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Week 11 · Chapter

Chapter 11 — Political Participation: Parties, Elections & Voting Systems

Introduction to Political Science Free sample edition

The module's primary reading · ~45–55 minutes · Objective 6 (explain American government and political participation — federalism, separation of powers, and the U.S. Constitution as one case of the survey's concepts; parties, elections, and voting systems and their effects; public opinion, polling, political behavior, and the media — reading election and polling data for what it documents and what it does not settle), Objective 3 (compare the major political ideologies and normative theories evenhandedly — liberalism, conservatism, socialism and its family, and the smaller families; justice, liberty, equality, and rights; the harm principle; Rawls and Nozick — stating each position in its strongest form and distinguishing empirical from normative claims) · Read it first this week; the lecture, the slides, the tutorial, the quiz, and the workshop all build on it.

In plain language

This week is about how votes turn into power. A political party (an organized group that competes under one label to win office) asks for your vote, and an electoral system (the rules that turn votes into seats) decides what your vote does. You will meet four families of rules, from "the most votes wins" to "seats should match votes", and you will see the same ballots produce different winners under different rules. Then you will read a real election, the 2024 UK general election, for what it shows and what it does not settle. No one here will tell you which rule is fairest; you will learn to argue it well.

Why you already care about this

Picture a student club with 30 members choosing a president. Three people run. When the ballots are counted, the votes come in 12, 10, and 8. The candidate with 12 wins, and the club moves on.

But stop and look at that result. The winner has 12 of 30 votes, which is 40 percent. Most members, 18 of them, voted for someone else. Under one counting rule the person with 12 votes is president. Under another rule the club would hold a second round between the top two, and the 8 people whose candidate came last would decide it. Same club, same ballots, and possibly a different president.

Every country faces that choice, on a much larger scale. In the 2024 UK general election, one party won 33.7 percent of the vote and 411 of the 650 seats in the House of Commons. Another party won 14.3 percent of the vote and 5 seats. Those are not errors in the count. They are what the counting rule does, and this chapter shows you exactly how.

By the end of this chapter you will be able to:

tell a plurality from a majority and compute both for a small election;
name the three jobs parties do and describe a party system without judging it;
name the family of an electoral system by asking how a winner is decided;
allocate seats with the D'Hondt method and compare the result with plurality rules;
state Duverger's law as a tendency, with its mechanism and its exceptions;
read an election result for what it documents and what it does not settle;
state the strongest case for first-past-the-post and for proportional representation.

Carry this line with you: the rule decides the winner — not only the votes.

Where we are

Already covered: Week 5 showed that free, fair, competitive elections are the floor of democracy. Week 7 built parliamentary and presidential governments, and named coalition government as normal. Week 8 gave you the three-layer read. Week 10 studied the U.S. case.
This week: parties and their jobs, party systems, the four electoral-system families, the D'Hondt method, Duverger's law, and your first real election data.
How they connect: Week 7 said a parliamentary government needs a majority behind it. This week shows how the electoral system shapes whether any one party gets that majority.
Where we're going next: Week 12 turns from seats to opinions: how a poll of a thousand people can speak for a whole country, and what its margin of error does and does not cover.

Week 1 taught you to sort a claim as empirical (checkable against evidence) or normative (argued from values). You will need that sort constantly this week, because election results are facts and their fairness is a judgment. Week 8 added the three-layer read. The design layer is what the rules say. The practice layer is what actually happens. The verdict layer is what ought to happen. An electoral rule sits on the design layer, a result sits on the practice layer, and "this rule is unfair" sits on the verdict layer.

1. Count to a winner: plurality versus majority

Go back to the club. Every number in it is illustrative, invented for teaching. Three candidates got 12, 10, and 8 votes from 30 members.

Two words describe two very different kinds of win.

A plurality is the largest number of votes among the candidates or parties, whether or not it is more than half of all the votes cast.
A majority is more than half of the votes cast or of the seats available.

Worked example — the club count (illustrative). Check the result step by step.

Add the votes. 12 + 10 + 8 = 30 votes, one from each member.
Find the winner's share. 12 ÷ 30 × 100 = 40 percent.
Find half of the club. 30 ÷ 2 = 15.
Find a majority. A majority means more than half, so it needs 15 + 1 = 16 votes.

In words: the winner has a plurality, the most votes, but not a majority. Under a rule that requires a majority, the top two, with 12 and 10 votes, would meet in a second round.

