Problem definition

The theoretical model developed in this section is concerned with deviations from proportionality in multi-district PR elections using the D’Hondt seat allocation formula or equivalent formulas (such as the Hagenbach–Bischoff formula used in Luxembourg and Switzerland, which leads to the same seat allocation as D’Hondt).

The model is based on two key elements. First, it translates the D’Hondt rounding rule into a simple and straightforward mathematical function for the partisan effect in the seat allocation. Simple algebra and straightforward assumptions result in a basic model that serves as the workhorse of my approach. The further parts discuss the model’s assumptions. These relate to small parties and four partially related problems of between-district variance. This discussion informs us about a series of adjustments to the operationalisation and allows for more precise predictions in the presence of small parties or in elections with variance on the four territorial dimensions.

This theoretically driven approach renders the model universal and comprehensive. I validate it relying on the largest available dataset of elections under D’Hondt in democratic countries. The model is comprehensive, as it analyses the interplay of several institutional and geographical components related to the allocation of seats in multi-district PR systems.

At the theoretical stage, my dependent variable is the absolute number of seats allocated to political parties. In the empirical tests, I use a transformed version (representation ratios) that is better suited to assessing the variance.

Constructing the model: the basic steps

The model introduces an algebraic approximation for the number of seats that a party can expect to win in a national parliamentary election under proportional (D’Hondt) rules. It is modular, and I introduce it in stages. Steps 1–3 examine the impact of the D’Hondt seat allocation formula based on national-level variables, such as national party vote shares. Steps 4–7 extend this to variance across districts. The main text discusses the main lines, while details on the operationalisation are provided in the Supplementary Material, Section A.

Step 1: the basic proportionality model

Any model of seat allocation under PR (or deviations from it) starts with the proportional rule as its basis (e.g. Kedar et al. Reference Kedar, Harsgor and Sheinerman2016: 682): in an assembly with S seats, the number of seats a party wins s_j is proportional to its vote share v_j, so that

\begin{equation*}s_j\,=v_j\,\cdot\,\,S\end{equation*}

Step 2: the D’Hondt rounding rule

The result of Step 1 is seat fractions, but eventually, they need to be rounded to full seats (non-negative integers). The centrepiece of any PR seat allocation rule (and of my model) is the rounding rule. The D’Hondt rule rounds them down. Its effect can be thought of as a tax system, in which each party pays a ‘head tax’ of 0.5 seats for each electoral list. The tax revenue is redistributed in favour of the larger parties, resulting in a net redistribution from small to large.

The 0.5 seat rule of thumb is not an arbitrary approximation. It is derived from the mathematical definition of D’Hondt as the divisor rule as with rounding down. The votes (here: vote share) of each party are divided by a divisor H_i, and then rounded down to the nearest integer. In the exemplary calculation in Table 1 for a 10-seat district in Portugal, this first divisor H ₁ amounts to 10%, equivalent to 1 out of 10 seats. The non-integer fractions of seats resulting from the division are rounded down. For example, the proportional allocation of seats of 4.71 for the PS is rounded down to 4. These rounding losses could be between 0 and 0.99 seats for each party or in any symmetric distribution an average of 0.5 seats per party (but see Step 3 for deviations). In a six-party competition, three seats in total are lost through rounding. In further iterations, the divisor H_i is lowered (here from 10% to 7.5%), and the non-integer fractions are rounded down again until these three seats are also allocated. Mathematically, lower divisors imply that the remaining three seats are reallocated, on average, in proportion to the parties’ vote share v_j. Each party loses 0.5 seats per district, but larger parties regain more.

Table 1.

D’Hondt Calculation Example (Portugal, 2005, Constituency Coimbra, 10 Seats)

Table 1 Long description

The table illustrates the D'Hondt method applied to the 2005 election results in Coimbra, Portugal, for allocating 10 seats among parties. PS, with 47.1% of the vote, initially received 4 seats but gained 2 more in the final allocation, totaling 6 seats. PSD, with 33.1% of the vote, initially received 3 seats and gained 1 more, totaling 4 seats. Esquerda, CDU, CDS-PP, and Other parties, each with less than 7% of the vote, did not receive any seats in either allocation phase. The table highlights the impact of vote percentage and divisor adjustments on seat distribution, emphasizing the advantage of higher vote percentages in gaining seats.

