What do we know about cabinet proportionality and deviations from it?

Any discussion of proportionality invariably begins with William Gamson’s early theory of coalition formation. ^{Footnote 2} According to Gamson (Reference Gamson1961: 376), ‘any participant will expect others to demand from a coalition a share of the payoff proportional to the amount of resources which they contribute to a coalition.’ Starting as an intuitive proposition, this statement was gradually recast as a ‘law’ in the wake of Browne and Franklin’s (Reference Browne and Franklin1973) classic observational study of European portfolio allocation in the 1950s and 1960s. The law-like reappraisal of Gamson turned entirely on Browne and Franklin’s operationalization of ‘resources’ in the quote above, which they measured as the share of legislative seats held (‘contributed’) by each party in the coalition. Within coalition governments in parliamentary systems, they found the correlation between portfolio shares and seat shares to be nearly perfect, a finding that (with various disclaimers attached) has been confirmed empirically time and time again, mainly in the Western European context (e.g., Schofield and Laver, Reference Schofield and Laver1985; Warwick and Druckman, Reference Warwick and Druckman2001). While this relationship may not be a ‘law’ in the strict sense of the term (Indridason, Reference Indridason2015), it has nonetheless been described as ‘perhaps the strongest empirical finding in political science’ (Warwick and Druckman Reference Warwick and Druckman2006: 635).

It is beyond the scope of this paper to review the vast secondary literature on proportionality in portfolio allocation. However, we draw attention to four core elements of this body of research. First, as noted above, Gamson’s Law has strong empirical support in parliamentary democracies, on which virtually all extant scholarship is based. Second, in parliamentary coalition governments, the observed deviations from proportionality have mostly come in the form of the ‘small party bias’, meaning that very small coalition partners tend to do slightly better than one would expect from a linear interpretation of Gamson’s Law (Browne and Franklin, Reference Browne and Franklin1973; Warwick and Druckman, Reference Warwick and Druckman2006). Some have referred to this small-party asterisk on proportionality as ‘modified Gamson’s Law’ (e.g., Cox, Reference Cox2021). Third, in parliamentarism there has been little or no empirical support for the idea of formateur advantage—which makes some sense given what we know about the minor overrepresentation of small parties (this practice already suggests some implicit forbearance by the formateur party).

The fourth and most intriguing aspect of this research program concerns the long-running attempt to uncover solid theoretical foundations for Gamson’s Law (Falcó-Gimeno and Indridason, Reference Falcó-Gimeno and Indridason2013). In the parliamentary literature, formal models of legislative bargaining have not fared well. So-called ‘proposer’ models, or alternating-offer games (e.g., Baron and Ferejohn, Reference Baron and Ferejohn1989; Ansolabehere et al., Reference Ansolabehere, Snyder, Strauss and Ting2005), suggest that the formateur should be advantaged by their take-it-or-leave-it control over the negotiation process. Empirical adaptations of this idea eschew the traditional focus on seat shares in the legislature (the Browne and Franklin interpretation of Gamsonian resources) in favor of ‘bargaining weights’ (the extent to which a given party is pivotal to a winning coalition). Yet using either or both operationalizations, empirical evidence of formateur advantage has generally been found lacking in parliamentarism, even in more recent work that attempts to correct for portfolio salience (Warwick and Druckman Reference Warwick and Druckman2001, Reference Warwick and Druckman2006; Bucur, Reference Bucur2018). Warwick and Druckman (Reference Warwick and Druckman2006) note this disconnect between empirical and theoretical work and name it the ‘portfolio allocation paradox’—in other words, proportionality shouldn’t exist but does.

