There is a comfortable belief that assumes a democratic election works like a transparent mirror. Citizens vote, the ballots are counted, and parliament faithfully reflects what the people want.
Kenneth Arrow, a mathematical economist, shattered that illusion in 1951 with his impossibility theorem [1]. He formally proved that no ordinal voting system can simultaneously satisfy a basic set of conditions for electoral fairness: no dictatorship, the party preferred by the electorate as a whole governs (Pareto efficiency), independence of irrelevant alternatives, inclusion of all preferences (unrestricted domain), and transitivity. In other words, as we already suspected, it is impossible to keep the entire electorate happy or to be fair to the preferences of every voter.
Proportional or majoritarian
It helps to start with the basics, because electoral jargon confuses even political scientists. A majoritarian system is one where the candidate with the most votes in a constituency takes the entire seat. It doesn’t matter whether they got 51% or 26%: the second, third and fourth place finishers go home with zero representation. The most famous model is the British and American First Past the Post: single-member districts, one winner per district.
A proportional system, by contrast, distributes seats according to the percentage of votes each list obtains in a larger constituency. If you get 30%, you win roughly 30% of the representatives. Not exactly, because allocation methods (D’Hondt, Sainte-Laguë, Hare) introduce small distortions, but that is the logic.
Maurice Duverger, a French jurist, formulated in 1951 what we now know as Duverger’s Law: single-member majoritarian systems tend to produce two-party systems, while proportional systems tend to produce multiparty systems [2]. The mechanism is twofold. On one hand, the mechanical effect: small parties don’t win districts, so they don’t get seats. On the other, the psychological effect: voters, knowing that a vote for a minor party is “wasted,” vote strategically for the lesser evil among the two big ones.
The cube rule
In two-party majoritarian systems, the relationship between votes and seats is not linear. It is cubic. Kendall and Stuart demonstrated this empirically in 1950 by analyzing British elections [3]. If party A gets 55% of the votes and B gets 45%, the vote ratio is 1.22 to 1. But the seat ratio shoots up to approximately 1.82 to 1, meaning A gets about 65% of the seats with only 55% of the vote. Majoritarian systems don’t seek to reflect public opinion: they are institutional machines designed to manufacture artificial absolute majorities at the expense of compressing the plurality of the popular vote, with the aim of facilitating governability.
The Anglo-Saxon laboratory: the United States and the United Kingdom
In the United States, the majoritarian system is combined with a practice that has the name of a mythological animal: gerrymandering. The term dates from 1812, when Massachusetts Governor Elbridge Gerry redrew a district in the shape of a salamander to favor his party. A journalist at the Boston Gazette drew the creature and christened it the “Gerry-mander.” Two centuries later, the salamander is still alive and kicking.
In the 2005 British election, Tony Blair’s Labour Party won 55.2% of the seats in the House of Commons with barely 35.2% of the popular vote [5]. A party with little more than a third of public support obtained an absolute majority to legislate. This is the system working exactly as designed.
Unintentional gerrymandering
Geographic bias doesn’t always come from deliberate map manipulation. Jowei Chen and Jonathan Rodden published a crucial finding in 2013: unintentional spatial bias [6]. Using automated computer simulations that generated thousands of random electoral maps free of partisan intent, they showed that even a completely blind algorithm drawing compact districts will naturally generate a bias favoring conservatives in majoritarian systems. The reason is purely geographic: progressive voters tend to cluster spontaneously in dense metropolitan cores, while conservative voters exhibit a much more homogeneous spatial distribution across suburbs and rural areas. Urban voters end up packed together by their own residential choices, inflicting on themselves an enormous territorial vote inefficiency.
Proportional examples: Scandinavia and Germany
At the other extreme are the Nordic countries. Sweden, Norway, Denmark and Finland use proportional systems with large multi-member constituencies and low thresholds (between 2% and 4%). The result is an almost millimetric correspondence between votes and seats. In 1991, Michael Gallagher designed an index bearing his name to measure disproportionality: the aggregated difference between the percentage of votes and the percentage of seats across all parties [7]. The Nordic countries consistently score below 2 on this index. The United Kingdom hovers around 15–20. It is the distance between a parliament that resembles its society and one that doesn’t.