Equation 1 presents the resulting model, consisting of two summands: first, each party j is allocated a number of seats proportional to its vote share v_j, but according to the second summand, each party loses the same amount. The rounding effect, and thus the redistribution, is larger with more competitors N. This is reiterated in every district d:

(1)\begin{equation}s_j=v_j\,S+\left(\frac{\left(Nv_j-1\right)}2\right)\,d\end{equation}

Incidentally, the equation identifies the vote share at which parties gain a perfectly proportional share of seats, which Rein Taagepera and Markku Laakso (Reference Taagepera and Laakso1980) defined as the ‘break-even point’. It is equal to the inverse of the number of competitors ${v_j} = \frac{1}{N}$. In the calculation example of an election with six parties and with no further information, parties with more than 17% of the vote tend to be overrepresented, and those with less than 17% tend to be underrepresented.

Other divisor or quota methods have different rounding rules, which makes some of them (Sainte-Laguë, Hare/Niemayer) free of systematic deviations from PR.

This parsimonious solution for dealing with rounding effects relies on simplifying assumptions. Steps 3 and 5 explain why the assumptions about the distribution of seat fractions may be violated, and Steps 4–7 discuss how to deal with variables that vary across the territory. See also Table 2 and the Supplementary Material, Section A.

Table 2.

Adjustments of the Basic Model

Table 2 Long description

The table details adjustments to an electoral model, focusing on correcting assumptions that impact party representation and vote allocation. Key issues include malapportionment, where voter-to-seat ratios vary across districts, favoring parties in less populated areas. Uneven district magnitude affects representation thresholds, giving smaller parties better chances in larger urban districts. The rounding effect benefits parties with strong performances in multiple districts, not accurately reflected by national averages. The number of competitors also influences party advantage, with more competitors in high vote-share districts skewing results. These corrections aim to improve the model's accuracy in reflecting electoral dynamics.

Step 3: assumption about seat fractions and rounding

The 0.5-seat deduction rule is based on the straightforward assumption that parties hold seat fractions after the point between 0.00 and 0.99. This applies to medium-sized or larger parties. Step 3 explains why and when this does not apply for very small parties and suggests an adjustment to the operationalisation of the model.

Very small parties rarely come close to winning a seat in a district, meaning their decimals tend to be closer to 0 than 1. For these parties, the average rounding-down effect (Step 2 of my model) is, therefore, smaller than 0.5. For example, in the 2006 Dominican Republic elections, 17 small parties with fewer than 3% of the vote were contesting across 47 constituencies, each with between 2 and 13 seats; none of them came close to winning a seat. According to the 0.5 seat rule-of-thumb, with 47 districts and 20 parties, no fewer than 470 seats would be redistributed to large parties. This is almost three times the total number of seats in the lower chamber. As a consequence, the model would predict a negative number of seats for small parties.

However, these predictions are wrong. This is because for very small parties with a vote share equivalent to fewer than 0.5 seats in a district, the average rounding-down effect amounts to less than 0.5 seats. By definition, parties with a vote share that is less than the natural threshold of representation T _R do not win any representation. Under the D’Hondt rule, this threshold of representation amounts to a vote equivalent to 0.5 district seats (T _R = 1/2m, where m stands for district magnitude).Footnote ²

I adjust the model by not applying it to these very small parties below the threshold of representation. Accordingly, I also remove their votes (also called ‘wasted votes’) from the calculation and remove these parties from the count of ‘viable’ parties (see Supplementary Material, Section A, for the details of the operationalisation). In this example, three viable parties remain, and the total number of seats reallocated due to the D’Hondt formula corresponds to this number.Footnote ³

Variance across districts: institutions and electoral geography

The remaining steps address elections in multiple districts and between-district variance. The basic model (Steps 1–3) is based on national averages, such as the number of competitors, national vote shares and average district magnitude. However, under the D’Hondt rule, seats are allocated in districts, and the national result is the sum of all district results. In the presence of variance across districts – whether due to electoral rules or party competition and votes not being uniform – national averages can be misleading when it comes to predicting the D’Hondt effect (Walter and Emmenegger Reference Walter and Emmenegger2023). Steps 4–7 account for this variance.