In the nearly two decades since Warwick and Druckman first posed this paradox, a number of scholars have attempted to resolve it. Several have considered the possibility that Gamson’s Law is nothing more than a social norm of fairness. ^{Footnote 3} In a paired comparison of Sweden and Germany, Bäck et al. (Reference Bäck, Meier and Persson2009) find little support for a social norm in the strict sense of the term, but do find that proportionality ‘serves as a bargaining convention for the actors involved, thus rendering proportional payoffs more likely’. In contrast, Cutler et al. (Reference Cutler, de Marchi, Gallop, Hollenbach, Laver and Orlowski2016) argue that there is indeed a proportionality norm in coalition formation, but that it is observed mainly as a conflict-defusing device after the difficult work of policy bargaining has already taken place. Ecker and Meyer (Reference Ecker and Meyer2019) take the policy payoff idea one step further. Matching the content of party manifestos to observed portfolio allocations, they argue that policy fairness is more likely to be achieved when parties already consent to apportionment of cabinet positions that are proportional to party size—in other words, to the terms of Gamson’s Law. Falcó-Gimeno and Indridason (Reference Falcó-Gimeno and Indridason2013) make a different but compatible claim about proportionality: that it is almost always the fastest and cleanest solution to bargaining situations characterized by high levels of complexity and uncertainty. More recently, Martin and Vanberg (Reference Martin and Vanberg2020) have focused on the optics of coalitional negotiations, developing a theory that they call ‘bargaining before an audience’. If parties are worried that their supporters may see them as ineffective negotiators, they may choose to accept proportionality of cabinet posts as a safely defensible outcome—while at the same time pushing hard for concessions on less easily observable issues.

All focusing on parliamentary regimes, these more recent contributions depict Gamson’s Law as something of a heuristic and/or a ‘default’ outcome in the composition of coalition governments: when the starting point is some notion of proportionality, then it is not surprising that the endpoint is something resembling proportionality. If a law is a default scenario and a default scenario is a law, then this circularity may bring us no closer to providing the desired theoretical microfoundations for Gamsonian proportionality.

Three logics of coalition formation: comparing constitutional formats

Scholarship on Gamson’s Law often assumes that there is a single institutional ecosystem for cabinet composition, a bias that may be forgiven given the postwar Western European origins of this literature. At the beginning of the Third Wave of democratization in the late 1970s, well over half of the world’s political democracies were parliamentary systems. However, by 2010 nearly two-thirds of existing democracies had directly elected presidents of some kind (Samuels and Shugart Reference Samuels and Shugart2010: 5). Instauration of new parliamentary democracies with ceremonial or indirectly elected heads of state is now exceedingly rare, while semi-presidential and presidential formats are on the rise across the globe. Theories of cabinet composition need to be updated to reflect this reality. Revising theory requires identification of the relevant actors (in this case, formateurs, political parties, and legislative assemblies) and recognizing how their motivations vary across three prominent constitutional types.

In discussing executive formats, we start with noncontroversial assumptions: formateurs should want to exaggerate their natural first-mover advantage, and parties should aim for the best deal they can get. Parties will want to maximize the number of ministers at the cabinet table and will oppose favoritism to other parties. The degree to which parties can influence the formateur will depend, at least in part, on broad constitutional characteristics pertaining to government formation, assembly confidence, and executive discretion.

Parliamentarism is familiar and its principles are depicted in the vast literature cited above (see also Lijphart, Reference Lijphart2012). As the name suggests, parliamentarism rests on the centrality of the elected assembly, of which both formateurs and ministers are working members. The head of state normally invites the leader of the largest party to attempt to form a government. In non-majoritarian configurations of seats (the modal outcome in the parliamentary world), automatic success in government formation is not guaranteed, and therefore the number of potential formateurs is greater than one. Governments generally depend on the support of a majority of the legislature in order to take office and remain in office. Minority governments can also survive, but normally with commitments from key opposition parties not to bring down the government. In both majority and minority scenarios, prime ministers are always subject to confidence motions. When coalition government becomes necessary under this constitutional format, political parties naturally find their bargaining power enhanced. Given that they are central to government formation and government termination, parties must be cultivated carefully. Each party will assess the essential fairness of any power-sharing arrangements, observing who gets what (the original insight of Gamson). Thus it is perhaps not surprising that when formateurs propose a coalition, they tend to treat their allies proportionally when allocating ministerial portfolios.

Presidential democracy works very differently (Linz, Reference Linz, Juan and Valenzuela1994). Elections for the head of government (president) and for the legislature are independent of each other, meaning that the executive and legislature have ‘separate origin and separate survival’ (Shugart and Carey, Reference Shugart and Carey1992). In contrast to parliamentary systems, the number of potential formateurs is consistently one, since only the directly elected president can exercise this role. The president does not require political support in the legislative assembly in order to take office, and once inaugurated enjoys a fixed term. In presidential systems, the absence of a vote of no confidence means that the head of government (president) is far less dependent on the support of coalition partners than is the equivalent figure (prime minister) in parliamentary systems (Silva, Reference Silva2023: 938). The incentive for the president to be ‘fair’ in portfolio allocation is consequently diminished. Moreover, presidential systems generally lack a requirement that ministers be members of the legislature. This means that nonpartisan or ‘technocratic’ ministers are quite common, and such individuals may often be awarded portfolios of high prestige and/or of great personal interest to the president (Altman, Reference Altman2000; Chaisty et al., Reference Chaisty, Cheeseman and Power2018). Finally, the president is by no means the ‘first among equals’ that is supposedly the case for prime ministers operating under collective responsibility. Rather, the president is a unipersonal executive who is directly elected in a nationwide constituency, and who individually composes and directs the cabinet, leading to an elevated level of discretion in portfolio allocation.

