COORDINATION PROBLEMS

Before diving into this much-debated subject, we must define several key terms.

A political body refers to any organizational entity that is granted a formal or informal role in governance. (We assume that organizations playing an important political role also play a representational role, regardless of how members are chosen [Rehfeld Reference Rehfeld2006].) Leaders, aka elites or representatives, are those chosen from a population of citizens to serve on these political bodies. Selectors appoint or elect leaders. (We leave aside questions about how the selectorate is formed.) Diversity, or heterogeneity, refers to the number of social groups and their relative size. The more equal in size, the greater the entropy across groups. Groups are defined by intrinsic traits such as ethnicity, race, caste, religion, language, gender, sexuality, age, disability, occupation, social class, education, or place of origin. Some of these identities are descent-based (Chandra Reference Chandra2006); others are more open-ended, but all are assumed to inform a group’s identity.

Descriptive representation refers to a representational capacity deriving from shared identity and shared life experience. It is who political representatives are that matters. Descriptive representation is achieved when the intrinsic traits of persons chosen for a political body mirror those found in the population they are intended to represent (Mansbridge Reference Mansbridge1999; Pitkin Reference Pitkin1967).

Standing in the way of this achievement is a formidable coordination challenge.Footnote ³ This arises from the dynamics of identity politics. While identities are fluid in many social contexts, in political contexts, they are usually treated as fixed categories. Political leaders are viewed as one of “us” or one of “them” (Goodyear-Grant and Tolley Reference Goodyear-Grant and Tolley2019). Although leaders have a certain amount of wiggle room to frame and reframe their public identity, such subtleties are difficult to communicate and may not be accepted by others. If cues are too subtle, representational bonds may fray. Constituents may not feel represented by someone whose identity is difficult to grasp (Lemi Reference Lemi2021) or whose identity undergoes frequent changes, either of which may call into question vital leadership qualities such as authenticity and credibility.

It follows that the task of descriptive representation centers on selection, for once leaders are selected, there is little room for adjustment. A leader’s ideology can be dialed up or dialed down, framed in various ways, and adjusted over time in response to events, new information, and changes in public opinion. Not so their identity.

At the point of selection, a two-level coordination challenge arises. On one level are social groups demanding representation; on the other, political bodies providing representation. Fitting these pieces together in the right proportion is complicated, even if there is political will to do so.

Imagine a game of musical chairs with millions of citizens and a few dozen chairs. Seats must be apportioned so that traits among the seated reflect traits in the general population. If there are fewer chairs than traits, the coordination problem is clearly unsolvable. Even if chairs out-number traits, it is a challenging task to align traits in their proper proportion. Note that errors may arise in both directions: a trait may be under- or over-represented.

This fiddly task is rendered more difficult in a typical political setting. Here, we are likely to find multiple bodies with different sizes, constituencies, and rules of selection—some appointive, some elective, and among the latter, a variety of complex electoral rules. How should selectors coordinate their efforts? Should each advocate for their own group, or should they support other groups—or only certain groups (e.g., minorities)? Note also that group identities are likely to cross-cut, so there are multiple dimensions of descriptive representation to juggle—ethnicity, gender, religion, and so forth (Collins and Bilge Reference Collins and Bilge2020). Achieving representation along one dimension may involve sacrifices along another. Finally, descriptive traits are not the only criteria of relevance for a top political job. Selectors must also consider ideology, experience, integrity, charisma, resources, and past performance—any of which may take precedence (Fisher et al. Reference Fisher, Fieldhouse, Franklin, Gibson, Cantijoch and Wlezien2018, Part 3).

Coordination challenges faced by descriptive representation are evidently immense. Yet, we argue that much of the action is contained in the core elements of the task: the composition of society and the composition of a political body. As a society becomes more diverse or a political body shrinks, obstacles to representation arise. It’s hard to fit multiple categorical traits into a small container in just the right proportions.

In principle, compositional effects apply to any political body, any set of descriptive traits, and any political context, whether appointive or elective, democratic or autocratic. They are perhaps most familiar in the context of electoral rules, where larger districts are often thought to allow more space for representation (Lijphart Reference Lijphart2004).Footnote ⁴ In this study, we explore a range of political bodies (executives, cabinets, political parties, supreme courts, and legislatures), four main descriptive traits (ethnicity, religion, language, and gender), and a global sample of countries.

Compositional effects are not universal, however. Note that because coordination presents an obstacle to the realization of a given preference set it is necessarily contingent upon that set of preferences. If preferences are weak or a trait is judged to be inappropriate for public office, that trait is unlikely to be represented. For example, if there is little inclination to represent youth, and age is correlated with other valued traits such as experience, the young are unlikely to be represented (Stockemer and Sundström Reference Stockemer and Sundström2018). In situations like this, preferences override compositional factors, a point discussed at length in Supplementary Materials I.

The resulting framework, summarized in Figure 1, suggests that preferences interact with compositional factors to generate a coordination problem. The resolution of this problem determines the level of descriptive representation achieved for a given trait. We will show that successes and failures follow those encountered when sampling randomly from a population.

Figure 1. A Compositional Model of Descriptive Representation

A Random Sampling Model

Although the inputs and outputs of the framework illustrated in Figure 1 are transparent and measurable, mechanisms are opaque. We cannot directly observe coordination. Helpfully, we can approximate it by examining how representation is affected by compositional changes under certain assumptions.