One of the most-discussed elements of between-district institutional variance is malapportionment (defined as an unequal number of voters per seat across districts). When unequal voters-per-seat ratios correlate with unequal party vote shares or numbers of competitors across districts, they introduce partisan bias in the national seat allocation (Kedar et al. Reference Kedar, Harsgor and Tuttnauer2021; Monroe and Rose Reference Monroe and Rose2002). However, focusing on malapportionment obscures three other sources of bias associated with between-district variance. More specifically, malapportionment usually correlates with several dimensions of electoral geography (uneven party strengths, uneven competitiveness and unequal percentages of wasted votes across districts), and it only has a partisan effect for parties whose vote shares vary across districts. However, all three dimensions of electoral geography affect seat allocation even in the absence of malapportionment. Steps 4–7 of the model disentangle these three variables from malapportionment.

I proceed by rewriting my model at the level of electoral districts, and then aggregate it across multiple districts, whereby I allow the above-mentioned parameters (number of voters per seat, district magnitude, parties’ relative vote strength, competitiveness) to vary across districts. This cumulative function is displayed in Equation 2, using the example of two districts. In analogy to the national-level model (Equation 1), the first summand is the proportional seat allocation for both my districts 1 and 2 (in analogy to Step 1); it also includes a parameter w_i for wasted votes by district (introduced in Step 3). The second part is identical to the national-level model for the rounding effect (Step 2, with the refinements from Step 3). However, at the district-level, instead of seats in parliament S and national vote shares, I calculate with district magnitude (m ₁ and m ₂) and district-level vote shares for parties above the threshold of representation (v_{j 1}′ and v_{j 2}′):

(2)\begin{equation}{s_{j1}}{\text{ }} + {s_{j2}}{\text{ }} = \frac{{{v_{j1{\text{ }}}}'}}{{1 - w{}_1}}{m_1}{\text{ }} + \frac{{{v_{j2}}{\text{ }}'}}{{1 - {w_2}}}{m_2} + \frac{{{N_1}\frac{{{v_{j1}}}}{{1 - {w_1}}} - 1}}{2} + \frac{{{N_2}\frac{{{v_{j2}}}}{{1 - {w_2}}} - 1}}{2}\end{equation}

This complex function can be simplified. The left summand is identical to the national-level proportional seat share under two conditions: (a) the number of voters per seat is identical in every district (i.e. no ‘malapportionment’); and (b) the share of wasted votes is identical in all districts. The right summand can be simplified if the party vote share, the number of competitors and the share of wasted votes do not vary across districts, or if this variance is accounted for.Footnote ⁴ For elections where these conditions are not met, Steps 4–7 introduce new parameters or modified variable operationalisations to account for variance across districts on the respective variables.

Step 4: malapportionment

Malapportionment is defined as the uneven allocation of seats to districts (disproportionally to the number of voters) (Flis et al. Reference Flis, Słomczyński and Stolicki2020: 207; Monroe and Rose Reference Monroe and Rose2002). For example, rural or peripheral areas (e.g. islands or sparsely populated regions) tend to have smaller districts and more seats per capita in parliament than urban or densely populated areas (Baker Reference Baker1955: 41–45). This can give parties an advantage in the seat allocation process, provided that their vote shares vary across districts. My national-level model (Equation 1) ignores this, as it builds on the assumption of an invariant number of voters per seat across districts (defined as no malapportionment; see Supplementary Material, Section A, Step 4).

An exemplary case of malapportionment is Spain, where in smaller constituencies (e.g. Soria in 2004) as few as 26,000 voters elect a representative, whereas at the national average, districts are assigned a seat for every 99,000 voters, almost four times as many. Consequently, parties that gain their votes from rural or peripheral areas are overrepresented in parliament.

I incorporate the direct effect of malapportionment (unequal seat-to-voters ratios) into my model in two steps. First, I calculate a district malapportionment index by comparing the voters per seat in each district to the national average. Second, I calculate a party malapportionment index by aggregating the district malapportionment indices based on the districts from which the party receives votes (calculation details are provided in the Supplementary Material, Section A).Footnote ⁵ Consequently, a Spanish party whose voters are based in Soria would be assigned an index of 3.9 and would be overrepresented in parliament by this factor. Such high scores are only obtained by local parties. The maximum for a larger party in my data is from Suriname, a 1.44 score for the General Liberation and Development Party. Spain’s much-cited malapportionment advantage for the Popular Party is around 1.02–1.04, equating to 2–4 extra seats for every 100 seats. Malapportionment affects the first segment of the national-level equation. Correlates of malapportionment, such as variance in district magnitude, are dealt with in the remaining steps.