Why does this matter? Because a club, a city, and a country all have to pick a rule before anyone votes. A "most votes wins" rule gives this club its president on the first count. A "more than half wins" rule sends the members back to vote again. Neither rule is a mistake; they answer the question "who should win?" in different ways.

Misconception check. Students often use "plurality" and "majority" as if they meant the same thing. That is tempting, because in everyday talk "most of the votes" can mean either one. But a plurality is the biggest pile, and a majority is more than half. The club's winner had the biggest pile, 12 votes, and fell short of a majority by 16 − 12 = 4 votes.

Memory hook: plurality is the most; majority is more than half.

2. Meet the parties: three jobs and the party system

Think about a group project with eight people and one deadline. If everyone pushes their own idea, nothing gets done. Usually two or three people pull the ideas into a plan, someone gets the others to show up, and when it goes badly the whole group can see whose plan it was. Parties do those same jobs for a whole country.

A political party is an organized group that competes under a common label to win public office and so shape government and policy.

Parties do three jobs.

Aggregation is the party function of gathering many separate interests and demands into one program that voters can support.
Mobilization is the party function of getting people to register, turn out to vote, and take part in campaigns and public life.
Accountability is the party function of giving voters a known label whose record in office they can reward or punish at the next election.

Worked example — which job is it? (illustrative) Sort each scene from a made-up country's campaign.

A party writes one platform that joins farmers' demands on water with city renters' demands on housing. That is aggregation: two interests become one program.
Volunteers drive older voters to the polls on election day. That is mobilization: people are helped to take part.
Voters remove the governing party after a failed budget, because its label was on the plan. That is accountability: a known label is rewarded or punished.

When you step back and look at a whole country, you can describe its party system: the usual pattern of competition among its parties.

In a two-party system, two large parties win nearly all the seats and take turns forming the government.
In a multiparty system, three or more parties win significant shares of seats, so governments are often coalitions.
In a dominant-party system, several parties compete in elections, but one party wins office again and again over a long period.

Each of these is a description of an equilibrium, a pattern that holds steady for a while. None is a verdict. A two-party system is not "better" or "worse" than a multiparty system by definition; that is a normative argument, and people make it both ways.

Misconception check. Students often hear "dominant-party system" and assume the country is a dictatorship. That is tempting, because one party keeps winning. But the label describes a pattern of competition, not a regime type. Whether the elections are free and fair is a separate question, the one Week 5 taught you to ask about electoral democracy.

Memory hook: gather, mobilize, answer for it.

3. Sort the four electoral-system families

An electoral system is the set of rules that turns the votes cast in an election into seats or offices won. Political scientists sort the world's many systems into a few families. These families are a classroom model, a simplification you can hold in one hand; real systems blend and adjust them. You will use four.

Plurality, or first-past-the-post

Picture a race where the first runner past the finish line wins, however close the others are. First-past-the-post is a plurality electoral system in which each single-member district elects the one candidate with the most votes, majority or not. A single-member district is an electoral district that elects exactly one representative.

Proponents argue it is simple to use and to count, that it ties every member to one place, and that it tends to produce single-party governments that voters can hold to account. Critics respond that many votes elect no one, and that a party whose support is spread thinly across the country wins far fewer seats than its vote share.

Majority-runoff

Majority-runoff is an electoral system that holds a second round between the two leading candidates whenever no candidate wins more than 50 percent in the first round. The club in Section 1 would have used it if its bylaws demanded a majority.

Proponents argue that the winner ends with majority support in the final round, so no one wins with 40 percent. Critics respond that it asks voters to come back a second time, and that it still squeezes smaller parties out of the final round.

Proportional representation

Proportional representation is a family of electoral systems that gives each party a share of seats roughly in line with its share of the vote. Most versions use a party list: a party's ranked list of candidates, from which its seats are filled in order. Most also set an electoral threshold: a minimum share of the vote a party must reach before it can win any seats.

Proponents argue that seats track votes, so a party with a fifth of the votes gets about a fifth of the seats, and smaller groups get a voice. Critics respond that it weakens the tie between a voter and a local member, and that it makes coalition government the usual result.

Mixed-member proportional

Mixed-member proportional is an electoral system in which each voter casts a district vote and a party-list vote, and list seats top up each party's total toward its share of the list vote.

Proponents argue it keeps a local member for every district and still makes the chamber track the votes. Critics respond that it is complex to explain, and that it creates two kinds of members, some elected in districts and some from lists.