Germany offers a fascinating hybrid that has just undergone a major reform. Each voter has two ballots: one for a district candidate (majoritarian) and one for a party list (proportional). The second ballot determines how many total seats each party receives. Historically, the Bundestag had a minimum of 598 members but could grow significantly because of overhang seats (Überhangmandate) and their compensations (Ausgleichsmandate), reaching 736 members in 2021. In 2023, Germany passed an electoral reform that eliminated that compensation mechanism and fixed the size of parliament at exactly 630 seats, applied for the first time in the 2025 elections [8]. The Federal Constitutional Court upheld almost the entire reform in July 2024, although it partially struck down the way the 5% threshold was to be applied, forcing the retention of a safeguard clause for parties with strong territorial support [8]. Not even the most carefully crafted design escapes constant litigation. This is not a flaw of the German system: it is what happens when you take proportionality seriously. There is a 5% threshold to enter the allocation, which filters out extreme fragmentation without preventing plurality. Lijphart, in his comparative analysis of 36 democracies, classifies Germany as an intermediate case that combines governmental stability with broad representation [9]. It works. It has been working since 1949, although now with stricter rules on the final size of parliament.
Spain: the most majoritarian of the proportional systems
The Spanish system is a laboratory curiosity. On paper, it uses the D’Hondt method with closed, blocked lists in provincial constituencies, with a legal threshold of 3%. It sounds reasonably proportional. But there is a structural trick that almost no one explains well: the constituency is the province, and some provinces elect only 2 or 3 deputies.
Penadés and Santiuste, in their 2013 study on inequality in the Spanish electoral system, showed that the system is not uniform but works as three distinct subsystems depending on district magnitude [10]. Small constituencies (1–5 seats), which number 27 districts and elect 99 deputies, function as majoritarian: high disproportionality and two-party dominance. Large ones (10+ seats), which are only 7 districts but elect 127 deputies, function as proportional: low disproportionality and multipartism. Intermediate ones (6–9 seats), with 18 districts and 124 seats, show mixed behavior. In small districts, the two largest parties historically won 79% of the votes and 93.5% of the seats. In large ones, the figures dropped to 71.2% and 77.9% [10].
The combined effect of territorial fragmentation and the D’Hondt method produces a disproportionality that benefits the two major parties and nationalists with concentrated votes, while punishing formations with support dispersed across the territory. In 2015 and 2016, with the emergence of Podemos and Ciudadanos, Spain’s Gallagher index rose above 7, one of the highest in Western Europe in that period [7]. Not because Spaniards voted strangely, but because the system was not designed to absorb four parties of similar size.
A curious fact: Madrid needs about 182,000 inhabitants per seat; Soria, about 44,000. A Soria voter’s ballot weighs four times more than a Madrid voter’s [10].
Vote location
Penadés and Santiuste identified a second mechanism of advantage that accumulates on top of the majoritarian effect: the premium for vote location [10]. Small constituencies not only function as majoritarian, but are also overrepresented in the allocation of seats. Large ones are proportional but underrepresented. Parties whose votes are concentrated in small, overrepresented areas get a double advantage: the majoritarian bias plus the apportionment bias. Historically, this mechanism has benefited the PP (and previously UCD) because their vote correlates negatively with district magnitude: they win more votes in small provinces than in large ones [10]. The PSOE, on the other hand, wins in the most proportional and underrepresented subsystem and loses in the most majoritarian and overrepresented one, so its victories are amplified little while its defeats are magnified [10].
The example Penadés and Santiuste use is damning: in Castilla y León, with 2.1 million registered voters (6% of the electoral roll), 32 deputies are elected (9.1% of parliament). The weight of the vote is 51% above the ideal of equal votes. If the whole region were turned into a single proportional district of 32 seats, the PP would lose two deputies, the PSOE one, and UPyD and IU would gain representation. But furthermore, if the overrepresentation were corrected by reducing the seats to the 21 that would correspond by population, the PP would go from 21 to 13 deputies and the PSOE from 11 to 6 [10].