Step 5: uneven district magnitude

Variance in district magnitudes across constituencies is a common side effect of malapportionment, but it also occurs independently. It matters in my model because low-magnitude districts (typically rural or peripheral) are associated with higher thresholds of representation, which makes it harder for smaller parties to enter parliament. In contrast, larger-magnitude districts (e.g. urban) allow small parties to win seats (Kedar et al. Reference Kedar, Harsgor and Sheinerman2016; Lago and Martínez i Coma Reference Lago and Martínez I Coma2024; Monroe and Rose Reference Monroe and Rose2002: 72). This difference is particularly significant for political parties with vote shares around the threshold of representation, as they are electorally viable only in higher-magnitude districts, where the threshold of representation is lower than their vote share.

To account for varying district magnitude, I need to adjust a series of variables for different thresholds of representation and calculate a new cumulative score across districts where parties are electorally viable (in particular, vote share above the threshold of representation and number of relevant competitors) or not (wasted votes) (see the Supplementary Material, Section A, for the modified operationalisation of these variables).

Step 6: uneven vote distribution

The third aspect of territorial variance in my quartet concerns uneven vote shares across districts (i.e. local party strongholds). As can be seen on the right-hand side of Equation 2, parties benefit from the D’Hondt rounding effect in proportion to their district vote share, irrespective of district size (i.e. district magnitude or the total number of voters). The national average vote share, which I use in Equation 1, underestimates the rounding advantage for parties that have their strongholds in (many) smaller districts (rural or peripheral parties) compared to equally strong parties with their strongholds in a few larger (urban) districts. This effect benefits the same parties as malapportionment does, in the same strongholds (districts with fewer voters) (Step 4), although it affects any party with uneven vote shares, even in the absence of malapportionment. The mathematical solution is straightforward: rather than relying on the national average vote share, the rounding effect should be calculated as a function of the average vote share across districts, which is slightly higher for parties that perform well in smaller districts.

Step 7: uneven number of competitors

The fourth variable in the quartet of territorial inequalities is the unequal number of competitors across districts. Variation in party strength and district magnitude implies that in some districts more electorally relevant parties are present than in others. The D’Hondt rounding effect yields more benefits for parties that face more competitors. Consequently, urban parties (with strongholds in districts of higher magnitude and greater political pluralism, see Step 6) tend to benefit more (Monroe and Rose Reference Monroe and Rose2002: 71). The Supplementary Material, Section F, illustrates this empirically. The factors driving this effect are associated with malapportionment (Step 4) but also occur independently. In order to adjust the D’Hondt rounding effect for the uneven number of competitors and in simplification of Equation 2, I first count the number of (electorally viable) competitors by district. Then, for each party, I calculate the average number of competitors across the districts in which it is running, weighted by its district vote share.

Aggregated effect: calculation example

I illustrate this calculation for two parties, the small Left Bloc in the 2011 elections in Portugal and the large People’s Party in the 2008 elections in Spain (Table 3). The D’Hondt rounding rule (Step 2) predicts 9.7 seats for the Portuguese Left Bloc and 151.6 seats for the Spanish People’s Party. My basic model predicts the actual seat allocation better than the proportionality model. However, I overestimate the representation of the small party (the Left Bloc) and underestimate that of the large party (People’s Party). With further adjustments (Steps 3–7), my model becomes less parsimonious, but now it exactly predicts 8.2 and 153.8 seats for the two parties. The only remaining difference is the inevitable need to round to an integer. For the small Left Bloc, the most important improvement to the prediction comes from uneven district magnitude (Step 5), whereas for the large People’s Party, all four dimensions of territorial variance (Steps 4–7) are important for the prediction.

Table 3.