In our model, intrinsic traits are chosen randomly from a population subject to conditions imposed by compositional factors. This setup allows us to hold background features constant while zeroing in on factors of theoretical interest—the size of the political body and the number and size of groups eligible for representation—which can be manipulated independently. In this fashion, simulations function as ersatz experiments that can be repeated infinitely and fine-tuned to test varying doses and interactions among treatments (De Marchi Reference De Marchi2005).

Of course, this depends upon the plausibility of assumptions. Modeling selection as a random draw from a population makes sense insofar as the items under consideration are understood as traits (carried by representatives), not the representatives themselves (who are selected randomly only in the rare case of sortition). The crucial assumption is that the relationship of theoretical interest—between compositional factors and representation—plays out under conditions of random selection much as it does in the real world. If the apparatus is realistic in this respect, it serves our purpose.

The logic of sampling implies that the larger a sample, and the more homogeneous a population, the more likely a given sample will accurately represent that population. Our model is designed to explore this dynamic by quantifying precisely the ways in which representation responds to exogenous changes in compositional factors.

The outcome of interest, $ {R}_b $ , is an index measuring how closely the distribution of a trait within a political body reflects the distribution of that trait in a population, the core idea behind descriptive representation (Pitkin Reference Pitkin1967, Chapter 4). This is similar to the intuition behind measures of vote/seat proportionality such as the Rose Index of Proportionality (Rose Reference Rose, Lijphart and Grofman1984). Just as parties should receive representation in accordance with their votes, groups should receive representation in accordance with their numbers. Following this logic, and building on Ruedin (Reference Ruedin2009), our proposed index of representation takes the following form:

(1) $$ \begin{array}{rl}{R}_b=1-\frac{1}{2}{\displaystyle \sum_{k=1}^K}|{g}_{p_k}-{G}_{b_k}|,& \end{array} $$

where $ {R}_b $ is the degree of representation present in a particular political body, $ {g}_{P_k} $ is group k’s share of the population, $ {G}_{B_k} $ is group k’s share of a political body, and K is the total number of groups in the population. The functional form of Equation 1 is chosen so that the representation index is bounded between 0 (no representation) and 1 (perfect alignment between the characteristics of a political body and a population).

The expected value of this representational index is probed as three compositional factors change—body size ( $ {n}_b $ ), the number of groups in society (K), and the relative size of those groups ( $ {g}_{p_k} $ ), which we will characterize using entropy. The expected representation under this random sampling process is described in Proposition I (see section S.II.1.12 in Supplementary Materials II for derivation).Footnote ⁵

Proposition I. Under random sampling of citizens to bodies, expected representation is

(2) $$ \unicode{x1D53C}\left[{R}_b\right]={\displaystyle \begin{array}{l}1+\frac{1}{2}\sum_{k=1}^K\Big\{{\left(1-{g}_{p_k}\right)}^{n_b-\lfloor {n}_b{g}_{p_k}\rfloor}\\ {}\times {g}_{p_k}^{\lfloor {n}_b{g}_{p_k}\rfloor +1}\left(\lfloor {n}_b{g}_{p_k}\rfloor +1\right)\left(\genfrac{}{}{0ex}{}{n_b}{\lfloor {n}_b{g}_{p_k}\rfloor +1}\right)\Big\},\end{array}} $$

where $ {n}_b $ denotes the size of body b, $ {g}_{p_k} $ denotes the fixed population share of group k, $ \left(\genfrac{}{}{0ex}{}{a}{b}\right) $ denotes the binomial coefficient, $ {}_a{C}_b $ , and $ \lfloor a\rfloor $ denotes the floor function (e.g., bringing 3.4 to 3).

Intuitively, $ \unicode{x1D53C}[{R}_b] $ represents the average representation score across myriad random draws with a body of size $ {n}_b $ and group population shares defined by the values of $ {g}_{p_k} $ —in effect, the degree of representation one would expect if representation was structured only by compositional factors.

This exercise is designed to capture overall representation, not the representation of particular groups. Note also that a group may be over-represented in some draws and under-represented in others. Our concern is with the average discrepancy across all of these draws (as opposed to the discrepancy of the average).

The complex dynamics of Equation 2 may be visualized by placing each dimension along one edge of a graph and observing how representation scores change as the parameters of these dimensions change. In these stylized scenarios, we assume that all members of the political body are drawn from a fixed population composed of one hundred persons. Since three explanatory elements are in play, two diagrams are required.

The left panel of Figure 2 focuses on body size and number of groups (of equivalent size). The bottom row is yellow, signaling perfect representation when society is composed of a single group. If all members of society are of Type A, all members of a political body drawn from that society must also be of Type A. The far-right column is also yellow, signaling perfect representation when the size of the political body equals the size of the population. If all members of society are included in a political body, that body (effectively, a popular assembly) must achieve perfect representation. Discrepancies increase as the number of groups increases or the body size shrinks, as depicted at the top-left of the diagram, where the representation score goes to 0.

Figure 2. A Model of Representation

Note: Color values indicate different levels of the expected value of the representation index as outlined in Equation 2. Left: Values of expected representation decrease with the number of groups, but increase with body size. Right: Expected representation index also gets smaller as the population entropy grows.