Semi-presidentialism is a mixed constitutional type that blends characteristics of both presidentialism and parliamentarism (Duverger, Reference Duverger1980; Schleiter and Morgan-Jones, Reference Schleiter and Morgan-Jones2009; Elgie, Reference Elgie2011). The requirement of parliamentary confidence means that the legislature is the relevant locus of coalition formation, thus enhancing the centrality of political parties. The prime minister is always important, but there is also a directly elected president with considerable constitutional powers, and whose popular legitimacy is consistent with some role in policymaking. The identity of the de facto formateur may vary. When coalitions are necessary and the president dominates the party proposing an interparty alliance for government formation, the president is likely to be the leading architect of the future government, thus elevating presidential discretion over portfolio allocation. This scenario of political alignment is closer to pure presidentialism, with its higher expectation of a ‘formateur bonus’. However, whenever the results of legislative elections require that the president appoint a prime minister from a different or even an opposition party (‘cohabitation’), the role of the president in cabinet composition is weak or nonexistent.Footnote ⁴ This scenario is much closer to parliamentarism, maximizing the negotiating potential of political parties and moving the system closer to a Gamsonian world. Thus, the relevance of Gamson’s Law to semi-presidentialism depends on who is the effective formateur, i.e., whether the system is ‘leaning presidential’ or ‘leaning parliamentary’. Empirically, the former scenario is far more common in the real world of semi-presidentialism.Footnote ⁵

Unpacking these three constitutional logics leads us to follow previous studies (e.g., Amorim Neto and Samuels, Reference Amorim Neto and Samuels2010) and propose a rather straightforward hypothesis about the binary comparison between presidentialism and parliamentarism:

We further expect that semi-presidential systems will stand in an intermediate position between pure presidentialism and pure parliamentarism. We note that some prior studies have assumed that the only institutional variable that really matters in shaping cabinet proportionality is the vote of no confidence, and therefore code parliamentary and semi-presidential systems (which share this feature) as belonging to a single category. ^{Footnote 6} However, to maintain a wide theoretical lens (and also to leverage the considerable cross-system variation in a large-N dataset), we continue to distinguish among our three constitutional formats in the analyses reported below, and hence we hypothesize:

Beyond constitutions: party-system and assembly-level predictors of proportionality

Key differences among presidentialism, semi-presidentialism, and parliamentarism are undoubtedly central to explaining variation in cabinet proportionality. Yet these predictors of proportionality are based entirely on intrinsic features of broad constitutional formats. Recently, a new wave of scholarship has begun to drill down below the level of the constitution to identify within-format factors that shape cabinet proportionality. For example, scholars of semi-presidentialism have identified a wide range of second-order institutional prerogatives afforded to the head of state (Kang, Reference Kang2009; Savage, Reference Savage2018; Bucur, Reference Bucur2020). These institutional levers affect the degree to which the formateur will need the cooperation of parties in parliament to pass legislation, and consequently, the degree to which the formateur will feel compelled to be fair to these parties. In parallel, scholars of presidentialism such as Silva (Reference Silva2023) have shown that cabinet proportionality is inversely related to the degree of legislative power held by presidents, which can vary immensely across presidential constitutions (Doyle and Elgie Reference Doyle and Elgie2016). These newer studies focus less on broad differences between constitutional types, and more on the ‘interior design’ of specific institutional settings.

Inspired by this newer work that looks ‘under the bonnet’, we also want to explore party-system and legislature-level predictors of cabinet proportionality, i.e., factors that vary not only across but also within constitutional formats. These factors, if correctly identified, should exert broadly similar effects on formateurs regardless of the constitutional rules under which they live. To take a simple example, party discipline exhibits cross-national variation that is independent of constitutional type. Formateurs in any system, when presented with parties with tight and predictable discipline on the floor, would see greater value in cultivating them by recruiting their representatives to the cabinet.