Misconception check. Students often treat these four families as the only systems in the world, each with a fixed form. That is tempting, because a neat list feels complete. But the families are a classroom model. Real countries change thresholds, district sizes, and counting details, and some systems sit between families. Use the families to ask the right question, not to file every country forever.

Memory hook: most votes, more than half, match the votes, mix the two.

4. Name the family: four made-up countries

The fastest way to name a system is to ask one question: how is a winner decided? Tavrenia, Halvenor, Lunmara, and Vessaly are invented countries, and every number in them is illustrative.

Worked example — name the family (illustrative). Read each made-up rulebook and ask how a winner is decided.

Tavrenia has 100 single-member districts, and in each one the candidate with the most votes wins, majority or not. In one district the winner has 38 percent. How is a winner decided? By the most votes. Family: plurality, or first-past-the-post.
Halvenor elects a president who needs more than 50 percent. The first round splits 41, 35, and 24 percent. Check: 41 + 35 + 24 = 100. No one clears 50, so the top two, at 41 and 35, meet in a second round. Family: majority-runoff.
Lunmara shares out 120 seats from party lists in proportion to the vote, with a 5 percent threshold. A party with 4 percent of the vote wins no seats, because 4 is below 5. Family: proportional representation.
Vessaly gives each voter a district vote and a list vote for a 120-seat chamber, with 60 district seats and 60 list seats. Check: 60 + 60 = 120. Party X wins 40 district seats and 35 percent of the list vote. Its proportional total is 35 ÷ 100 × 120 = 42 seats. So it receives 42 − 40 = 2 list seats. Family: mixed-member proportional.

Tally: 1 plurality + 1 majority-runoff + 1 proportional + 1 mixed-member proportional = 4 countries.

Look at Vessaly again, because it is the hardest. Party X won many district seats, so it needed only a small top-up. A party that won few districts but a large list vote would get many list seats. The list seats work like a correction: they pull each party's total toward its share of the list vote.

Misconception check. Students often call any system with districts "first-past-the-post". That is tempting, because Vessaly has districts too. But Vessaly's list seats top up each party toward its list share, so the overall result tracks the list vote. Ask how the final chamber is decided, not whether districts exist.

5. Work the D'Hondt table

Imagine four friends splitting eight slices of pizza in proportion to how much each chipped in. You cannot cut a person in half, so you need a fair rule for whole slices. Proportional representation has the same problem: seats come in whole numbers. One common rule is the D'Hondt method.

The D'Hondt method is a seat-allocation rule that divides each party's votes by 1, 2, 3, and so on, pools every result, and awards the seats to the largest results until the seats run out.
A quotient is the result of dividing one party's votes by a D'Hondt divisor.

Worked example — the D'Hondt table (illustrative). Four parties share 100,000 votes in one eight-seat district: Party A 48,000; Party B 30,000; Party C 14,000; Party D 8,000. Check the total: 48,000 + 30,000 + 14,000 + 8,000 = 100,000 votes.

Step 1, generate every party's quotients first. Divide each party's votes by 1, 2, 3, 4, and 5.

D'Hondt quotients for four illustrative parties sharing 100,000 votes in an eight-seat district
Divisor	Party A	Party B	Party C	Party D
÷1	48,000	30,000	14,000	8,000
÷2	24,000	15,000	7,000	4,000
÷3	16,000	10,000	—	—
÷4	12,000	7,500	—	—
÷5	9,600	—	—	—

The dashes mark quotients too small to matter: Party C's 14,000 ÷ 3 and Party D's 8,000 ÷ 3 fall below every quotient that wins a seat.

Step 2, pool the quotients and rank the eight largest: 48,000 (A), 30,000 (B), 24,000 (A), 16,000 (A), 15,000 (B), 14,000 (C), 12,000 (A), 10,000 (B). The ninth, 9,600 (A), wins nothing, because the seats have run out.

Step 3, count the seats: A 4, B 3, C 1, D 0. Check: 4 + 3 + 1 + 0 = 8.

Step 4, compare vote shares with seat shares. Party A has 48 percent of the votes and 4 ÷ 8 × 100 = 50 percent of the seats. Party B has 30 percent of the votes and 3 ÷ 8 × 100 = 37.5 percent of the seats. Party C has 14 percent of the votes and 1 ÷ 8 × 100 = 12.5 percent of the seats. Party D has 8 percent of the votes and 0 percent of the seats.