The right panel of Figure 2 focuses on body size, as previously, along with group entropy, defined as

(3) $$ \begin{array}{rl}\mathrm{Entropy}({g}_{p_1},{g}_{p_2},\dots, {g}_{p_K})=-{\displaystyle \sum_{k=1}^K}{g}_{p_k}\log ({g}_{p_k}),& \end{array} $$

where again $ {g}_{p_k} $ represents the population share for group k. Entropy is bounded theoretically at 0, where only one group is present in the population. Maximum entropy is achieved when all groups are of equal size. This value is unbounded as it depends on the total number of groups in play—more groups lead to higher values if all groups are of equal size.

For heuristic purposes, we limit our exercise to six groups (the median value of ethnic groups in our global dataset, described below). With the number of groups held constant, we can observe the impact of changes in their relative size. Entropy is low where there is extreme inequality, that is, one group encompasses nearly the entire population. We draw different group share values (which constitute a six-dimensional simplex) from a Dirichlet distribution with the $ \alpha $ parameters all set to 1 so that all group share combinations are equally likely. We then average across the randomness inherent in this process of generating the group shares. For any body size (except where the size of the body equals the size of the population), increasing entropy decreases representation. Again, the upper left quadrant signals the point of worst representation—where body size is smallest and group entropy greatest.

This simulation exercise establishes a baseline for understanding the problem of coordination in descriptive representation. In subsequent sections of the paper, we test these theoretical expectations. It turns out that point estimates based on simulations are successful in explaining a good deal of the variance in representation across the world, reinforcing our theoretical expectation that failures in representation are often due to compositional factors.

DATA COLLECTION

Having examined our topic abstractly, we turn to the real world of representation. To do so, we extend the Global Leadership Project (Gerring et al. Reference Gerring, Oncel, Morrison and Pemstein2019) with new coding focused on the demographic characteristics of leaders and a second round of data collection. Details pertaining to the recruitment of experts and the collection of data are discussed in Supplementary Materials IV. In this section, we explain the coding and discuss challenges posed by missingness and measurement error.

In accordance with the scope conditions of our theory, we focus on four dimensions of identity where there appear to be strong preferences for descriptive representation. (Several additional traits including age and education are explored in Supplementary Materials I.) For each leader, country experts code gender (male or female),Footnote ⁶ language (mother tongue), religion (by birth), and ethnicity. The latter is understood as an uber-identity, describing the most important cleavage existing in a country at a particular point in time—which might be defined by race, religion, language, caste, region, cultural practices, or some combination of the foregoing.

In making coding decisions, country experts draw on a variety of sources including parliamentary websites, Wikipedia entries, country-specific sources, and clues implicit in a leader’s name, place of birth, and so forth. Coding thus rests largely on each leader’s presentation of self, their public persona—presumably crafted for political purposes. Since representation is a public act, it is appropriate to focus on how leaders present themselves. Fortuitously, this information is usually readily accessible.

After compiling the data, we arrive at a total of 2 genders (male and female), 11 religions, 280 languages, and 807 ethnicities. (A complete list of categories is posted in section S.IV.1.3 in Supplementary Materials IV.) To ascertain the size of each group in the general population, we consult censuses or surveys for each country—limiting our purview to legal citizens. (Since nonvoting aliens are generally disqualified from holding office, they are excluded from our analysis.) Only groups with more than one hundred thousand individuals or composing more than 1% of the population are included. (Groups falling short of this threshold are removed from the analysis even if they achieve representation, as happens on a few occasions.)

Leaders are classified into seven political bodies: (a) executive (including all persons who perform an executive function such as president or prime minister, but not those whose role is purely symbolic), (b) cabinet (with and without portfolio), (c) parliamentary party group, (d) upper chamber of the legislature (if bicameral), (e) lower (or unicameral) chamber of the legislature, (f) legislature at-large (both upper and lower chambers), and (g) supreme (or constitutional) court.

Since these categories are partially overlapping, leaders may belong to multiple political bodies, which means that units of analysis are not entirely independent. Supplementary analyses employ hierarchical bootstrap sampling to provide standard errors that account for the nonindependence of data points (see section “Additional Robustness Tests”).

The resulting dataset, summarized in Table 1, incorporates 156 countries, two rounds of coding (2010–13 and 2017–19), 1,099 social groups, 1,552 political bodies, and 53,560 leaders. Coverage along these dimensions is more extensive than any comparable dataset, though limited to two points in time.

Table 1. Descriptive Statistics

Even so, coverage is uneven. Political bodies, the main units of analysis in the tests that follow, are included only if 75% of their members are coded along the relevant social dimension. For a core group of 120 countries, coverage is fairly strong across all dimensions; for other countries, only one or two identities are successfully coded. There is also unevenness across rounds. Linguistic and religious identities are coded only in the first round, while upper chambers are coded only in the second round.

To cope with problems of missingness, we take several steps to ensure that the main results are robust. First, we replicate benchmark analyses with no threshold of inclusion: all political bodies for which $ any $ members are coded (for the relevant social dimension) are included. Second, we replicate benchmark analyses using datasets where missing values are imputed. Third, we conduct analyses limited to a single round, or a single dimension of social identity. Results from these tests are very close to those obtained from the benchmark sample, as reported in tables and figures to follow.

In addition to problems of missingness, we must consider potential measurement error. Highly subjective features of human identity are sometimes difficult to define and to code. Reassuringly, an intercoder reliability analysis shows a high level of agreement across expert coders (section S.IV.1.2 in Supplementary Materials IV). Moreover, when the same leader appears in rounds 1 and 2, their ethnicity is coded identically 97% of the time. This constitutes an informal inter-coder reliability test since most countries are coded by different experts across rounds.

