. This variable is a widely used measure to make comparisons across different contexts (e.g., see Weldon 2006;Van Biezen and Poguntke 2014;Kölln 2015). 13 Moreover, we control for party leader selection rules. In the previous section, we have shown that our measure of VIF was built starting from party leadership races. We now consider how party leaders are selected. More specifically, we focus on the 11 For details about the rationale behind the score attribution, see Johnson and Wallack (2006). For a similar classification, see Seddon et al. (2002). On the link between electoral systems and personal vote, see also Mershon (2020). For a critical review of extant classifications, see André et al. (2016). 12 Data for this variable are taken from Gallagher (2019). 13 Data on party members mainly come from the MAPP project data archive (Van Haute and Paulis 2016), integrated with party-related sources. Instead, data on each country's electorate come from Nohlen and Stöver (2010) until 2009 and, for the most recent years, from each country's electoral authority.
Selectorate, the body of people that selects the leader, in line with the related literature (see Kenig, 2009a;Boucek, 2012;Pilet and Cross, 2014;Kenig et al., 2015). 14 Moreover, we also control for the Candidacy, namely, the requirements to make a bid in the party leadership race (Kenig 2009b). 15 Trivially, a wider body of people selecting the party leader and lower barriers to become a candidate for the party leadership increase the likelihood of finding heterogeneous political stances that, in turn, might coagulate around different intraparty fractions (e.g., see Kenig 2009a;Kenig et al. 2015).
Furthermore, we also include three other standard party-related controls, namely party size, party age, and party's left-right placement. Party size is nothing but the percentage of votes received by a party in a given parliamentary election, while Party age is the number of years elapsed since party formation. 16 The underlying idea is that larger and older parties are less likely to be unitary actors, as, on the one hand, bigger parties are more likely to be internally heterogeneous (e.g., see the discussion in Giannetti and Pinto 2020), and, on the other hand, there is more time for intraparty disagreements to show themselves. Then, we include Left-right placement under the assumption that right-wing parties are less internally divided that left and center parties (Kenig et al. 2015) and we measure it through the 'Rile' variable of the Comparative Manifesto Project (Volkens et al. 2019). Finally, we include in the model a trend variable (Time), operationalized as the number of years since 1965. 17 Table 1 below reports the descriptive statistics of the dependent, independent, and control variables.

Intraparty fractionalization across time and space
After having presented our measurement of VIF and the independent and control variables, it is time to delve deeper into its temporal and spatial analysis. We start by 14 The variable ranges from 0 to 2. In particular, 0 means that the selectorate in a party leadership race is restricted to the parliamentary party group, the party elite, or even a single person (usually the party leader); 1 means the selectorate consists of the party convention or congress; and finally, 2 means the selectorate is made by party members or even party voters or sympathizers (i.e., open primary elections). Data for this variable come from the COSPAL (Comparative Study of Party Leaders) dataset (see below) up to 2012; for the latest years, as well for Greece and Ireland, excluded from the COSPAL, we have relied on our own calculations based on party-related sources. 15 The variable goes from 0 to 2, where 0 means no requirements at all; 1 means being a party member; 2 means there are stricter requirements, such as being an MP. The data sources are the same as for the Selectorate variable. 16 Data on party size are taken from Nohlen and Stöver (2010) until 2009 and from each country's electoral authority for the most recent years. Conversely, data on party age come from Casal Bértoa and Enyedi (2022) or other party-related sources. 17 Indeed, 'the trend variable helps to avoid the problem of spurious correlation arising when the values of the dependent variable and those of one or more independent variables vary independently but in a consistent direction over time. This variable also makes it possible to identify secular trends in the dependent variable itself' (Roberts and Wibbels 1999: 581-582; see also Tavits 2005: 290 on the same point).
considering the evolution over time of intraparty fractionalization. Figure 1 reports the boxplots of VIF by decade. 18 By looking at Fig. 1, an increasing trend over time in the level of intraparty fractionalization is detectable. Indeed, the VIF increases, on average, from 1.25 to 1.54 between the 1960s, our starting decade, and the 2010s. However, a closer inspection of the data reveals that the VIF has remained substantially steady until the end of the 1990s, when its average value is still the same as in the 1960s. Then, a limited increase is visible in the 2000s (average VIF is 1.31), although this is mainly due to outliers, as shown in the graph below. Subsequently, in the last decade, Western European parties have shown growing fractionalization, but this time the increase in the average VIF is more consistent across units, as the great bulk of observations records a higher VIF than the previous decade, almost without the presence of outliers.