The New Zealand experiment
In 1993, New Zealand did something almost no other country has done: it changed its electoral system by referendum. Until then it had used the British First Past the Post. A Royal Commission in 1986 recommended the German mixed proportional system, and after a second referendum in 1993, the first election under the new MMP (Mixed Member Proportional) system was held in 1996 [11].
The effective number of parties in parliament rose from 2.1 to 3.5 in a single legislature. Women’s representation went from 21% to 29% in the first MMP election. The number of Māori deputies grew. No party has since won an absolute majority with less than 45% of the vote. New Zealanders did not become more progressive or more diverse overnight. Parliament simply started to look more like the country. And in 2011, when they were asked whether they wanted to go back to the old system, 57% voted no [11].
Singapore: how to force hegemony
Singapore illustrates the extreme use of institutional engineering to shield electoral hegemonies through Group Representation Constituencies (GRCs). In these macro-constituencies, citizens don’t elect an individual deputy but closed teams of three to six members of parliament under a simple plurality rule in which the most-voted list wins all the seats in the district [12]. In the 2011 elections, this mechanism allowed the incumbent People’s Action Party (PAP) to convert 60.1% of the popular vote into 81 of the 87 available seats (93% of parliament), almost completely nullifying the representation of an opposition that had harvested four out of every ten votes in the country [13].
Possible modifications
If Spain were to replace its 52 provincial constituencies with a single national constituency of 350 seats using the Sainte-Laguë method, the party system would restructure itself within two or three electoral cycles. Peripheral nationalist parties would lose their veto power as coalition partners, because their structural advantage (votes concentrated in a few provinces) would disappear. In exchange, formations with 8–10% of dispersed votes would no longer be underrepresented. This is not a prediction; it is an extrapolation based on the simulations that outlets like elDiario.es have run with the results of the July 2023 elections, where with a single constituency the PP would have won 16 fewer seats and parties like Sumar would have gained representation [17]. It would be a clear step forward if what we want is for deputies to represent citizens’ votes.
And moving on to the American empire: if the United States adopted ranked-choice voting at the federal level, the two-party system would erode within about three or four legislatures. It wouldn’t disappear, because political culture and campaign financing have enormous inertia, but Duverger’s psychological effect would weaken by allowing voters to express ordinal preferences without fear of “wasting” their vote. Australia, which has used this system since 1918, has a functional multipartism with two large blocs and several pivot parties [18]. The hypothesis is that the shift would be similar, though slower given the size and polarization of the United States.
What is this good for: a practical guide
Understanding the electoral system has concrete uses that go beyond curiosity.
If you are a voter, knowing how your constituency works lets you calculate whether your vote carries real weight or is statistically irrelevant. In a safe district where one party always wins by 20 points, your vote changes nothing. In a marginal district, it moves a seat. Party campaign strategy is designed on exactly that logic: they don’t ask you for your vote if you live in a stronghold. They ask the undecided voter in district number 47.
If you are interested in civic participation or activism, knowing the barriers to entry explains why certain parties don’t take off even when they have social support. It’s not that people don’t want them; the mathematical formula demands 25% in Soria for them to win a single deputy. That guides where to apply pressure: it is not enough to ask for “more democracy”; you have to ask for larger constituencies, lower thresholds, or less biased allocation methods.
If you work in journalism or communications, understanding the Gallagher index or the efficiency gap lets you put electoral results in context without repeating the winning party’s narrative as if it were a divine mandate. “It won with 35% of the vote” and “it obtained an absolute majority” are two sentences that describe different realities.
Now then: is there any combination of rules that maximizes both representativeness and the capacity to govern, or are they objectives that trade off against each other in any possible design? Ultra-proportional systems like the Dutch one produce the longest coalition negotiations on the continent: with fifteen parties in parliament, assembling a majority is an exercise in patience, not simple arithmetic. Maximum representativeness and maximum difficulty in governing tend to travel together. Lijphart believes that consensus democracies match or slightly outperform majoritarian ones in economic management, and clearly outperform them in the representation of women and minorities [9]. But later reviews, with additional statistical controls, found that several of those “quality of government” differences lose significance depending on which other variables are included. Here there is real and active debate. No one has settled it yet.