Of course, one must also be concerned with the definition of social categories, which may be aggregated in different ways. For example, religion may be classified with coarse categories (e.g., Christian) or differentiated categories (e.g., Protestant, Catholic, and Orthodox). Additional complexities arise when one considers the intersection of multiple dimensions of identity, which may be interactive rather than simply additive (Celis and Erzeel Reference Celis and Erzeel2017).

We cannot test all possible measures of identity—an essentially infinite set. However, we do explore different dimensions (gender, linguistic, religious, and ethnic), one intersectional identity (ethnic+gender), and regions of the world with different identity configurations. We also conduct analyses in which observed ethnic groups are combined in randomly chosen superordinate groups. Results from all of these tests (described below) are robust, offering reassurance that our findings are not hostage to arbitrary choices in categories or occasional measurement errors. Further reassurance is offered by a convergent validity test that compares our country-level index of ethnic representation with a comparable index from Ruedin (Reference Ruedin2009) (see Figure S.II.3 in Supplementary Materials II).

AN INDEX OF DESCRIPTIVE REPRESENTATION

Having introduced our data, let us consider how it might be aggregated into an overall index of representation, one suitable for comparisons across varied dimensions and contexts.

Traditionally, work on descriptive representation focuses on groups facing special discrimination or economic hardship, for which the term minority is loosely employed. Unfortunately, exclusion is not easy to measure, being a matter of degree and subject to differing perceptions. Moreover, discrimination and economic hardship do not always overlap (groups may face discrimination despite being relatively affluent) and are not always associated with relative size (complicating the idea that an oppressed group must necessarily be a statistical minority). Finally, one must reckon with societies where groups are not clearly ranked by social or socioeconomic status, a pattern common in sub-Saharan Africa. What does adequate representation mean in these contexts?

To sidestep these obstacles, we accept the reality that groups may be defined in any number of ways—by ethnicity, gender, and so forth—and that these categories may be understood in different ways and combined in various ways, generating intersectional categories.

However defined, we propose an encompassing approach to measurement. All members of society must fit somewhere within the chosen dimension; they must count as members of some group when calculating overall representation across that dimension. If ethnicity is the dimension of interest, for example, all persons must be placed within an ethnic category, not just those deemed “minorities.” Helpfully, representation is a zero-sum outcome: one group’s representation must be achieved at the expense of another’s. If one ethnic group is over-represented, another must be under-represented. Accordingly, we need not agonize over whom to identify as minorities and majorities or attempt to ascertain which groups are subject to what degree of discrimination.

In aggregating across groups (along a single dimension of identity), we employ the representation index outlined in Equation 1, replacing theoretical quantities with data collected for this project:

(4) $$ \begin{array}{rl}{\hat{R}}_b=1-\frac{1}{2}{\displaystyle \sum_{k=1}^K}|{\hat{g}}_{p_k}-{\hat{G}}_{b_k}|.& \end{array} $$

Here, $ {\hat{R}}_b $ is the estimated degree of representation present in a particular political body, $ {\hat{g}}_{P_k} $ is an estimate of group k’s share of the population from country-level sources, $ {\hat{G}}_{B_k} $ is the estimate of group k’s share of a political body from our data, and K is the total number of groups in the population.

To illustrate the dynamics of Equation 4, imagine a society with three equal-sized groups (33%, 33%, 33%). At one extreme, each group achieves equal representation (33%, 33%, 33%), rendering a representation score of 1.00, a perfectly proportional relationship between body and population group shares. At the other extreme, all offices are controlled by individuals who are not members of the population (0%, 0%, 0%), for example, foreign colonizers or an occupying power. Here, the representation score is precisely 0. In a more typical scenario, one group achieves twice the number of offices as the other two (50%, 25%, 25%), generating a representation score of 0.83.

Country Scores

So measured, how does descriptive representation vary across countries? To answer this question, we must aggregate representation scores across offices within each country (executive, parliamentary, and judicial) for each chosen dimension of social identity (gender, language, religion, and ethnicity), and then aggregate across those dimensions to obtain a summary score.

Since missingness across countries may introduce bias, we first impute missing values for political bodies that are partially coded in rounds 1 or 2. Values are imputed by fitting a nonlinear prediction model for each variable and iteratively predicting missing values with that model until convergence (Stekhoven Reference Stekhoven2015). Background factors in the imputation stem include covariates from Model 4 in Table 2. (Imputations for expected representation in Figure 3 are conducted separately.)

Table 2. Main Analysis

Note: Outcome: representation index (where 1 = perfect representation), measured across various identities—ethnicity, religion, language, and gender. Estimator: ordinary least squares, t-statistics in parentheses, standard errors clustered by country. * denotes $ p<0.05 $ . Missing values were imputed in Model 1 to ensure compatibility across country and standard errors calculated via the country-level block bootstrap (with the imputation model re-fit on every bootstrap draw); see main text. In the unit of analysis row, “C” denotes country, “G” denotes group, and “B” denotes body. Full model results are given in Table S.V.1 in Supplementary Materials V.

Figure 3. The Relationship between Observed and Expected Representation, Aggregated to the Country Level

Note: Missing representation values have been imputed to ensure comparability across country as described in the main text. A regression model summarizing this relationship can be found in column 1 of Table 2 (see Table S.V.1 in Supplementary Materials V for full model specification).