This finding is also interesting from another viewpoint. Indeed, one of the most important recent findings in the party system literature is that the number of parties in contemporary Western European democracies has been increasing over time (Bolleyer 2013; Emanuele and Sikk 2020). In this context, according to Sartori (1976Sartori ( [2005: 79), we could have expected that 'the more numerous the parties, the greater the homogeneity of each party and the lesser the need for intra-party fractionism', but what data tell us is that, even in a context of increasing party system fractionalization, intraparty fractionalization has risen as well. Moreover, this also tells us that the trend toward party system destructuring highlighted so far (Chiaramonte and Emanuele 2017; Lisi 2019) is even more pronounced if we consider not simply the 'visible' part of a party system (i.e., political parties themselves), but also its 'invisible' one (party fractions), in line with what was advocated by Sartori (1976). By moving to cross-country comparisons, Fig. 2 above shows the evolution of the average VIF over time for the 11 Western European countries under consideration. 19 Data tell us that, while some countries show that their parties are, on average, unitary actors (Austria, Germany, and Norway), in some other cases, a higher average VIF can be detected (Greece, Ireland, and the United Kingdom). Concerning temporal variations, while Ireland and Italy show an increasing trend over time, some other countries show no significant variations (Germany and Norway) or erratic trends with notable ups and downs (Greece and the United Kingdom).
All in all, Fig. 2 shows different patterns of intraparty fractionalization in Western Europe. How can we account for this variation? Do-and to what extent-institutional factors influence the visible number of fractions? The next section tries to answer these questions.

Analysis and results
The dataset built for the empirical analysis has a multilevel hierarchical data structure. Our observation is the party in a given parliamentary election. The partyelection units (667) are nested within parties (70), which are nested within countries (11). Therefore, we have a hierarchical data structure, where problems of heteroskedasticity might emerge, as errors might not be independent and identically distributed. We have performed an intra-class correlation (ICC) test that confirms that we need to take into account the multilevel hierarchical nature of the data. This is why we have opted for a multilevel mixed-effects model (Rabe-Hesketh and Skrondal 2008;Schmidt-Catran and Fairbrother, 2016) with parties in countries set as random intercepts. 20 Table 2 below reports the results of the regression analysis. 21 The table consists of four models. The first is the null model, where only party in country random intercepts are reported. Then, Model 1 only includes our two main independent variables  20 We have also performed a Variance Inflation Factor test that reports no problems of multicollinearity. Notice that, for the sake of robustness, we have also replicated our main regression analysis through different model specifications and alternative techniques. See later in the text and Tables A1-A3 in the Supplementary material. 21 For the sake of comparability, we have rescaled all our variables in the models to a 0-1 range. Notice that we have only considered those elections in which data on intraparty fractionalization were available for parties representing more than 50% of the electorate. This decision has been driven by the necessity to test the effect of systemic-level factors on party-level observations representing a sufficiently large portion of the electorate. However, we have rerun Model 3 presented in Table 2 by including all available observations (N = 681). The results confirm the findings shown in Table 2. Notice that given that Personal vote is almost time-invariant, the inclusion of fixed effects is not recommendable (Beck and Katz, 2004;2007;Plümper, et al. 2005; see also Mainwaring et al. 2017). However, we have run a leave-oneout test by excluding one party at a time. Our findings prove robust to individual unit exclusion. Finally, we have accounted for the fact that one of the problems of multilevel models is that bottom-level coefficients include both within-and between-panel variation. Following, Enders and Tofighi (2007), we have replicated the models presented in Table 2 by mean-centering all the related variables and the results do not significantly change.
(Disproportionality and Personal Vote), while Model 2 adds all the control variables, and Model 3 also adds the trend variable Time. The results of the regression analysis confirm both the hypotheses formulated in the previous section. Indeed, all else equal, both disproportionality and personal vote are, as expected, substantive predictors of the visible intraparty fractionalization in Western Europe. First, the effect of Gallagher's index is positive and significant (p < 0.001). This means that a less permissive electoral system gives fractions more incentives to structure themselves and organize their 'voice' within the party, given the opposite strategy of 'exit' would surely be riskier, as their chances of getting representation would be lower. This is not merely in line with our expectations but also in line with the literature on party system fractionalization, which has emphasized the constraining role of the electoral system in limiting the number of parties (Duverger, 1954;Lijphart,  Multilevel mixed-effects model; b coefficient and standard errors (se) reported; + p < .10; *p < .05, **p < .01, ***p < .001 1994; Sartori, 1994;Cox, 1997). 22 So, we can consider Hp1 as confirmed by the empirical analysis. Second, the incentives for cultivating personal vote are a positive and statistically significant (p < 0.01) predictor of VIF as well. As expected, the larger the role allowed to individual candidates and intra-party choices in the electoral system, the higher the intraparty fractionalization. The presence of a more personalized electoral system provides higher incentives for 'people' to make the difference at the expense of 'structures', in line with Karvonen's definition of the personalization of politics (2010: 4). Thus, all else equal, it might be easier for fraction leaders to organize their army and possibly be more legitimized thanks to the possibilities offered by the electoral system. All in all, Hp2 finds empirical confirmation.