What the evidence says
Comparative research is quite clear on one point that converges from multiple methodologies: proportional systems produce higher electoral turnout, greater representation of women and minorities, and a more faithful correspondence between citizens’ preferences and the composition of parliament [7][9][19]. Blais and Carty, in an analysis of 132 elections in 20 countries, found that turnout is between 5 and 8 percentage points higher under proportional systems [19]. This does not mean that proportionality is “better” in an absolute moral sense, nor that the governments that emerge from such elections are better.
REFERENCES
[1] Arrow, K. J. (1951). Social Choice and Individual Values. John Wiley & Sons. Label: Theoretical framework. Formal mathematical proof in decision theory that no ordinal voting system can simultaneously satisfy basic conditions of fairness. Founded the field of social choice theory. Methodology: Axiomatic modeling and formal mathematical proof in abstract microeconomics.
[2] Duverger, M. (1951). Les partis politiques. Librairie Armand Colin, Paris. Label: Theoretical framework. Foundational work of electoral sociology. “Duverger’s Law” has been empirically confirmed in dozens of later studies, although with nuances (the psychological effect is more variable than the mechanical one). Methodology: Qualitative historical-comparative and empirical analysis of party systems and electoral laws in Western democracies.
[3] Kendall, M. G. & Stuart, A. (1950). “The Law of the Cubic Proportion in Election Results.” The British Journal of Sociology, 1(3), 183–196. Label: Reliable. Methodology: Statistical modeling and quantitative empirical analysis of historical series of UK general elections. Link: https://www.jstor.org/stable/588113
[4] Stephanopoulos, N. O. & McGhee, E. M. (2015). “Partisan Gerrymandering and the Efficiency Gap.” University of Chicago Law Review, 82(2), 831–900. Label: Reliable. Methodology: Quantitative analysis of electoral results in all U.S. states between 1972 and 2012, design of the efficiency gap metric, and validation against Monte Carlo simulations with thousands of random electoral maps. Fun fact: the metric was cited in the case Gill v. Whitford before the Supreme Court in 2018, although the court avoided ruling on the merits due to standing issues. Link: https://www.jstor.org/stable/23884463
[5] Official data from the 2005 British general election, House of Commons Library, Research Paper 05/33. Label: Reliable. Official electoral data from the UK Parliament. Link: https://researchbriefings.files.parliament.uk/documents/RP05-33/RP05-33.pdf
[6] Chen, J. & Rodden, J. (2013). “Unintentional Gerrymandering: Political Geography and Electoral Bias in Legislatures.” Quarterly Journal of Political Science, 8(3), 239–269. Label: Reliable. Methodology: Automated computer simulation based on geographic information systems (GIS) to generate thousands of random electoral maps free of partisan bias and isolate the effect of the spatial concentration of voters. Link: http://dx.doi.org/10.1561/100.00012033
[7] Gallagher, M. (1991). “Proportionality, Disproportionality and Electoral Systems.” Electoral Studies, 10(1), 33–51. Label: Reliable. Methodology: Construction of a disproportionality index (the Gallagher index) applied to electoral results from 21 democracies. The index has become the standard measure in the comparative literature and has been replicated and used in hundreds of later studies. Link: https://doi.org/10.1016/0261-3794(91)90004-W
[8] Bundesverfassungsgericht (July 30, 2024), ruling on the 2023 electoral reform; Bundesministerium des Innern, Elections to the German Bundestag. Label: Institutional source. Methodology: Ruling of the German Federal Constitutional Court and official documentation from the Ministry of the Interior on the 2023 electoral reform that fixed the Bundestag at 630 seats.