Table S.III.1 in Supplementary Materials III provides a complete list of countries and their representation scores across all four dimensions, combining results for all (national) political bodies in each country. A summary score for each country is derived by averaging across the four dimensions—gender, language, religion, and ethnicity.

So calculated, the mean representation score across all countries in our sample is 0.73, the standard deviation is 0.07, and the minimum/maximum values are 0.50 and 0.88. The highest levels of overall representation are achieved in Iceland, Poland, Norway, Denmark, and Finland, while the lowest levels are registered in Sierra Leone, Central African Republic, Solomon Islands, Indonesia, and Congo (DRC).

To demonstrate the contribution of compositional factors to these country-level results, Figure 3 plots observed values against values we would expect under conditions of random sampling given the size of each political body and the number and relative size of social groups, as described by Equation 4.

Evidently, random sampling offers a very reasonable approximation of the selection process if compositional factors are taken into account—a point confirmed in the first analysis of Table 2. As we shall see, the fit between them is not substantially improved when institutional, sociological, and economic factors are added to the model.

Note, however, that all data points fall below the diagonal line in Figure 3. This means that all countries are less representative than they would be if the selection of political leaders were entirely random. (The size of this representational gap is calculated for each country in Table S.III.1 in Supplementary Materials III.) Indeed, the maximum observed value (0.88) is only slightly greater than the average value of expected representation (0.87).

MAIN TESTS

Having looked at representation in a descriptive fashion, we turn to a series of analyses that attempt to probe the causal effect of compositional factors. Initial tests are shown in Table 2 (for descriptive statistics, see Table S.II.1 in Supplementary Materials II).

Model 1 replicates the bivariate scatterplot shown in Figure 3. Here, country-level representation scores are regressed against those predicted by Equation 4 (in which leaders are chosen randomly, taking into account the size of political bodies and the distribution of social groups). More formally, we assume

(5) $$ \begin{array}{rl}{\hat{R}}_c=\gamma \times \unicode{x1D53C}[{\hat{R}}_c]+{\epsilon}_c,& \end{array} $$

where c is a country index, as described in the previous section. When the $ \gamma $ coefficient in this single-parameter model is greater than $ 1 $ , observed representation will be on average higher than expected under random sampling; when below 1 it is lower than expected. This bivariate model explains over half of the variance in representation across countries.

In further analyses, the outcome is disaggregated. Rather than country-level scores, we look at the representation score for each political body across each identity dimension (gender, language, religion, and ethnicity) and each round of data collection. An individual observation is therefore composed of a political body, a social identity, and a coding round. For this purpose, we employ raw (unimputed) data, with the provision that a political body is included if 75% of its members are coded across a particular dimension of identity. The total size of the body is therefore defined as the number of members whose identity is known. For example, if 90 members of a one hundred-member legislature are coded for ethnicity, n=90.

While offering greater empirical leverage, this disaggregated approach is likely to increase stochastic error. Numerous factors may affect the level of representation achieved for a particular political body along a particular dimension of social identity at a particular point in time. These stochastic factors are minimized in country-level aggregate scores, as instances of over- and under-representation cancel each other out. In body-level analyses, they remain, and thus are likely to weaken the overall fit of the model.

Model 2 in Table 2 includes only the expected representation index, that is, the prediction issued by Equation 4 under conditions of random sampling and using the same structure as in Equation 5. This bivariate model (no intercept) accounts for over one-third of the variance. As expected, overall model fit is attenuated relative to Model 1, which we attribute to increased stochastic error. However, the point estimates for expected representation are nearly identical.

In further tests, we distinguish two compositional factors. The size of each political body is understood as its membership, transformed by the natural logarithm (to account for the diminishing marginal impact of larger membership). The dispersion of groups is measured with the Herfindahl index of fractionalization, which captures the probability that two randomly chosen individuals belong to the same social group (calculated separately for gender, linguistic, religious, and ethnic groups). Model 3, including only these variables, explains over two-fifths of the variability in representation.

Model 4 adds dummies for each social identity (ethnicity, religion, language, and gender), for the existence of a gender quota, for body type (executive, cabinet, et al.), for the selection rule applicable to that office (appointive, proportional, majoritarian, mixed, indirectly elected, and other), for each round of data collection (1 and 2), and for each country.

Model 5 drops country fixed effects in favor of country-level covariates. In choosing covariates, we rely on extant work and our own hunches about factors that could plausibly affect representation. Chosen covariates include the Lexical index of electoral democracy (Skaaning, Gerring, and Bartusevicius Reference Skaaning, Gerring and Bartusevicius2015), population (log), per capita GDP (log), and inequality, measured by the Gini coefficient of income inequality.

Estimates for the two compositional factors are extremely close across Models 3–5, despite dramatic changes in specification. Moreover, additional covariates scarcely improve model fit and also generally exhibit small t-statistics (see Table S.V.1 in Supplementary Materials V), suggesting that they are relatively minor influences on representation in this global sample.

As a further specification probe, we assess the out-of-sample predictive importance of right-side predictors from Model 5 via Lasso regression, which imposes a penalty on the absolute magnitude of coefficients, thereby setting some coefficients exactly to 0 unless they meaningfully improve predictive out-of-sample performance (approximated by cross-validation). We find that the regularizing Lasso model sets many of the model coefficients to exactly 0 but leaves as nonzero both body size and fractionalization, indicating that these quantities meaningfully improve out-of-sample predictions (see Table S.II.10 in Supplementary Materials II).