The Substantial Effect for a Small and a Large Party: Real Results and Model Predictions

Table 3 Long description

The table compares real election results with model predictions for two political parties: Portugal's Left Bloc in 2011 and Spain's People's Party in 2008. It shows the percentage of votes, actual seats won, and representation ratios, alongside predicted seat counts and ratios across seven modeling steps. For the Left Bloc, real results show 8 seats with a representation ratio of 0.65, while predictions range from 5.65 to 12.22 seats, indicating varying levels of underrepresentation. For the People's Party, real results show 154 seats with a representation ratio of 1.10, and predictions range from 146.88 to 168.54 seats, suggesting slight overrepresentation. The modeling steps include proportional representation, D'Hondt method, threshold adjustments, malapportionment, districting, geographic factors, and competitor effects, each affecting seat predictions differently. The data highlights the complexities in predicting electoral outcomes and the impact of different electoral systems and methodologies on representation ratios.

Dataset and descriptives

I selected democratic parliamentary elections (first chamber) under the D’Hondt rule, for elections for which complete election results (votes and seats) are available at the constituency level, for all political parties, including smaller ones, which are omitted from many comparative compendiums of election results.Footnote ⁷ The CLEA dataset offers complete election results, which for most countries include both votes and seats, also for the smallest parties, as well as district-level information. Where necessary, I supplemented this data with information from national electoral commissions.

In some countries, the D’Hondt rule is used in conjunction with other elements, such as multiple tiers of seat allocation or legal thresholds. Technically, the model may also be applied to analyse the lower-tier seat allocation in multi-tier elections or to upper-house elections under PR. However, as the seat allocation is partially or entirely based on the upper-tier rules, the parties’ nomination strategies and the vote distributions will be influenced by the latter. Similarly, upper-house elections are contaminated by the lower-house election and election rules. In order to restrict the sample to the most similar and comparable cases, I only analyse lower house single-tier electoral systems. I incorporate legal thresholds into the model by setting the number of electorally viable votes in a district to zero for parties that do not cross a district-level legal threshold (Albania, Croatia, Spain) or a national legal threshold, including differentiated thresholds for coalitions (Czech Republic, Poland). For elections involving list apparentments, the parties running in apparentments and their votes are dealt with as a single entity, where possible.Footnote ⁸ There are practical difficulties in using national aggregated variables when the set of parties participating in apparentments varies across districts. Because of such incongruent apparentments, I cannot include Argentina or Switzerland. All elections, as well as the reasons for exclusion, are listed in the Supplementary Material, Section B.

The first visual display of the data (Figure 1, left panel) can best be described in one term: proportionality. The empirical match between parties’ vote shares and seat shares in D’Hondt elections is almost perfect, with almost all points falling in the area around the bisecting line, and a correlation coefficient of r = 0.99. However, non-PR electoral systems also lead to such a near-perfect correlation. It is due to large parties gaining more representation than small parties, and the correlation is largely insensitive to over- or under-representation. A more meaningful measure for our purpose is the ‘representation ratio’ (Taagepera and Laakso Reference Taagepera and Laakso1980). It is the ratio of seat shares to vote shares by party. As can be seen in the right part of Figure 1, the representation ratio reveals the variance that is barely visible in the left panel. The bisecting line now becomes a horizontal line, at y = 1. Values above 1 stand for the degree of overrepresentation, and values below 1 for underrepresentation. In particular, for parties below 15% of the vote, the representation ratio varies across parties. Large parties with more than 20% of the national vote tend to be overrepresented, in the range of 1–1.35, thus winning up to 35% more seats than their actual vote share. This variance is more pronounced for PR systems with low district magnitudes (<5 seats or 5–10 seats, national average). In my sample, low-magnitude cases such as these are found in Latin American countries, as well as Norway and Spain. By default, the proportionality model (Step 1) predicts a representation ratio of 1 for all data points. Thus, the explanatory power of my first step shrinks from 99% to 0%.

Figure 1.

Vote and Seat Distributions (Left) and Representation Ratios (Right)

The results vary somewhat from country to country: Colombia, Peru and Spain have many small regional political parties, some of which win a proportional or even above-proportional seat share (see Figure 2). The distribution is more even in Luxembourg, Finland and Suriname, all of which are among the countries with fewer districts (4, 14 and 10, respectively), and almost flat in countries with a single district (Moldova, the Netherlands, Israel, San Marino, Serbia and Timor-Leste).

Figure 2.