In words: the seats track the votes closely, though not perfectly. With only eight seats, whole numbers force some rounding, and the larger parties come out slightly ahead.

Now run the plurality contrast. Split the same 100,000 votes into eight single-member districts, and suppose Party A leads in every one. Under first-past-the-post, Party A wins 8 of 8 seats, which is 8 ÷ 8 × 100 = 100 percent of the seats on 48 percent of the votes. Parties B, C, and D win none. Same votes, different rule, different chamber.

In practice (your tools). You can build this table in any spreadsheet. Type the four party names in column A and their votes in column B, starting in row 2. In cell C2 type =B2/1, in D2 type =B2/2, in E2 type =B2/3, in F2 type =B2/4, and in G2 type =B2/5. Fill those five formulas down to row 5. Then copy every quotient into one column and sort it from largest to smallest. The top eight are your seats.

Misconception check. Students often allocate D'Hondt seats one party at a time: give Party A its seats, then Party B, and so on. That is tempting, because it feels orderly. But it gives the wrong answer, because the seats go to the largest quotients across the whole pool. Generate every party's quotients first, then rank the pool.

Memory hook: every quotient first, then rank the pool.

Honesty note: D'Hondt is one rule among several. Other divisor rules split the same votes a little differently, so a real country's result depends on which rule its law names.

6. Explain Duverger's law: a tendency, not a guarantee

Suppose your town holds a race with three candidates, and you like the one polling a distant third. Many people in your spot vote for their second choice instead, so their vote is not "wasted". Multiply that choice across a country, and you get one of the most famous ideas in political science.

Duverger (1951), Political Parties (English translation 1954), is taught here in words, because the book is still in copyright.

Duverger's law is the tendency, described by Maurice Duverger in 1951, of single-member plurality rules to produce competition between two main parties, with named exceptions.

It works through two effects.

The mechanical effect is the way a counting rule itself converts a party's votes into seats, so that under plurality rules a third party with scattered support wins few seats.
The psychological effect is the way voters, expecting a party to win few seats, desert it for a party they think can win, which changes the votes themselves.

The first effect happens at the count. The second happens before the vote, inside voters' heads. Together they push plurality countries toward two big parties.

The key word is tendency: a pattern that holds in most cases and has a mechanism behind it, but that real cases can break. Duverger's law has its own named exceptions. The United Kingdom, Canada, and India all use plurality rules, and all still seat significant third or regional parties. A party whose voters live close together can win districts even with a small national share, because plurality rules reward votes that are concentrated in a place.

Worked example — which effect is it? (illustrative) Sort two scenes from a made-up plurality country.

A party wins votes spread evenly across every district but finishes second or third in each, so it wins no seats. That is the mechanical effect: the counting rule turned its votes into nothing.
Before the next election, polls show that party far behind, and many of its supporters switch to a bigger party they also like. That is the psychological effect: expectations changed the votes.

Misconception check. Students often read Duverger's law as a guarantee that plurality rules always produce exactly two parties. That is tempting, because "law" sounds like physics. But Duverger stated a tendency with a mechanism, and the United Kingdom, Canada, and India are named exceptions. A tendency predicts what usually happens and explains why; it does not forbid the exceptions.

Memory hook: the count squeezes, and the voters see it coming.

7. Read the 2024 UK result with the data scaffold

Now for real data. The House of Commons Library published a briefing on the 2024 UK general election. You will read it with the same five questions every time: what is measured, over whom and when, what it shows, what it does not, and whether it shows correlation or cause.

What is measured. The briefing reports each party's share of the votes cast and its number of seats. The election used 650 single-member constituencies under plurality rules, and turnout was 59.7 percent.

Over whom and when. Voters across the United Kingdom, in the 2024 general election. The figures are reported at the party level, with no candidate named.

Worked example — vote share against seat share. Read each line and do the arithmetic.

Labour won 33.7 percent of the vote and 411 of 650 seats. Its seat share is 411 ÷ 650 × 100 ≈ 63.2 percent. The gap is 63.2 − 33.7 = 29.5 percentage points.
A majority of 650 seats needs more than 650 ÷ 2 = 325, so 326 seats. Labour's 411 is above that line.
Reform UK won 14.3 percent of the vote and 5 seats. Its seat share is 5 ÷ 650 × 100 ≈ 0.8 percent. The gap is 14.3 − 0.8 = 13.5 percentage points, in the other direction.
The Liberal Democrats won 12.2 percent of the vote and 72 seats. Reform UK's 14.3 percent beat them by 14.3 − 12.2 = 2.1 percentage points. Yet the Liberal Democrats won 72 − 5 = 67 more seats.
The Scottish National Party won 2.5 percent of the vote and 9 seats, more seats than Reform UK on a far smaller share.