All of the available evidence suggests that a principal driver of representation is compositional: larger bodies generate better representation, and more heterogeneous countries generate worse representation. To get a sense of these effects, Figure 4 plots the expected values generated by the benchmark specification (Model 3, Table 2). Across our sample, fractionalization has a slightly steeper curve; nonetheless, a shift along the x-axis from minimum to maximum values translates into a substantial shift in representation for both regressors. Estimates are also precise, signaled by the extremely tight confidence bounds around these estimates.

Figure 4. Predicted Representation Index Values Based on Model 3 in Table 2 with 95% Confidence Intervals

Note: Mean/median/SD values across the sample: body size (35/6/123), fractionalization (0.43/0.50/0.21). Above the x-axis labels for both plots, we display rug plots illustrating the empirical density of data points in our sample.

INSIDE THE BOX

Having offered a parsimonious account of compositional effects, we are now in a position to disaggregate the treatment, thereby shedding light on potential mechanisms and also on further implications of our theory.

Based on the idea that larger bodies provide better representation, we infer that the representational capacity of various bodies follows their size. In our sample, average membership is as follows: executive (n = 1–2), supreme court (n = 9), parliamentary party (n = 18), cabinet (n = 20), upper house (n = 80), and lower house (n = 232). Accordingly, we expect the degree of representation to increase in a monotonic fashion from the smallest body to the largest using this indirect measure of body size.

To test this expectation, the first model in Table 3 regresses the representation index against these body types (with executive as the excluded category) along with fixed effects for identity, quota type, selection rule, round, and country. Results accord with theoretical expectations insofar as larger bodies are generally more representative. The notable exception is parliamentary parties, which may reflect the concentrated support that some parties obtain from specific identity groups.

Table 3. Implications of the Main Analysis

Note: Outcome: representation, measured for each identity—ethnicity, religion, language, and gender. Higher values indicate better representation. Estimator: ordinary least squares, t-statistics in parentheses, standard errors clustered by country. * denotes $ p<0.05 $ . Full model results are given in Table S.V.2 in Supplementary Materials V.

A better research design compares parties of different sizes to each other, thereby sidelining a great many background factors that might serve as confounders. Following our theory, larger parties should be more representative than smaller parties. Model 2 is therefore limited to parliamentary party groups, along with the usual vector of controls (excluding selection rule, which is collinear with country fixed effects). Coefficients for the compositional factors of theoretical interest are nearly identical to that of the benchmark model in Table 2, confirming that parties follow the pattern established for other political bodies.

Finally, our theory suggests that both the number of groups (log) and group entropy exert independent effects on representation. In our benchmark specification, these are combined into a single fractionalization measure. In Models 3 and 4, we differentiate these factors, tested separately by virtue of their collinearity. As expected, the number of groups and their entropy (similarity in size) both reduce representation.

DIMENSIONS OF IDENTITY

In this section, we explore compositional effects across different identity categories. To the four familiar dimensions—ethnicity, religion, language, and gender—we add a measure of intersectionality, formed from the intersection of ethnicity and gender (following Weldon Reference Weldon, Goertz and Mazur2008). There are of course many other potential intersectionalities one might explore (Collins and Bilge Reference Collins and Bilge2020). However, since our construction of ethnicity aims to represent the most important cleavage in each society, and gender is orthogonal to ethnicity, this seems a logical choice.

Density plots for these five dimensions of identity, displayed in Figure 5, show the empirical distribution of representation scores across each dimension. The curves for language, ethnicity, and religion are similar, with modes just below 1 (perfect representation) and long left tails. Evidently, many political bodies achieve decent representation along these dimensions, while some are horribly askew.

Figure 5. The Shape of Descriptive Representation

Note: Descriptive statistics (mean/median/SD): gender (0.64/0.60/0.16), ethnicity (0.68/0.77/0.28), language (0.73/0.85/0.28), religion (0.65/0.77/0.28), ethnicity–gender intersection (0.47/0.48/0.23). Group-level means are represented as tick marks at the bottom of the figure (with random jitter added along the y-axis to make the lines distinguishable).

Gender has an accentuated mode at 0.5, marking the point where political bodies are dominated by a single gender (male). The truncated left tail is a product of the distribution of gender in populations across the world. Because men and women compose roughly half of the population everywhere, the greatest possible violation of equal representation—that is, the total exclusion of women from public office—is not as extreme a violation as the total exclusion of a linguistic, religious, or ethnic group comprising a super-majority. For example, the exclusion of Blacks from representation in Apartheid South Africa, where they composed roughly eighty percent of the population, would render a lower representation score than the exclusion of women.

Among the four core dimensions of identity—gender, religion, language, and ethnicity—the mean values of representation, illustrated across the x-axis of Figure 5, are remarkably close. This is surprising given the disparate nature of these identities, their disparate histograms, and their weak intercorrelations (see Figure S.I.1 in Supplementary Materials I and also Ruedin Reference Ruedin2010). One would have thought that some identities would be better represented than others. Yet, we find little variation across sample means.

By contrast, the intersectional index has a lower mean, and wider dispersion, than other dimensions of representation. A plausible explanation, consistent with our theory, is that the multiplication of categories introduces greater social diversity, and with it additional coordination problems that translate into lower overall representation.