On the significance of party-system variables, we are influenced by Kellam’s (Reference Kellam2015: 515) recent work on presidentialism, in which she shows that particularistic parties are ‘relatively cheap coalition partners for presidents in policy terms’ and their representation in parliament makes it easier for formateurs to change the composition of their legislative coalition throughout their term of office. Conversely, programmatic parties will be motivated to capture the levers of policy that are afforded by ministerial portfolios.

Adapting Kellam’s logic, we can expect that more cohesive and programmatic party systems will force the formateur to ‘behave better’, i.e., to be more ‘fair’ in assigning cabinet posts to coalition parties on a proportional basis. We also hypothesize that formateurs will be more likely to treat parties fairly if they believe these parties can command significant support in civil society, for example by demonstrating capacity for social mobilization. Here we are inspired by the recent work of Martínez (Reference Martínez2021) on the relationship between party institutionalization and presidential interruptions in Latin America. ^{Footnote 7} Bringing together three dimensions of party system development, we therefore hypothesize that:

Variation in the distribution of power between executives and assemblies is also found to have an impact on cabinet proportionality in recent research. Both Bucur (Reference Bucur2020) on semi-presidentialism and Silva (Reference Silva2023) on presidentialism have highlighted the role of executive dominance over the legislature in shaping the applicability of Gamson’s Law. The less presidents need the support of parties to pass legislation, the less they will practice proportionality in portfolio allocation. While highly creative and with strong empirical results, both analyses are largely restricted to a single constitutional type or world region ^{Footnote 8} and are difficult to test comparatively given limitations in existing measures of executive power both longitudinally and across different executive formats.

Nonetheless, we can generalize this idea with existing measures of legislative power. Assemblies with greater capacity to monitor and sanction the behavior of the executive would be expected to raise the probability that formateurs will practice proportionality in the composition of cabinets. In line with existing research (e.g., Martínez-Gallardo, Reference Martínez-Gallardo2012; Camerlo and Pérez-Liñán, Reference Camerlo and Pérez-Liñán2015), we would expect formateurs to be more likely to distribute cabinet seats proportionally if the advice and consent of the assembly are required for the investiture of ministers. ^{Footnote 9} Also, the more the power base of the chief executive is determined by the confidence of the legislature, the more we should expect formateurs to compose their cabinets on a fair and proportional basis. Hence, taking together three aspects of legislative influence over politics, we hypothesize that:

We therefore have identified at least six different party-system and assembly-level variables that might affect the portability of Gamson’s Law across broad constitutional formats. These include party-system characteristics such as party programmaticity, party cohesion on the floor, and the institutionalization of parties in society. They also include variables shaping the executive-legislative relationship such as the power of the assembly to oversee and constrain the executive; the assembly’s power to confirm ministerial appointments by the head of state; and a measure for the extent to which the political standing of the executive rests on legislative support. Party-system and assembly-level predictors of cabinet proportionality can only be captured in a large-N design that covers all three main constitutional types.

Data and results

To estimate the extent to which different executive formats affect cabinet proportionality and why, we analyze cabinet composition data compiled by WhoGov (Nyrup and Bramwell, Reference Nyrup and Bramwell2020). The WhoGov dataset contains over 8,000 country-years of information on cabinet members across 177 countries for the period 1966 to 2019. We augment this dataset with legislative seat shares of cabinet-represented parties. ^{Footnote 10}

We confine our analysis to competitive polities, which we define broadly as political systems with a positive score (1–10) on the Polity IV scale. ^{Footnote 11} Given our research question, we also limit the analysis to governments formed by cross-party coalitions. The exclusion of single-party cabinets removes approximately one-third of the cases of competitive polities from the WhoGov dataset. Furthermore, we dropped cases of coalitions that had incomplete cabinet party data. This amounts to approximately 400 cases. We include these data for robustness checks only. We are left with a final sample of 1551 country-years of competitive politics in 97 countries in the period from 1966 to 2019. Table 1 lists the countries covered by the analysis.