The briefing itself calls Labour's 33.7 percent "the lowest for any party winning a general election in the post-war period". A percentage point is worth a word here. Percentage points are the unit for the difference between two percentages. A seat share of 63.2 percent against a vote share of 33.7 percent is a gap of 29.5 percentage points, not a "29.5 percent" gap.

What it shows. Seat shares did not track vote shares. A party whose votes were spread across the country, Reform UK, came second in many constituencies and won few seats. Parties whose votes were concentrated in particular places, the Liberal Democrats and the Scottish National Party, won more seats with fewer votes. The read is that a geographically efficient vote converts into seats and a dispersed one does not.

What it does not show. It does not show whether the result was fair, and it does not show whether the rules should change. It also does not show what a winning party's voters wanted it to do.

Correlation or cause? The table shows that seat shares and vote shares went together only loosely. The claim that the plurality rule produced the gap is a causal claim, and it rests on an explicit argument, the mechanical effect applied to where each party's votes were. The table alone cannot show how people would have voted under a different rule. The psychological effect means some voters may have chosen differently because of the rule itself.

In practice (your tools). In this week's workshop you will enter these party figures in a spreadsheet. With the vote share in column B and the seats in column C, the formula =C2/650*100 in column D gives the seat share, and =D2-B2 in column E gives the gap in percentage points.

Now apply the three-layer read. "The 2024 election used 650 single-member constituencies under plurality rules" is empirical, on the design layer. "Labour won 411 seats on 33.7 percent of the vote" is empirical, on the practice layer. "The UK ought to switch to proportional representation" is normative, on the verdict layer, and the count does not settle it.

Misconception check. Students often read a vote share as if it were a seat share, or swap the two. That is tempting, because both are percentages about the same party. But 33.7 percent is Labour's share of the votes, and 63.2 percent is its share of the seats. A vote share has votes as its base; a seat share has seats as its base.

Caveat: these are the House of Commons Library's figures for one election. They show how one plurality election came out, and they are not a law about every election.

8. Cure the week's traps

Three wrong turns catch almost everyone the first time. You have already seen the evidence for each one; this section names them.

Trap one: "the party with the most votes always wins the most seats." Under plurality rules, a party's seats depend on where its votes are, not only on how many there are. The 2024 result shows a party with more votes winning far fewer seats: Reform UK's 14.3 percent brought 5 seats, and the Liberal Democrats' 12.2 percent brought 72.

Trap two: "proportional representation produces stable single-party government." The opposite is more typical. Because seats track votes, one party can win a majority of the seats alone only if it wins about half of the votes. So under proportional rules, coalitions are the more typical result. Week 7 taught you that a coalition government is a normal mode of parliamentary government, not a breakdown. This week shows one reason coalitions are so common in some countries.

Trap three: "a seat majority proves a mandate." A mandate is the claim that an election result gives the winner the voters' authority to carry out its program, which is a normative judgment a seat count does not settle by itself. Labour's 411 seats are a fact. Whether 411 seats won on 33.7 percent of the vote amount to a mandate is an argument. Proponents of the mandate reading argue that the voters chose a government under known rules, so the government should act. Critics respond that a minority of the votes cannot speak for the majority who voted otherwise.

Worked example — empirical or normative? (illustrative) An invented commentator writes three sentences about a made-up election.

"The governing party won 58 of 100 seats." Empirical, on the practice layer: you can check it against the official count.
"The election used single-member districts under plurality rules." Empirical, on the design layer: you can check it against the electoral law.
"So the voters ordered the government to carry out every promise." Normative, on the verdict layer: it is an argument about what the result means, and the count does not settle it.

Misconception check. Students often treat "they won, so they have a mandate" as a plain fact. That is tempting, because winning is a fact. But the step from "won the seats" to "has the voters' authority for every policy" is an ought hidden inside an is, the smuggling move from Week 1. Name the layer before you judge.

Memory hook: seats are counted; mandates are argued.

