Elections, Parties & Public Opinion: Rules That Turn Votes into Power

At the United Kingdom's general election on 4 July 2024, the Labour Party won 33.7% of the votes and 411 of the 650 seats, about 63% of the House of Commons. Reform UK won 14.3% of the votes and 5 seats, under 1%. Turnout was 59.7%. Nobody cheated and nobody miscounted. That is simply what the rule, one winner per constituency, does with votes.

In a country that uses proportional representation, the same votes would have produced a very different parliament. And in both cases, in the months before the vote, pollsters had tried to tell the country what would happen using a few thousand phone calls or online panels. This piece is about those two machines: the one that turns votes into seats, and the one that tries to measure votes before they are cast. Plus the oldest question of all: why people vote the way they do, or at all.

This describes how elections and polling work. It does not say which party or system is better; each electoral system trades something off, and every example is given to show a mechanism.

Seats are made by rules: plurality versus proportional

Every electoral system answers one question: how do votes become seats? There are three big families. In plurality systems (the United Kingdom, Canada, India, the United States for most offices), each district elects one person, and whoever gets the most votes wins, even with less than half. In proportional representation (PR) systems (the Netherlands, Spain, Brazil's lower house, South Africa, Israel and most of continental Europe), a party's share of seats follows its share of votes, using a formula. Mixed systems combine the two (Germany, New Zealand, Mexico).

Two PR formulas are common. The D'Hondt method (Victor d'Hondt, 1878; used for example in Spain and Portugal) gives seats one at a time to the party with the highest quotient votes ÷ (seats won + 1). Sainte-Laguë (André Sainte-Laguë, 1910; used for the German Bundestag and in New Zealand) divides by 1, 3, 5 and so on, which is a little kinder to small parties. A threshold (5% in Germany) excludes parties that fall below it.

Political scientists measure the damage a system does to proportionality with the Gallagher index (Michael Gallagher, 1991): the square root of half the sum of squared gaps between each party's vote share and seat share. Zero is a perfect match. They also count how many parties matter with the effective number of parties (Laakso and Taagepera, 1979): one divided by the sum of squared shares. Four equal parties give 4; one party with 90% gives about 1.2. Below, ten districts, four parties, and an invented set of votes show all three systems side by side.

At the default setting, party A has 33.5% of the votes and wins 7 of 10 seats under plurality, while party C, with 24.2% spread evenly, wins none. The PR systems give A 4 seats (D'Hondt) or 3 (Sainte-Laguë) and C 2 or 3. Slide the shift and watch a small change in votes swing several seats. That sensitivity is the mechanism behind the "landslide on a minority of votes" in the UK example. Raise the threshold to 15% and see D drop out. Real systems add many details (district size, open or closed lists, bonus seats), and each has its defenders: plurality gives clear single-party governments and a local member; PR gives fairer representation and more coalitions.

Duverger's law: why plurality squeezes parties

The French jurist and sociologist Maurice Duverger noticed, in Les partis politiques (1951), that countries with plurality elections in single-member districts tend to have two main parties, while PR tends toward many. He gave two reasons. A mechanical effect: a party that wins 15% of the votes everywhere wins no seats, as party C just did. And a psychological effect, which is the stronger one in modern work: voters see this and avoid "wasting" a vote on a party that cannot win, so they back their favourite of the two who can. The tendency is not a law; India and Canada have regional parties that win locally, and Duverger's logic works district by district rather than nationally.

You can see the psychological effect in a small model. Voters sit on a left-to-right line (more are near the middle). Candidate A stands at −1.0 and candidate B at 0.3. Candidate C's position is yours to set. Voters sincerely back the closest candidate. Then the voters of whichever candidate is last in the polls start to doubt: a share of them switch to their preferred of the other two. Try a plurality vote, and then a two-round runoff, in which the top two meet again in a second round.

With C at 0.8 and nobody switching, A wins with 36% under plurality, even though B, the candidate closest to the middle, would beat either rival in a head-to-head vote. Let half of C's voters switch, and B wins. A runoff produces the same outcome without anyone needing to guess: C's voters back B in round two. Move C to 0.5, next to B, and it is B that gets squeezed out. This is why parties that share a space worry about "splitting the vote", why many countries pair presidential elections with runoffs (France, Brazil, Argentina, Chile, Colombia), and why electoral rules change how parties form.