Table 1 Sample summary of cases by region and country, 1966–2019

Table 2 classifies our cases by constitutional format. Parliamentary country-years comprise the largest percentage of this sample: 45% of all observations, compared to 31% in the semi-presidential category and 24% in pure presidentialism. Consistent with theoretical expectations about prime ministers in parliamentary systems, these data show that parliamentary coalitions tend to include more parties than is the case in either semi-presidential or presidential regimes—suggesting greater limitation of the formateur’s discretion in parliamentary systems. While the relative share of two-party coalitions is similar across all constitutional formats, coalitions of three or more parties are preponderant in parliamentary systems.

Table 2 Frequencies of party coalitions across different executive formats (percentages in parentheses)

These data form the focus of our analysis, which proceeds in two stages. First, we estimate the extent to which cabinet proportionality is achieved across the broad constitutional formats. This entails the analysis of a Gamsonian measure of fairness ranging from zero to one: it measures the extent to which the formateur distributes cabinet portfolios among coalition parties proportionally to their seat shares within the coalition’s legislative contingent. ^{Footnote 12} We also estimate whether the formateur is compelled to share prestigious cabinet posts by calculating the degree to which the formateur’s party or nonpartisans are underrepresented relative to the share of seats held by the formateur’s party within the legislature. Data for these highly salient posts are taken from WhoGov, which uses Krook and O’Brien’s (Reference Krook and O’Brien2012) typology of four policy areas assumed to enjoy universal prestige. The names of the ministerial portfolios naturally vary across countries, but the four easily recognizable categories are: (1) Defence/Military/National Security; (2) Foreign Relations; (3) Interior/Home Affairs; and (4) Finance/Budget/Treasury. ^{Footnote 13}

In the second stage of our study, we seek to account for the observed variation within each executive format using our party-system and assembly-level predictors. This involves analysis of party-level and institutional (legislature-specific) factors that feature in system-level studies of cabinet composition. To estimate their effects on cabinet proportionality, we consider a range of V-Dem variables. They include party-system covariates that measure the magnitude of programmatic partisan cohesion, ^{Footnote 14} linkages with voters, ^{Footnote 15} and the institutionalization of parties in terms of the wider society. ^{Footnote 16} They also include legislative factors that affect executive power: the power of the assembly to oversee and constrain the executive; ^{Footnote 17} the assembly’s power to confirm ministerial appointments by the head of state; ^{Footnote 18} and a measure for the extent to which the political authority of the executive rests on the support of the legislature. ^{Footnote 19}

Cross-format results

We begin with a simple visual inspection of the data, comparing observed cabinet allocations in presidentialism, parliamentarism, and semi-presidentialism to the relationship posited by Gamson’s Law. Gamsonian proportionality is represented by the diagonal line in each of the three panels in Fig. 1, which shows the relationship between the share of cabinet posts held by the formateur’s party and the share of coalition seats controlled by this party across our three broad constitutional formats.Footnote ²⁰ The panels provide strong visual support for HYPOTHESIS 1 (cabinet proportionality should be lower in presidential systems than in parliamentary systems) and HYPOTHESIS 2 (proportionality in semi-presidential systems should be lower than in parliamentary systems but higher than in presidential systems). As can be seen in panel (a) of Fig. 1, country-year observations in parliamentary systems behave largely according to expectation and are arrayed close to the Gamsonian line, with a linear association of 0.742; semi-presidential observations (panel b) exhibit greater dispersion with a correlation between seat shares and cabinet shares of 0.618; meanwhile, pure presidential observations (panel c) are more scattered with a coefficient of 0.598. ^{Footnote 21} Globally, the linear association for parliamentary systems (0.742) is much lower than one finds in the large body of research on postwar European parliamentary systems.

Figure 1. Formateur portfolio shares in three constitutional formats vs Gamsonian line.

Notes: Total N = 1551. Correlation for parliamentary systems only: r = 0.742, N = 697. Correlation for semi-presidential systems only: r = 0.618, N = 478. Correlation for presidential systems only: r = 0.598, N = 376. All associations significant at p < 0.01. Diagonal line represents perfect Gamsonian proportionality (r = 1.00).

The exceptional levels of disproportionality in presidential systems are also illustrated by the greater proportion of country-year cases above the Gamsonian line. Observations above the line indicate the relative size of the formateur bonus: here, the proportion of cabinet posts awarded to the formateur party exceeds its contribution to coalition seats on the floor. The results provide clear support for HYPOTHESIS 1 and HYPOTHESIS 2. In panel (a), observations above the line are 35.7% for parliamentarism, in panel (b) 43.7% for semi-presidentialism, and an astounding 82.4% for pure presidentialism (panel c). Presidential formateurs are thus more than twice as likely as prime ministers to exaggerate their party’s share of the cabinet.

