science

Largest remainder method

Largest remainder method is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Largest remainder method rather than just read about it. In short: The quota or divide-and-rank methods make up a category of apportionment rules, i.e. algorithms for allocating seats in a legislative body among multiple groups (e.g. parties or federal states). The quota methods begin by calculating an entitlement (basic number of seats) for each party, by dividing their vote totals by an electoral quota (a fixed number of votes needed to win a seat, as a unit).

Key takeaways

Largest remainder method belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Largest remainder method to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Largest remainder method from memory before moving on to harder problems.

Reference excerpt

The quota or divide-and-rank methods make up a category of apportionment rules, i.e. algorithms for allocating seats in a legislative body among multiple groups (e.g. parties or federal states). The quota methods begin by calculating an entitlement (basic number of seats) for each party, by dividing their vote totals by an electoral quota (a fixed number of votes needed to win a seat, as a unit). Then leftover seats, if any, are allocated by rounding up the apportionment for some parties. These rules are typically contrasted with the more popular highest averages methods (also called divisor methods). By far the most common quota method are the largest remainders or quota-shift methods, which assign any leftover seats to the "plurality" winners (the parties with the largest remainders, i.e. most leftover votes). When using the Hare quota, this rule is called Hamilton's method or the Hare-Niemeyer method, and is the third-most common apportionment rule worldwide (after the d'Hondt and Sainte-Laguë highest averages methods). Despite their intuitive definition, quota methods are generally disfavored by social choice theorists as a result of apportionment paradoxes. In particular, the largest remainder methods exhibit the no-show paradox, i.e. voting for a party can cause it to lose seats. The largest remainders methods are also vulnerable to spoiler effects and can fail resource or house monotonicity, which says that increasing the number of seats in a legislature should not cause a party to lose a seat (a situation known as an Alabama paradox).

Method The largest remainder method divides each party's vote total by a quota. Usually, quota is derived by dividing the number of valid votes cast, by the number of seats. The result for each party will consist of an integer part plus a fractional remainder. Each party is first allocated a number of seats equal to their integer. This will generally leave some remainder seats unallocated. To apportion these seats, the parties are then ranked on the basis of their fractional remainders, and the parties with the largest remainders are each allocated one additional seat until all seats have been allocated. This gives the method its name - largest remainder. Largest remainder methods produces similar results to single transferable vote or the quota Borda system, where voters organize themselves into solid coalitions. The single transferable vote or the quota Borda system behave like the largest-remainders method when voters all behave like strict partisans (i.e. only mark preferences for candidates of one party).

Quotas

There are several possible choices for the electoral quota. The choice of quota affects the properties of the corresponding largest remainder method, and particularly the seat bias. Smaller quotas allow small parties to pick up seats, while larger quotas leave behind more votes. A somewhat counterintuitive result of this is that a larger quota will always be more favorable to smaller parties. A party hoping to win multiple seats sees fewer votes captured by a single popular candidate when the quota is small. The two most common quotas are the Hare quota and the Droop quota. The use of a particular quota with one of the largest remainder methods is often abbreviated as "LR-[quota name]", such as "LR-Droop". The Hare (or simple) quota is defined as follows:

total votes total seats {\displaystyle {\frac {\text{total votes}}{\text{total seats}}}}

LR-Hare is sometimes called Hamilton's method, named after Alexander Hamilton, who devised the method in 1792. The Droop quota is given by:

total votes total seats + 1 {\displaystyle {\frac {\text{total votes}}{{\text{total seats}}+1}}}

and is applied to elections in South Africa. The Hare quota is more generous to less-popular parties and the Droop quota to more-popular parties. Specifically, the Hare quota is unbiased in the number of seats it hands out, and so is more proportional than the Droop quota (which tends to give more seats to larger parties). The Hare suffers the disproportionality that it sometimes allocates a minority of seats to a party with more than half the votes in a district and sometimes allocates a majority of seats to a party with less than a majority of votes

Examples The following example allocates 11 seats using the largest-remainder method by Hare quota.

Pros and cons It is easy for a voter to understand how the largest remainder method allocates seats. Moreover, the largest remainder method satisfies the quota rule (each party's seats are equal to its ideal share of seats, either rounded up or rounded down) and was designed to satisfy that criterion. However, this comes at the cost of greater inequalities in the seats-to-votes ratio, which can violate the principle of one man, one vote. However, a greater concern for social choice theorists, and the primary cause behind its abandonment in many countries, is the tendency of such rules to produce erratic or irrational behaviors called apportionment paradoxes:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Largest remainder method

Start with the simplest possible case. Write down what Largest remainder method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Largest remainder method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Largest remainder method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Largest remainder method

In research: Largest remainder method appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Largest remainder method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Largest remainder method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Apportionment methods, Party-list proportional representation, Voting theory, so understanding it makes those chapters shorter.
In everyday life: Look for Largest remainder method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Largest remainder method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Largest remainder method means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Largest remainder method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Largest remainder method in simple terms?

Why does Largest remainder method matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Largest remainder method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Largest remainder method.