Representation Ratios by Country

Deviations from proportionality can also be associated with the number of competing parties. However, as the case of the Dominican Republic shows, the reallocation effect will be vastly overestimated if all parties – including those below the threshold of representation – are included in the count (see above, Step 3).

Problem: electoral districts with no electoral coordination

Before moving on to the systematic analysis, I identify a few peculiar parties that gain a tiny national vote share but are vastly overrepresented. Generally, representation ratios range between 0 and 1.3 and reach up to 1.5 for a few very small regional parties. Very few parties reach representation ratios of 7 or 8; these are all tiny, local parties, mostly from Colombia. In the 2006 elections, Afrounincca won one of the two seats in the special constituency for Afro-Colombians, with just 4.3% of the constituency electorate. With 27 parties running, the number of votes to win a seat in this two-seat constituency was very low. I exclude this party and 11 others with a regionally concentrated electorate running in constituencies with a hyper-fractionalised result from further calculations.Footnote ⁹

Validating the model

How well does my model explain the deviations from proportionality? And which of Steps 2–7 are relevant for this purpose? Figure 3 provides a visual comparison of the results, both for the basic model (Step 2, left panel) and for the fully developed model (all seven steps, right panel). The x-axis shows the predicted representation ratio, and the y-axis the actual representation ratio. In the left panel, many parties are located around the bisecting line, meaning that the basic D’Hondt rounding rule has some explanatory power, even when applied with no further additions. However, most of the other parties, especially the small parties, are far away. The empirical fit is much better for the full model (right panel).

Figure 3.

Representation Ratios, Predictions and Empirical Values: Basic D’Hondt Model and Full Model

To assess the contribution of these elements more systematically, I have run these models for all 2⁶ combinations of explanatory elements and calculated the explanatory power for each of these models. Since the different elements of the model affect the success of small or large parties in winning seats in parliament differently, I analyse two subsamples with parties below 5% and above 5% of the national vote. Small parties, with a vote share of no more than 5% of the national vote, make up the bulk of the data, accounting for 2,931 out of 3,641 cases, and thus dominate the results when the full sample is analysed.

Figure 4 plots the empirical explanation for all steps of the explanatory model. On the left-hand side of Figure 4, I distinguish between single-district elections and those with multiple districts. For the latter (panels on the right-hand side), I separate small parties (below 5% of the vote) and non-small parties. I use two different metrics of explanatory power. The first metric, the conventionally used R ², is shown in the top panel.Footnote ¹⁰ Elections in single-district systems are almost perfectly predicted by the proportionality rule and the D’Hondt rounding rule (Steps 1 and 2), which is unsurprising given that these elections involve very large districts, resulting in high proportionality. On the other hand, these elections lack variance in elements associated with uneven districts or electoral geography and are thus, by default, not affected by Steps 4–7. Multi-district elections are, therefore, key to explaining the results. For parties above 5% of the national vote, the full model provides 88% explanatory power in multi-district elections, with the D’Hondt rounding rule and average thresholds of exclusion explaining most of the variance (Steps 2 and 3, R ² of 0.40 and 0.53), followed by uneven district magnitude and electoral geography (Steps 5 and 6, R ² of 0.66 and 0.83). Conversely, malapportionment and the number of district competitors add little to the explanatory power.

Figure 4.

Fever Curve for R ² and Cumulative Squared Deviations

The empirical importance of each element of the solution is highly country-specific (see Supplementary Material, Section C, for results by country). This is a result of different institutional rules (district magnitude and its variance), of the party’s electoral geography, and of the context in which it competes (number of relevant competitors, wasted votes). In countries such as Colombia, Finland, Macedonia and Paraguay, because of the large number of small parties, the D’Hondt rounding rule alone, without taking into account thresholds of representation, massively overestimates the benefits for large parties.

I complement the R ² with a second measure of empirical explanatory power: the index of cumulative squared deviations (Figure 4, lower panel). The classic metric of explanatory power, R ², has its limitations when used to assess the quality of the model in subsamples with systematic deviations from the prediction. For example, the subsample R ² fails to detect the substantial overestimation of the large parties’ benefits from the D’Hondt rounding rule (Step 2) when this bias is systematic for the entire subsample. Instead, my second measure is the average deviation from the prediction, and it is calculated for the entire sample and for subsamples (e.g. by country or by party size). The results are broadly in line with the R ² for the full sample, except for Step 2.