As can be seen in Table 3, our inclusion of parliamentary systems in transitional and developing democracies outside of Europe results in much weaker correlations, which are nonsignificant in some regional clusters. Moreover, the result for parliamentary systems in Europe is lower than typically assumed (it is 0.90 or higher in many post-Browne and Franklin studies), and this may also be a consequence of the eastward expansion of political democracy since the 1990s.

Table 3 Pearson correlations of the formateur’s share of cabinet posts and coalition seats, parliamentary systems only, by world region

** Correlation is significant at the 0.001 level (two-tailed).

Nonetheless, despite this complexity, basic constitutional differences are still key to understanding the extent of cabinet proportionality. Figure 2 summarizes the average size of cabinet proportionality across all executive formats. In contrast with Fig. 1, which reports data for formateur parties only, these data summarize mean cabinet proportionality for all parties in the coalition.Footnote ²² In line with HYPOTHESIS 1, these results once again depict the strong formateur advantage under the separation of powers. Whereas formateurs in parliamentary and semi-presidential systems are more likely to assign cabinet positions in proportion to the legislative support commanded by their coalition partners, this declines significantly in pure presidentialism. The pairwise comparison of means between presidential cases and the other executive formats is much stronger than between parliamentary and semi-presidential systems (see note in Fig. 2).

Figure 2. Mean cabinet proportionality estimates by executive format.

Note: Pairwise comparisons of means with equal variances: semi-presidential vs parliamentary (contrast −0.062; st. error 0.010; p = 0.000), presidential vs parliamentary (contrast −0.226; st. error 0.011; p = 0.000); presidential vs semi-presidential (contrast −0.164; st. error; p = 0.000).

Similarly, co-partisans of the formateur in presidential systems are much more likely to be overrepresented in prestigious cabinet posts relative to their presence in the assembly (Fig. 3). Once again, the most significant pairwise comparisons are between presidentialism and the other two constitutional types.

Figure 3. Mean representation of formateur contingents in the most prestigious cabinet posts, by executive format.

Notes: These data summarize mean differences between the share of cabinet and legislative seats held by the formateur in each constitutional format. Positive and negative mean values indicate overrepresentation and underrepresentation of the formateur in prestigious cabinet posts. Pairwise comparisons of means with equal variances: semi-presidential vs parliamentary (contrast 0.020; st. error 0.016; p = 0.426), presidential vs parliamentary (contrast 0.147; st. error 0.017; p = 0.000); presidential vs semi-presidential (contrast 0.127; st. error 0.018; p = 0.000).

Why are the constraints on the formateur bonus significantly weaker in presidential systems? Is this only because of intrinsic constitutional features or can it result from other aspects of institutional design? We now proceed to the within-format analysis of the party-system and legislative factors that are theorized to account for variance in Gamson’s Law in constitution-specific studies. What is the relative significance of party-system characteristics and legislative influence?

Within-format results

In this analysis, we first estimate the effects of party and assembly-level predictors in simple bivariate models, and then include the most salient predictor for within-system variation—party-system institutionalization—in multivariate analysis, while controlling for world region and level of political democracy. The panel data used in this analysis present some common challenges. Our dataset is unbalanced due to attrition and missing data and is affected by the problem of autocorrelation, i.e., repeated observations for the same countries that are likely to be positively correlated and thus not independent of each other. To correct these problems, we estimate each model using Generalised Estimating Equations (Ghisletta and Spini, Reference Ghisletta and Spini2004). This statistical method is designed to correct unbalanced panel data and autocorrelation, producing more efficient estimates. We also include robust standard errors to mitigate sensitivity to the correlation structure.

Key results of the analysis are presented visually in Fig. 4. They summarize the average marginal effect of each party-system and assembly-level predictor on our measurements of cabinet proportionality for each broad constitutional format. ^{Footnote 23} Values for variables to the left of the vertical line indicate depressing effects on cabinet fairness, while values to the right are indicative of effects increasing cabinet proportionality. Confidence intervals represented by horizontal lines either side of the point estimate indicate nonsignificance when they overlap the vertical line.

Figure 4. Party-system and assembly-level variables shaping cabinet proportionality (with 95% confidence intervals).