Further notice that, by including the control variables, the goodness-of-fit of the models increases, as shown by the values of the Wald Chi-squared statistics, while our variables of interest do not significantly reduce their effect on the explanandum.
For a better visualization and comparability of the results, we report below the coefficient plot of the regression analysis of Model 3 of Table 2.
All in all, the analysis of the effects of Disproportionality and Personal Vote on VIF shows that not only has the electoral system an effect on the 'visible' part of the party system (i.e., the number of parties) but also on its 'invisible part' (Sartori, 1976). The b coefficients in the regression table tell us that, as Disproportionality and Personal Vote move from 0 to 1, the visible number of fractions increases by, respectively, 0.52 and 0.38. Moreover, as shown in Fig. 3 above, our two variables of interest appear as the strongest predictors of intraparty fractionalization, even including party-level characteristics on which the literature has hitherto focused.
From the combination of the two predictors, we can draw a synthetic piece of information about the constraint exerted by the electoral system on intraparty fractionalization. Indeed, in a permissive and lowly personalized electoral system, namely when both independent variables are below their mean, VIF is very limited compared to the opposite situation, where both Disproportionality and Personal Vote are above their mean (average VIF equals 1.2 against 1.8, respectively).
For the sake of robustness and given the sensitivity of hierarchical models to different specifications, we have replicated the results in Table 2 across different model specifications and estimating techniques. First, we have run Model 3 in Table 2 by adding the Candidacy variable. As shown in the descriptive statistics in Table 1 above, there is a substantial number of missing cases compared to the other variables (667 vs. 594 observations), and this is why we have excluded Candidacy from the main regression. However, we report the model including such variable in Model A in Table A2 in the Supplementary material. Moreover, to control whether the results are sensitive to the kind of leadership race, we have rerun Model 3 by excluding the races held with the most open selectorate, that is, party members or voters (Model B in Table A2). Furthermore, we have replicated Model 3 in Table 2 by replacing the linear trend variable Time with a categorical variable for decades (Model C in Table A2). Then, we have replicated Model 3 by setting only countries as random intercepts (Model A in Table A3), while, to control for the time-series nature of the data structure and potential issues of serial correlation, we have also run: 1) a Prais-Winsten regression with panel-corrected standard errors (PCSE) and a first-order autoregressive parameter (AR1) (Beck and Katz 1995;and, 2) a Generalized Estimating Equations (GEE) regression with AR1 (Liang and Zeger 1986). The two models are reported in Table A3 (Models B and C) in the Supplementary material. All our substantive results hold across such various models, thus confirming the robustness of our findings.
By turning back to our main regression model, surprisingly enough, the variables related to the party level do not show relevant effects of the VIF. Intraparty fractionalization seems somewhat independent of party size, 23 party age, the left-right placement, and even the organizational density of the party. 24 The only exception is the degree of openness of the selectorate, which appear to exert a nonlinear effect on our dependent variable. Indeed, compared to the most diffused case of leader selection by the party congress (our reference category), intraparty fractionalization increases when the selectorate is limited to a small group of people but also when several thousands or even millions of people select the party leader. 25 Finally, we find a negative and significant effect of the effective number of parties. As expected, given the parallelism we made above between the effects of electoral system features on intraparty and interparty fractionalization, we find that the two aspects of the 'manifest' (ENEP) and 'hidden' (VIF) fractionalization are inversely related. 26 This is consistent with the previous literature about the effects of electoral systems on party system fractionalization (Duverger 1954;Lijphart 1994;Sartori 1994;Cox 1997). As a result, in contexts of higher ENEP, we find lower VIF and vice versa. In other words, fractionalization tends to stay either inside or outside parties, depending on electoral systems' incentives. Conversely, it is less likely to find parties that are internally fractionalized and surrounded by a high number of 23 This means that larger parties do not necessarily have more intraparty fractions than smaller parties, all else equal. Moreover, we have also run an additional model with an interaction between Disproportionality and Party size to check whether incentives deriving from the permissiveness of the electoral system vary according to the size of the party. The result of this test shows a positive and significant coefficient of the interaction, meaning that the effect of Disproportionality is higher for larger parties. One possible explanation is that, as larger parties are advantaged from higher disproportionality, intraparty fractions within larger parties have more incentives to 'voice' inside the party rather than 'exit' compared to intraparty fractions in smaller parties. 24 Notice that the effect of our main predictors is robust by replicating Model 3 with Membership/Electorate replaced by the absolute number of party members. 25 For further discussion about the relationship between selectorate and leadership races, see Kenig (2009a) and Kenig et al. (2015). 26 For a similar view, see Belloni and Beller (1976: 547). Notice that if we exclude the control for ENEP from the regression analysis, the results of our predictors are almost identical to those presented in Model 3 in Table 2, and both hypotheses find empirical confirmation. external competitors or to find parties that act as unitary actors in a context of limited party system fractionalization.