The results for small parties (below 5% of the vote) differ from those for larger parties. The D’Hondt rounding rule itself does little to explain their level of representation. The average penalty (0.5 seats per constituency) is overestimated, even leading to negative seat predictions for 764 out of 3,642 data points (truncated in Figure 3), especially in countries with no legal thresholds and many constituencies. Thus, for small parties, the D’Hondt rounding rule should only be used when there is information about the electoral geography. When including Steps 3 and 6, the model has high explanatory power (R ² = 0.76 or R ² = 0.87 for the full model).Footnote ¹¹

Substantial effects (simulation)

In this section, I discuss the substantial effects of each of the steps of the model for parties of different sizes (multi-district elections only). For each of the seven elements of the model, I simulate how it changes the predicted electoral results of parties according to their size (Figure 5) and between countries (Figure 6).Footnote ¹²

Figure 5.

Representation Ratios for Each of the Seven Steps (First Row) and Marginal Effects (Second and Third Row), Parties by Size. Only Multi-district Elections Included

Notes: First row: Representation ratio, as the model is gradually extended from Step 1 (PR) to all seven steps. Second and third row: change in the number of seats (%, absolute), as new steps are added to the model.

Figure 6.

Marginal Effects by Countries (Only Parties with More Than 5% of the National Vote Share)

Note: Change in the number of seats (%), as new steps are added to the model.

The largest substantial effects are associated with the D’Hondt rounding effect (Step 2) and thresholds of representation, which partially reverse the effects of the rounding rule (Step 3). The rounding effect of the D’Hondt rule rewards large parties with more than 10% of the vote and has a negative effect on small parties with less than 10% of the vote. Conversely, when the average thresholds of representation are taken into account, the dividing line between net winners and net losers moves from 10% to 20%.

Malapportionment (Step 4) affects the representation of larger parties above 10%, although the effect is heterogeneous and can be either positive or negative. For small parties, uneven district magnitude (Step 5) and electoral geography (Step 6) are crucial, as they can positively affect small parties’ representation. However, the effect is highly heterogeneous within this category of parties.

Discussion: district magnitude and uneven districts

Surprisingly, the model presented in this articledoes not include district magnitude, the variable most frequently addressed to characterise the proportional effects of PR and other electoral systems (Rae et al. Reference Rae, Hanby and Loosemore1971; Taagepera and Shugart Reference Taagepera and Shugart2017). Nor does other similar work either (Schuster et al. Reference Schuster, Pukelsheim, Drton and Draper2003: 658).

One explanation for this is that several other variables are either correlated with or transformed from district magnitude. First, the rounding effect of the D’Hondt formula is multiplied by the number of districts, and district magnitude is the ratio of seats in parliament to the number of districts. The number of seats in parliament largely reflects a country’s population size (Taagepera Reference Taagepera1972). Thus, smaller district magnitudes result in more districts and amplify the rounding effect. Second, the threshold of representation, which determines which parties are viable competitors, is a transformation of district magnitude. These are direct (mechanical) effects of the seat allocation rule.

Furthermore, district magnitude and the prospect of winning a seat have behavioural implications, also known as the ‘psychological effect’. In the model, this is measured by the number of competitors N, which tends to be lower in low-magnitude districts. A lower N, in turn, increases the break-even point (the vote share determining above which parties become overrepresented).

My results also contrast common beliefs, suggesting that malapportionment plays a somewhat less important role compared to other variables associated with between-district variance. The method used in this article differs conceptually and methodologically from earlier work (Kedar et al. Reference Kedar, Harsgor and Tuttnauer2021; Monroe and Rose Reference Monroe and Rose2002) in that it separates malapportionment from related phenomena, namely the political geography of the party voters and uneven district magnitude. Both factors have a sizeable impact on the seat allocation, the latter in particular for mid-sized and larger parties in countries, even in elections without malapportionment. The method employed allows me to distinguish between the direct (mechanical) effect of malapportionment, i.e. the consequence of some districts being overrepresented, and other indirect (psychological) effects. Notably, unequal district magnitudes are associated with variance in the number of competitors. This, in turn, alters the size of the D’Hondt rounding effect.