IA idea · Voting, fairness & game theory

Turning votes into seats: D'Hondt, Sainte-Laguë and largest remainder

AI SLAA SLAI HL Solid Also in: Statistics

Research question

Applied to real vote shares from a proportional election, how differently do the D'Hondt, Sainte-Laguë and largest remainder methods allocate seats, which favours large parties, and can any of them produce a paradox?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

The methods are simple to state and give different results from the same votes. Measuring disproportionality with an index makes the comparison quantitative.

The mathematics you'll need

Each method defined and applied
Quotients and rounding
A disproportionality index (for example the Gallagher index, explained)
Constructing a paradox example
Comparing methods systematically

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Use official results from a proportional election (a national electoral commission publishes them); cite the source.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain each method.
Apply them to real results.
Measure disproportionality.
Find which parties gain and lose.
Construct or find a paradox and reflect.

Pitfalls that cost marks

Unofficial results.
Not explaining the index.
Political opinions instead of mathematics.

Showing personal engagement

Use an election in your own country.
Predict which party benefits before calculating.
Design your own fair method.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Each method defined and applied; Quotients and rounding
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Each method defined and applied; Quotients and rounding
AI HL	Good fit	Each method defined and applied; Quotients and rounding

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Run a real vote or game with people you know, and choose the methods or rules to compare. Predict the outcome before you analyse it.

Reflection (D)

Reflect on the gap between the mathematically rational choice and what people did, and on what each fairness method gains and gives up. For this idea, start with: unofficial results — say how it affects your answer.

Use of mathematics (E)

SL: Each method explained with a worked example, then analysed with probability, expected value or counting; results compared systematically rather than case by case.

HL: Mixed strategies found by solving equations or with calculus, a proof that a method has (or lacks) a fairness property, or a probability model of how often methods disagree.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Investigate the Alabama paradox for largest remainder by increasing the number of seats.

Extending it for HL

Prove a fairness property in general, or model random ballots and calculate how often two methods pick different winners.

See a complete IA, marked

Our annotated exemplar Should I lease or buy my first car? (AI SL) asks a different question, but shows how a complete voting & game theory exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).