9. Weigh it: first-past-the-post or proportional representation?

Here the facts stop and the argument starts. The question is perennial: should a democracy elect its legislature by first-past-the-post or by proportional representation?

Proponents of first-past-the-post argue three things. First, accountability: one party usually governs alone, so voters know exactly whom to reward or remove. Second, a local member: every place has one representative who answers for it. Third, decisive government: a single-party majority can carry out its program without bargaining with partners. Look at the plurality contrast in Section 5: the system turns a lead into a working majority.

Proponents of proportional representation argue three things too. First, fairness to voters: seats should match votes, so a party with a seventh of the votes should not end up with a handful of seats. Second, no large group of voters should elect no one. Third, coalition government forces parties to bargain and so represents more of the country in each decision.

Each side names a real cost in the other. First-past-the-post can hand a large seat majority to a party most voters did not choose, as the 2024 gap of 29.5 percentage points shows. Proportional representation can give a small party the power to make or break a coalition, and it weakens the tie to a local member. Mixed-member proportional is one attempt to keep the strengths of both; its critics say it inherits complexity instead.

Notice what kind of question this is. The facts about each system are empirical, and you have checked several of them. Which costs are worth paying is normative. It depends on what you value most: a clear line of accountability, or a chamber that mirrors the votes. This course hands that verdict to you.

Pause and reflect. Choose one option and write a few sentences.

(a) Imagine you are advising a new democracy. State the strongest case for the system you find less appealing, as its best defender would state it, before you give your own view.
(b) Think of a group you belong to (a club, a team, a workplace council). Does it choose leaders by plurality, by majority, or by something proportional, and what would change if it switched?

Misconception check. Students often think the 2024 UK result proves that first-past-the-post is unfair. That is tempting, because the gap between votes and seats is large. But the result is a fact about the practice layer, and "unfair" is a verdict. Defenders of first-past-the-post accept the gap as the price of accountable single-party government. The data show the trade-off; they do not decide it.

Key terms

plurality — The largest number of votes among the candidates or parties, whether or not it is more than half of all the votes cast. (first used: 1. Count to a winner: plurality versus majority)
majority — More than half of the votes cast or of the seats available, so 16 of 30 votes or 326 of 650 seats. (first used: 1. Count to a winner: plurality versus majority)
political party — An organized group that competes under a common label to win public office and so shape government and policy. (first used: 2. Meet the parties: three jobs and the party system)
aggregation — The party function of gathering many separate interests and demands into one program that voters can support. (first used: 2. Meet the parties: three jobs and the party system)
mobilization — The party function of getting people to register, turn out to vote, and take part in campaigns and public life. (first used: 2. Meet the parties: three jobs and the party system)
accountability — The party function of giving voters a known label whose record in office they can reward or punish at the next election. (first used: 2. Meet the parties: three jobs and the party system)
party system — The usual pattern of competition among a country's parties, described by how many parties seriously compete for power; it describes a steady pattern, not a judgment of quality. (first used: 2. Meet the parties: three jobs and the party system)
two-party system — A party system in which two large parties win nearly all the seats and take turns forming the government. (first used: 2. Meet the parties: three jobs and the party system)
multiparty system — A party system in which three or more parties win significant shares of seats, so governments are often coalitions. (first used: 2. Meet the parties: three jobs and the party system)
dominant-party system — A party system in which several parties compete in elections but one party wins office again and again over a long period. (first used: 2. Meet the parties: three jobs and the party system)
electoral system — The set of rules that turns the votes cast in an election into seats or offices won. (first used: 3. Sort the four electoral-system families)
first-past-the-post — A plurality electoral system in which each single-member district elects the one candidate with the most votes, majority or not. (first used: 3. Sort the four electoral-system families)
single-member district — An electoral district that elects exactly one representative, as each constituency in the UK House of Commons does. (first used: 3. Sort the four electoral-system families)
majority-runoff — An electoral system that holds a second round between the two leading candidates whenever no candidate wins more than 50 percent in the first round. (first used: 3. Sort the four electoral-system families)
proportional representation — A family of electoral systems that gives each party a share of seats roughly in line with its share of the vote. (first used: 3. Sort the four electoral-system families)
party list — A party's ranked list of candidates under proportional representation, from which its seats are filled in order. (first used: 3. Sort the four electoral-system families)
electoral threshold — A minimum share of the vote a party must reach before it can win any seats under proportional representation. (first used: 3. Sort the four electoral-system families)
mixed-member proportional — An electoral system in which each voter casts a district vote and a party-list vote, and list seats top up each party's total toward its share of the list vote. (first used: 3. Sort the four electoral-system families)
D'Hondt method — A seat-allocation rule that divides each party's votes by 1, 2, 3, and so on, pools every result, and awards the seats to the largest results until the seats run out. (first used: 5. Work the D'Hondt table)
quotient — The result of dividing one party's votes by a D'Hondt divisor, such as 48,000 divided by 2 giving 24,000. (first used: 5. Work the D'Hondt table)
Duverger's law — The tendency, described by Maurice Duverger in 1951, of single-member plurality rules to produce competition between two main parties, with named exceptions. (first used: 6. Explain Duverger's law: a tendency, not a guarantee)
mechanical effect — The way a counting rule itself converts a party's votes into seats, so that under plurality rules a third party with scattered support wins few seats. (first used: 6. Explain Duverger's law: a tendency, not a guarantee)
psychological effect — The way voters, expecting a party to win few seats, desert it for a party they think can win, which changes the votes themselves. (first used: 6. Explain Duverger's law: a tendency, not a guarantee)
tendency — A pattern that holds in most cases and has a mechanism behind it, but that real cases can break, unlike a guarantee. (first used: 6. Explain Duverger's law: a tendency, not a guarantee)
percentage points — The unit for the difference between two percentages, so 63.2 percent of seats against 33.7 percent of votes is a gap of 29.5 percentage points. (first used: 7. Read the 2024 UK result with the data scaffold)
mandate — The claim that an election result gives the winner the voters' authority to carry out its program, which is a normative judgment a seat count does not settle by itself. (first used: 8. Cure the week's traps)