Source: drawn from V-Dem (Coppedge et al., Reference Coppedge, Gerring, Knutsen, Lindberg, Teorell, Altman, Bernhard, Cornell, Fish, Gastaldi, Gjerløw, Glynn, Grahn, Hicken, Kinzelbach, Marquardt, McMann, Mechkova, Paxton, Pemstein, von Römer, Seim, Sigman, Skaaning, Staton, Tzelgov, Uberti, Wang, Wig and Ziblatt2022) and authors’ dataset building on Nyrup and Bramwell (Reference Nyrup and Bramwell2020); see text for details.

In general, the results for parliamentary and semi-presidential systems support all variants of HYPOTHESIS 3, namely, that cabinet proportionality is likely to increase in line with the floor-level cohesion (HYPOTHESIS 3a), tendency toward programmaticity (HYPOTHESIS 3b) and overall institutionalization (HYPOTHESIS 3c) of party systems. They show that party-system institutionalization in particular is far more significant in explaining proportionality in parliamentary and semi-presidential systems than in presidential polities.

In Fig. 4, positive values for the marginal effects of party predictors on cabinet proportionality in parliamentary and semi-presidential systems indicate the extent to which they contribute towards Gamsonian ‘fairness.’ This is especially evident for the party-system institutionalization variable. A one-unit increase in party-system institutionalization increases cabinet representativeness by 35 percentage points in parliamentary systems, and by 39 percentage points in semi-presidential systems. A one-unit increase in voting cohesion on the assembly floor also raises the representativeness of cabinets by 14 percentage points in semi-presidential systems and by 9 percentage points in parliamentary systems. Similarly, programmatic parties are significantly more likely to improve the representativeness of cabinets, with a one-unit increase in programmaticity increasing cabinet fairness by 6 percentage points in parliamentarism.

However, no version of our HYPOTHESIS 3 finds support among presidential systems: the average marginal effects of our party-system covariates are either small or nonexistent. The negative results for party cohesiveness, programmaticity and institutionalization in Fig. 4 suggest that these predictors might actually decrease the proportionality of cabinets in pure presidentialism, although these results are not statistically significant.

Assembly-level predictors are less consistently significant. Levels of oversight capacity have a significant impact in raising the proportionality of cabinets in semi-presidential systems only. A one-unit increase in oversight increases Gamsonian proportionality by 35 percentage points. And, in parliamentary systems, proportionality increases in line with the power of assemblies to confirm ministerial appointments.

These findings are robust to alternative specifications including incomplete party data. ^{Footnote 24} We also re-estimated our models with a thinned dataset that includes only ‘consolidated’ democracies—i.e., country-years classified by Polity at point 6 and higher—and observe the same general patterns.

Finally, multivariate tests (see Table 4) confirm the positive effect of party-system institutionalization on cabinet proportionality, but in parliamentary systems only. Even when controlling for regional variation and levels of democratic development in parliamentary systems, party institutionalization is strongly significant in differentiating levels of cabinet proportionality. Interestingly, however, the effect of party institutionalization is vacated in semi-presidential systems. This appears to be largely a consequence of the inclusion of regional controls, which capture weaker party-system development in many regions outside of Europe. This result reaffirms the limited geographic portability of Gamson’s Law, with semi-presidential cabinets in Asia and the former Soviet Union significantly less likely to be representative of coalition parties in the legislature than is the case for European semi-presidential systems.

Table 4 Multivariate regressions of cabinet proportionality on party-system institutionalization, by constitutional format (generalised estimating equations with robust standard errors)

Notes:

^a Excluded reference category is Europe; other minor exclusions explained in note (b).

^b Not applicable. Regions are excluded when there are no cases or no variation at the country level (e.g., just one country was classified as parliamentary in the Americas regional category: Trinidad and Tobago).

^c Measured by the V-Dem Electoral Democracy Index. V-Dem asks: ‘To what extent is the ideal of electoral democracy in its fullest sense achieved?’ (Coppedge et al., Reference Coppedge, Gerring, Knutsen, Lindberg, Teorell, Altman, Bernhard, Cornell, Fish, Gastaldi, Gjerløw, Glynn, Grahn, Hicken, Kinzelbach, Marquardt, McMann, Mechkova, Paxton, Pemstein, von Römer, Seim, Sigman, Skaaning, Staton, Tzelgov, Uberti, Wang, Wig and Ziblatt2022: 43).

Significance levels:

* P < 0.05;

** P < 0.01;

*** P < 0.001.