For our purposes, the most important issue is whether compositional effects vary across different dimensions of social identity. To assess this question, we replicate the benchmark specification (from Table 2) across each dimension in Table 4. Samples are focused on ethnicity (Model 1), religion (Model 2), language (Model 3), gender (Model 4), and intersectionality (Model 5).Footnote ⁷

Table 4. Analysis by Group Identity

Note: Outcome: representation index. Higher values indicate better representation. Estimator: ordinary least squares, t-statistics in parentheses, standard errors clustered by country. * denotes $ p<0.05 $ . Full model results are given in Table S.V.3 in Supplementary Materials V.

There is some evidence of causal heterogeneity across the five measures of identity. Of course, these may be stochastic or a product of varying samples. One must resist the temptation to over-interpret small differences.

The varying precision of these estimates can be probed in our random sampling framework, following section “A Random Sampling Model.” This analysis, visualized in Figure S.II.1 in Supplementary Materials II, explores the residual standard deviation of the theoretical model, $ \sqrt{\unicode{x1D53C}[{({R}_b-\unicode{x1D53C}[{R}_b])}^2]} $ , as we vary body size, group number, and group entropy. We find that the residual standard deviation shrinks as the number of groups increases. This could explain why compositional features are more precisely estimated for intersectional identities than for other social identities (as captured by t-statistics across these models).

In any case, the main takeaway is that body size is always associated with increased representation, while fractionalization is always associated with reduced representation. Most estimates are similar to the benchmark model in Table 2.

Of particular note are results contained in Model 1. Recall that ethnicity represents the most salient cleavage in a country at the time the data were collected. This is of course a judgment, and we must rely on our expert coders to perceive which dimensions are most important in a particular context. Nonetheless, it suggests that compositional factors matter as much for the most important cleavage as they do for other cleavages. These relationships are probed at greater length in section S.II.1.5 in Supplementary Materials II.

CONTEXTS

Having examined compositional effects across different identity categories, we now explore background factors that might impact compositional effects on representation. To test potential moderators, the full sample is divided into sub-samples according to the background factor of interest. Results of these paired tests are shown in Tables 5 and 6.

Table 5. Analysis in Varying Contexts

Note: Outcome: representation, measured for each identity—ethnicity, religion, language, and gender. Higher values indicate better representation. Estimator: ordinary least squares, t-statistics in parentheses, standard errors clustered by country. * denotes $ p<0.05 $ . Full model results are given in Table S.V.4 in Supplementary Materials V.

Table 6. Heterogeneity Analysis by Region

Note: Outcome: representation, measured for each identity—ethnicity, religion, language, and gender. Higher values indicate better representation. Estimator: ordinary least squares, t-statistics in parentheses, standard errors clustered by country. * denotes $ p<0.05 $ . Full model results are given in Table S.V.5 in Supplementary Materials V.

First, we explore the nature of the office, categorized as elective (Model 1, Table 5) or unelected (Model 2). The latter category includes supreme courts, cabinets, and a few parliamentary parties where there is no apparent elective process. We find that effects persist across both samples. The impact of body size and fractionalization is somewhat weaker across unelective bodies, though this may simply reflect the limited variability of our sample, composed largely of small bodies.

Second, we differentiate democracies (Model 3) and autocracies (Model 4). Democracies are understood as polities with minimally competitive multiparty elections for the legislature and the executive, operationalized as a score of 4–6 on the Lexical index (Skaaning, Gerring, and Bertusevicius Reference Skaaning, Gerring and Bartusevicius2015). We find virtually no difference in estimates for our two variables of theoretical interest across these sub-samples; compositional effects are equally strong in autocracies and democracies.

Third, we compare two periods in time corresponding to Round 1 (2010–13) and Round 2 (2017–19) of the data collection process. To make this comparison exact, we include only representation by ethnicity and gender, which were coded for both rounds. Changes in leadership and representation between rounds scarcely affect estimates of compositional effects across the two samples, as shown in Model 5 (Round 1) and Model 6 (Round 2).

Table 6 continues the exercise with another set of comparisons. First, we compare rich, industrialized countries with poorer, less developed countries. To differentiate the two groups, the sample is divided into OECD countries (Model 1) and non-OECD countries (Model 2). (The Lexical index is excluded from Model 1 as there is no variability in regime type within this sub-sample.) There is little difference across these sub-samples, suggesting that compositional effects are not moderated by economic development.

Next, we compare various regions of the world: the Americas (Model 3), Asia (Model 4), Europe (Model 5), and the Middle East and North Africa (Model 6). Again, coefficient estimates for the variables of theoretical interest are stable.

INSTRUMENTAL VARIABLES

Neither of the key variables in this study is randomly assigned, so one must consider whether the data-generating process might in some way confound estimates reported in previous tables. It is possible, for example, that the degree of diversity in a country—or the degree of representation achieved in a country—affects its institutions, including the size of political bodies. It is possible that the degree of representation and recognition achieved by social groups affects their self-definition (Liu Reference Liu2011). It is even possible that deep-seated social norms affect the shape of society, the shape of institutions, and the representation of social groups.

We are at pains to work out all the possible ways in which confounders might affect the analyses presented in previous tables—though we take some comfort in the stability of the results across different specifications (which include controls for economic development, democracy, inequality, and country fixed effects) as well as the theoretical derivations presented in section “Coordination Problems.” In this section, we approach the challenge of causal identification with instruments.

In order to serve their intended function, the chosen instruments must be exogenous and must affect the outcome only through the treatment variable. Because these assumptions are impossible to prove, we regard estimates posted in Table 6 as robustness tests rather than baseline models.