Why people vote as they do

Anthony Downs's An Economic Theory of Democracy (1957) gave the field its simplest model. If voters sit on a line and each backs the closer candidate, two vote-seeking candidates are pulled toward the position of the median voter, the voter with half the electorate on either side. Duncan Black proved the underlying result in 1948. Try it, and then add the realistic twist: a voter who feels too far from both candidates stays home.

Drag candidate 1 toward the middle: they gain, and the other candidate must follow. If both end at the same spot, the vote splits evenly. Then lower the "stays home" slider: the candidates at the extremes lose voters who prefer neither, which is why turnout and the pull toward the centre trade off. Real elections do not collapse to the middle, for many reasons: parties have activists and loyal supporters, voters care about more than one issue, and in multiparty systems each party aims at its own group.

Political scientists also ask what drives the vote in practice. The Michigan school's The American Voter (1960, Campbell, Converse, Miller and Stokes) found that party identification, a long-term emotional attachment often learned in the family, predicts the vote better than issue positions. Philip Converse (1964) found many voters held opinions that were not tightly connected across issues. Morris Fiorina (1981) emphasised retrospective voting: reward or punish the government for how things went. Christopher Achen and Larry Bartels (Democracy for Realists, 2016) argued that voters respond to events they should logically ignore, and cited a study suggesting that shark attacks on the New Jersey shore in 1916 cost Woodrow Wilson votes. That claim did not hold up: Anthony Fowler and Andrew Hall (2018) re-examined every fatal shark attack in the US and presidential returns from 1872 to 2012 and found little compelling evidence of an effect. Achen and Bartels have replied that their broader argument stands. It is a good example of why the field now checks striking findings against more data.

Turnout: why anyone votes at all

Downs raised an awkward problem. Voting costs time, and the chance that one vote decides an election is tiny. William Riker and Peter Ordeshook (1968) wrote the voter's calculation as R = pB − C + D: the chance of being decisive times the benefit, minus the cost, plus a sense of duty or expression (D). Without D, almost nobody would vote. A rough size: even in the most favourable case, an election dead-level on average, the chance that one vote is the tie is about √(2 ÷ (π N)) for N voters. For a million voters that is about 0.08%, or 1 in 1,250, and real elections are rarely so close, so the true number is far smaller.

Yet hundreds of millions vote. Turnout differs sharply between countries, and rules explain part of that. In Australia, voting at federal elections has been compulsory since 1924, and turnout has rarely fallen below 90%. Belgium, the oldest compulsory system (1893), has about 90%. Argentina adopted compulsory voting in 1912, Brazil in 1932, and Peru, Bolivia and Uruguay also enforce it. Fines, or small bureaucratic penalties such as the difficulty of getting a passport, are the enforcement. In the UK in 2024, with voluntary voting, turnout was 59.7%. Scholars debate whether compulsory voting changes results or only adds reluctant voters, and whether people grow into the habit.

Polling and its errors

A poll asks a few hundred or thousand people and infers the opinions of millions. The inference works because of a result you met in Statistics: if you draw a truly random sample of size n from a population in which a share p holds an opinion, your estimate will land within about 1.96 × √(p(1 − p) ÷ n) of the truth 95% of the time. For p = 50% and n = 1,000, that is ±3.1 points; for n = 4,000 it is ±1.55. Halving the margin costs four times the sample. And the margin on the lead between two candidates is roughly double.

The catch is the word "truly random". The classic failure is the Literary Digest poll of 1936, which mailed ballots to about 10 million people drawn from its own subscribers, telephone directories and car registrations, got 2.38 million back, and predicted a landslide win for Alf Landon. Franklin Roosevelt won with 60.8% of the vote. A huge sample did not help, because it was a biased one, and the people who strongly disliked Roosevelt were more likely to answer. A modern version is non-response: people who do not pick up the phone differ from those who do. The American Association for Public Opinion Research (AAPOR) task force on the 2020 US election found that national polls overstated Joe Biden's lead by about 3.9 points and state polls by 4.3, the largest national polling error in about 40 years, and could not pin the cause on a single factor (non-response was the leading suspect). The UK's 2015 polling inquiry, led by Patrick Sturgis, blamed unrepresentative samples too.