Summary

A plurality is the most votes; a majority is more than half.
Parties aggregate interests, mobilize voters, and give voters someone to hold accountable.
Two-party, multiparty, and dominant-party systems describe patterns, not verdicts.
Ask how a winner is decided to name the family: most votes, more than half, match the votes, or mix the two.
D'Hondt: generate every party's quotients first, then rank the pool; the illustrative table gives A 4, B 3, C 1, D 0.
Duverger's law is a tendency with a mechanical and a psychological effect, and it has named exceptions.
In 2024, Labour won 33.7 percent of the vote and 411 of 650 seats; Reform UK won 14.3 percent and 5 seats.
A seat count is a fact; whether it is fair or a mandate is a verdict you argue.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A neighborhood association of 45 voting members elects a chair. The three candidates get 20, 15, and 10 votes. Does the winner have a plurality, a majority, or both? Show the steps.
A made-up city council has 5 seats shared by the D'Hondt method. Party P gets 12,000 votes, Party Q 7,000, and Party R 3,000. How many seats does each party win?
A made-up country uses plurality rules. A regional party wins 30 seats with 6 percent of the national vote, all from one region. A classmate says this disproves Duverger's law. Is the classmate right?
An invented commentator writes: "Our party won 52 percent of the seats, so the voters handed us a mandate for every promise." Sort the two parts of the sentence by kind and layer.

Answers (worked reasoning — read after attempting)

The winner has a plurality but not a majority. Check the parts: 20 + 15 + 10 = 45. The winner has the most votes, 20, so it is a plurality. Half of 45 is 45 ÷ 2 = 22.5, so a majority needs at least 23 votes, and 20 falls short. The winner's share is 20 ÷ 45 × 100 ≈ 44.4 percent.
Party P wins 3 seats, Party Q wins 2, and Party R wins none. Generate every quotient first. Party P: 12,000, 6,000, 4,000, 3,000. Party Q: 7,000, 3,500. Party R: 3,000. Rank the pool: 12,000 (P), 7,000 (Q), 6,000 (P), 4,000 (P), 3,500 (Q). The five seats are gone before Party R's 3,000 comes up. Check: 3 + 2 + 0 = 5.
No. Duverger's law is a tendency, not a guarantee. A party whose voters are concentrated in one region can win districts there even with a small national share. That is the same pattern as the named exceptions, where plurality countries still seat regional parties.
"Our party won 52 percent of the seats" is empirical, on the practice layer, because you can check it against the official count. "So the voters handed us a mandate for every promise" is normative, on the verdict layer. The count does not settle what the result authorizes.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, the quiz, the discussion, and the assignment. Next week the margin-of-error rule shows how a poll of a thousand people can speak for a whole country.

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