[9] Lijphart, A. (2012). Patterns of Democracy: Government Forms and Performance in Thirty-Six Countries (2nd ed.). Yale University Press. Label: With reservations. Methodology: Study of 36 democracies with quantitative indicators of institutional performance. The part on the representation of women/minorities is more robust than the part on “quality of government,” which loses strength depending on the statistical controls used in later reviews. Link: https://yalebooks.yale.edu/book/9780300172027/patterns-of-democracy/
[10] Penadés, A. & Santiuste, S. (2013). “La desigualdad en el sistema electoral español y el premio a la localización del voto” (Inequality in the Spanish electoral system and the premium for vote location). Revista Española de Ciencia Política, 32, 89–116. Label: Reliable. Methodology: Quantitative analysis of Spanish electoral results (1977–2011) classifying constituencies into three subsystems by district magnitude and measuring differential effects on fragmentation, proportionality, and partisan bias through Gallagher and Loosemore-Hanby disproportionality indices, proportionality profiles, and advantage rates adjusted for vote weight. Link: https://dialnet.unirioja.es/descarga/articulo/4358911.pdf
[11] Royal Commission on the Electoral System (1986). Towards a Better Democracy: Report of the Royal Commission on the Electoral System. Government Printer, Wellington, New Zealand. And 2011 referendum data, Electoral Commission of New Zealand. Label: Reliable. Official report and official electoral data. The change to MMP was implemented after two binding referendums (1993, and confirmation in 2011). Link: https://www.elections.org.nz/dmsdocument/95-royal-commission-on-the-electoral-system-report-1986
[12] Elections Department Singapore (ELD). Types of Electoral Divisions: Group Representation Constituencies. Label: Institutional source. Official documentation of the Singaporean electoral system. Link: https://www.eld.gov.sg/elections_type_electoral.html
[13] Elections Department Singapore (2011). 2011 Parliamentary General Election Results. Label: Institutional source. Official electoral results data. Link: https://www.eld.gov.sg/elections_past_parliamentary2011.html
[14] Christensen, R. V. & Johnson, P. E. (1995). “Toward a context-sensitive approach to electoral systems: The Japanese case.” Comparative Political Studies, 28(3), 395–423. Label: Reliable. Methodology: Quantitative and qualitative case study on the impact of institutional change on Japanese politics after the 1994 electoral reform. Link: https://doi.org/10.1177/0010414095028003003
[15] Norris, P. (2004). Electoral Engineering: Voting Rules and Political Behavior. Cambridge University Press. Label: Theoretical framework. Comparative analysis of 191 elections in 72 countries using data from the Comparative Study of Electoral Systems. Includes analysis of parallel and mixed systems. Link: https://doi.org/10.1017/CBO9780511790973
[16] Apportionment Paradox, U.S. Census Bureau / OpenStax Contemporary Mathematics. Label: Historical source. Documentation of the 1880 incident in the context of the census and the apportionment of House of Representatives seats. Link: https://openstax.org/books/contemporary-mathematics/pages/11-5-fairness-in-apportionment-methods
[17] elDiario.es (2023). “Así quedaría el Congreso con circunscripción única” (This is how Congress would look with a single constituency). Simulation using the results of the July 23 elections. Label: Recent. Journalistic, not academic, simulation, but based on official data from the Ministry of the Interior. Link: https://www.eldiario.es/politica/asi-quedaria-congreso-ley-electoral-circonscripcion-unica_1_10377829.html
[18] Norris, P. (2004). Op. cit. [15]. Section on Australian preferential voting. Label: Theoretical framework.
[19] Blais, A. & Carty, R. K. (1990). “Does Proportional Representation Foster Turnout? A Cross-National Reanalysis.” European Journal of Political Research, 18(6), 679–690. Label: Reliable. Methodology: Regression analysis of 132 national elections in 20 Western democracies between 1960 and 1982, controlling for socioeconomic variables. The finding of 5–8 points higher turnout under proportional systems has been replicated in later studies with expanded samples. Link: https://doi.org/10.1111/j.1475-6765.1990.tb00251.x
Note on sources: All the references cited support the central conclusion of the text: electoral design produces measurable and distinct effects on representation, participation, and party structure. There is no real academic debate over whether majoritarian systems generate greater disproportionality than proportional ones; this is well established through multiple methods (disproportionality indices, counterfactual simulations, comparative analysis, natural experiments such as the New Zealand one). The only points where legitimate discussion exists are the exact size of the effect on turnout, which some more recent studies with additional controls reduce to 3–5 points instead of the original 7–8 from Blais and Carty, and Lijphart’s section on “quality of government,” which loses significance depending on the controls used. The direction of the effect is not questioned in either case.