Simulate it. Each click draws 1,000 polls from a population where one candidate really has the share you choose. Then add a bias, the tilt that non-response introduces.

Leave the bias at zero: about 95% of polls land within the margin of error, and with a 52% truth and n = 1,000 around one poll in ten calls the wrong winner. Now press "Huge but biased": five thousand people per poll, bias of 4 points. The margin of error shrinks to under 1.4 points, yet almost no poll lands within it and nearly all call the wrong winner. Sampling error shrinks as samples grow; bias does not. Good pollsters weight their samples to match known population figures, and aggregators average many polls, but a shared bias passes through both.

Polarisation, measured

"Polarisation" is used for several different things. Ideological polarisation is how far apart people's positions on issues are. Affective polarisation is how much people dislike and distrust the other side. They need not move together. The most-cited cross-country study, by Levi Boxell, Matthew Gentzkow and Jesse Shapiro, measured affective polarisation (negative feeling toward opposing parties) in twelve OECD countries over four decades. The United States showed the largest rise; five other countries rose by less, and six fell. In other words, there is no worldwide trend that everyone shares, and the American pattern is the exception more than the rule, though the causes are debated.

Other measures exist: the V-Dem project, for example, publishes an expert-coded indicator of political polarisation for each country-year, and survey projects ask people how warmly they feel toward the parties. Different indices count different things, so check what an index measures before comparing countries. Researchers also disagree about causes: some point to the sorting of parties by identity, some to media, some to electoral rules (plurality tends to create two big opposing camps; PR, more cross-cutting coalitions), and none of these explains everything.

What a graduate just sees

• The electoral rule is part of the result. Ask how votes become seats before arguing about who "won".
• Plurality tends toward two parties (Duverger), and small shifts in votes can swing many seats.
• Proportional systems track votes closely, and the formula (D'Hondt, Sainte-Laguë) and threshold decide who is just in or out.
• A party that splits a camp can hand the seat to the other side; runoffs and ranked ballots exist to fix that.
• Candidates are pulled toward the median voter, and held away from it by activists and abstention.
• For one voter the chance of deciding is tiny, which is why turnout is partly duty and partly rules.
• A poll's margin of error covers only sampling luck. Bias, from non-response or an unrepresentative sample, is a separate and often bigger risk.
• "Polarisation" can mean disagreement or dislike, and measures differ across countries.

Where it shows up next

• Public Policy (35/16): coalitions, parties and veto players shape what gets proposed and passed.
• Research Methods in Political Science (35/18): sampling, bias and causal claims, the tools behind every poll and every "voters do X" finding.
• Probability & Statistics (34/05): margin of error, sample size and bias are the same ideas you meet there.
• In the world: campaign and polling firms, election administration and observation (the OSCE, the Carter Center), journalists who report polls, civil servants who must govern with a coalition, and anyone who has to read a result and a forecast sensibly.

If you want to go deeper

• 📘 Textbook. Michael Gallagher and Paul Mitchell (eds.), The Politics of Electoral Systems (Oxford University Press, 2005). Country-by-country coverage of how each system works.
• 📘 Textbook. Arend Lijphart, Patterns of Democracy (Yale University Press, 1999; 2nd edition 2012). Compares 36 democracies on electoral systems, party systems and executives.
• 📘 Gentler. Nate Silver, The Signal and the Noise (Penguin, 2012), a readable account of forecasting, including election forecasts and why they miss; and Achen and Bartels, Democracy for Realists (Princeton, 2016), a harder read but written for general audiences, for the case that voters are less issue-driven than we assume.
• 🌐 Free course. MIT OpenCourseWare 17.50 Introduction to Comparative Politics covers electoral systems, party systems and voting alongside regimes. For electoral systems in particular, the ACE Electoral Knowledge Network (run by International IDEA and other partners) has a free encyclopaedia with an Electoral Systems section.