As instruments for the size of political bodies, we employ political body types, categorized as executive, upper house, or lower house in countries where membership to those bodies is determined by voter input. (Other types are excluded.) The assumption is that these body types are predictors of body size (as shown in Table 3) but are not for other reasons likely to be more or less representative. (This assumption would be violated if selectors relate to the identity dimension of candidates differently across political body types.)

To instrument for ethnic fractionalization, we employ a geographic feature—dispersion in elevation across regions of a country—grounded in work on the long-run sources of ethnic diversity (Michalopoulos Reference Michalopoulos2012). (Other dimensions of identity are here excluded.) Geography is assumed to be exogenous and unrelated to representation, except through its influence on diversity. Figure S.II.8 in Supplementary Materials II visualizes the assumptions of these IVs.

The first-stage analyses in Table 7 offer a reasonably good fit to the data, and the IV diagnostics are favorable. (For example, we reject the null hypothesis in the weak instruments test; we also reject the null in the Wu–Hausman test, indicating that the use of instruments is helpful in accounting for endogeneity.) The second-stage analyses report estimates for the key variables that are comparable to those reported in Table 2, though slightly stronger for fractionalization. These analyses offer some reassurance against threats to inference stemming from nonrandom assignment.

Table 7. IV Analysis

Note: First stage outcomes: log(body size) and fractionalization. Second stage outcome: levels of representation (where 1 = perfect representation). Estimator: two-stage least squares, t-statistics in parentheses, standard errors clustered by country. * denotes $ p<0.05 $ . Full model results are given in Table S.V.6 in Supplementary Materials V. Bold values are those of theoretical relevance and interpreted in the main text.

ADDITIONAL ROBUSTNESS TESTS

In this section, we briefly discuss additional robustness tests whose full results are posted in Supplementary Materials II.

First, in section S.II.1.6 in Supplementary Materials II, we assess the extent to which our results are robust to the potentially arbitrary aggregation of ethnic categories, an issue discussed in section “Data Collection.” To do so, we take the ethnic groups for a given country, randomly aggregate them into higher-order categories, and replicate our main analyses.

Second, we return to the problem that our use of clustered standard errors can account for error covariances within countries but not the hierarchical structure of our data, where information about a single leader can contribute to multiple observations. In section S.II.1.9.2 in Supplementary Materials II, we replicate the main analysis employing a hierarchical bootstrap procedure described in section S.II.1.7 in Supplementary Materials II that accounts for multiple levels of uncertainty.

Third, we evaluate the degree to which our results may be affected by post-treatment bias due to the inclusion of parliamentary parties in the analysis. These bodies are post-treatment in the sense that the composition of a party could affect its popularity and, thus, its representation in the legislature. To obviate this issue, we replicate the main analysis excluding parties. Results are presented with clustered standard errors (section S.II.1.10.1 in Supplementary Materials II) and with significance assessed via the hierarchical bootstrap described above (section S.II.1.10.2 in Supplementary Materials II).

Finally, we assess whether results might be affected by our inclusion criterion, whereby political bodies are included only if we are able to gather information on the identity of more than 75% of its members along a particular dimension of identity. Relaxing this criterion, we include all political bodies for which any member can be coded. Results are presented with clustered standard errors (section S.II.1.11.1 in Supplementary Materials II) and a hierarchical bootstrap (section S.II.1.11.2 in Supplementary Materials II). In Figure S.II.4 in Supplementary Materials II, we plot the main regression coefficients using Model 4 in Table 2 as we vary the coverage threshold between 0 and 1. Section S.II.1.8 in Supplementary Materials II replicates the main analysis when randomly perturbing ethnic identities for which coders were uncertain.

These tests offer some reassurance with respect to the robustness of our main findings, as deviations from the benchmark model are generally small.

LIMITATIONS

Although capacious, the present study encounters several empirical limitations.

First, many traits—including sexual orientation, disability, and social class—are unexamined due to the difficulty of collecting data on these subjects on a global scale. Our theory suggests that these dimensions of identity are subject to compositional effects insofar as they are valued by selectors, a point that we develop in Supplementary Materials I but do not have an opportunity to test.

Second, our data sample the world at two points in the contemporary era. Accordingly, we are unable to directly address longer-term historical patterns. We theorize that as views about social groups change these changes should be reflected in compositional effects. Systematic tests must await better evidence.

A third limitation concerns the national focus of our data. We see no reason to suppose that subnational bodies are exempt from compositional effects, and a smattering of studies focused on local councils in the United States (cited at the outset) mostly supports that interpretation. Further research is needed, particularly outside the United States.

A fourth limitation stems from our global approach, which is unable to capture nuances of descriptive representation—arising, for example, from laws, norms, history, and geography—that pertain to specific countries or social groups. Helpfully, a rich body of research focuses on these issues in specific contexts, especially in Europe and the United States.Footnote ⁸ The present study should be viewed as a complement, not a replacement, for these focused studies.

Finally, we have little ground for speculating upon compositional effects in nonpolitical bodies such as firms, labor unions, and other nongovernmental organizations. Insofar as there is growing pressure to represent society in these bodies, we would not be surprised if a similar dynamic applies. So far as we know, the impact of compositional factors on representation in these venues has not been studied; nor is it entirely clear how relevant constituencies should be defined in these contexts. We leave these matters for future research.