Conclusion
The study of parties' internal divisions is extremely important for party politics from several viewpoints, even if it has been partly neglected until recent years. Party factions have been hitherto studied via case-study or small-N analyses including only a limited number of parties or countries, and with an intra-party level approach. This has brought about two side effects. On the one hand, the absence of a truly comparative perspective and large-N diachronic and synchronic analyses. On the other hand, the emphasis put on intraparty matters has meant overlooking the potential influence of contextual factors.
More specifically, one of the most critical research questions overlooked by the literature is the role exerted by institutional constraints on intraparty fractionalization in Western Europe. This matter is highly relevant in that, as initially recognized by Giovanni Sartori, intraparty groups constitute a sort of invisible element of a party system and, therefore, we might expect that, precisely like its visible counterpart (i.e., political parties), such invisible element is influenced as well by institutional factors among which, in particular, the characteristics of the electoral system.
In this article, we have addressed this matter by proposing three theoretical and empirical innovations. First, by reassessing the extant research on intraparty divisions, we have shown the problems of conceptual stretching and empirical detection in the literature about factions. We have proposed to resort to the Sartorian fraction  Table 2) as a more generic term that acknowledges and encompasses the fundamental variability of intraparty groups in their level of structure, attitude, stability, and so forth. Accordingly, and in line with the few existing empirical studies on party factions, we have operationalized the level of visible intraparty fractionalization by looking at party leadership races, and we have devised a measure to allow for large-N crosscountry and cross-time comparisons.
Second, we have built an original dataset including 70 parties in 11 Western European countries since 1965 by accounting for about 700 party cases, and data show that Western European parties have become increasingly fractionalized, particularly in the last decade.
Third, we have put forward an empirical analysis to detect the effect played by institutional constraints on intraparty fractionalization. Empirical evidence has shown the effect of electoral systems' characteristics, namely, their level of disproportionality and the incentives for personal vote. Our finding is that both factors exert a positive and significant effect on intraparty fractionalization.
On the one hand, as far as the disproportionality of the electoral system increases, intraparty fractionalization increases as well. This is consistent with our expectations and, more importantly, clearly tells us that the permissiveness of the electoral system is not only an important element in explaining the number of parties-as the literature about party system fractionalization has widely shown-but also a substantive predictor of the fractionalization detectable inside them.
On the other hand, the more an electoral system is personalized, the higher the intraparty fractionalization. This is because, in closed-list systems, voters are only allowed to mark the party symbol. This brings about two consequences. First, the party presents itself as a unitary actor both during the electoral campaign and in the ballot box, given that it is not possible for intra-party groups to weigh their strength, not even indirectly via the parliamentary election. Having this in mind, politicians opposing the national party leadership will have few incentives to openly challenge it by fostering intra-party divisions, given that their chances of candidacy or re-candidacy in future elections will largely depend upon the decision of the national party leadership that has to put their name on the party's electoral list. Second, and conversely, when preferential vote is allowed, or candidates' names are directly put on the electoral ballot, there is higher visibility granted to individual candidates and leaders vis-à-vis their party, and, consequently, intra-party fractions have greater potential power and opportunities to emerge.
In a nutshell, the results of the empirical analysis tell us that to have a more comprehensive assessment of intraparty fractionalization, we should account-exactly as in the case of party system fractionalization-for institutional determinants. Indeed, according to Sartori's definition of a party system as 'the system of interactions resulting from inter-party competition ' (1976 [2005]: 39), our analysis seems to suggest that such interactions are not just those taking place amongst the visible actors of the system (namely, political parties), but shall also include the patterns of interactions among intraparty groups and between them and parties.
Summing up, this article can open up a new strand of research on the role of intraparty groups and their interactions with the surrounding institutional and political environment, especially in times of increasing party organizational destructuring. Moreover, our results may also bring about implications for policy-makers. As the latter usually pass electoral reforms with the underlying rationale of producing a direct effect on the inter-party competition (e.g., a more disproportional system is usually introduced with the explicit goal of reducing party system fragmentation), they have to be aware that such reforms also have a relevant impact on the number of intra-party groups (e.g., a more disproportional system increases the number of intra-party fractions). Therefore, as intra-party divisions are becoming increasingly important from several viewpoints and parties are less and less conceivable as unitary actors, electoral reforms may have unintended consequences that policy-makers shall carefully assess.