IA idea · Voting, fairness & game theory

Same ballots, different winners: comparing voting methods in a class vote

AA SLAI SLAA HLAI HL Accessible Also in: Statistics

Research question

When your class ranks the options for a trip (or a film, or a charity), do plurality, run-off, Borda count and Condorcet methods choose the same winner, and how often would they disagree for random ballots?

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

You collect real ranked ballots, apply four methods by hand, and often find they disagree. Then you can ask how common that is, which needs probability or simulation.

The mathematics you'll need

Each voting method defined precisely with a worked example
Pairwise comparisons and Condorcet winners
Counting possible rankings (permutations)
Simulation of random ballots
Proportion of disagreements

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

Run an anonymous ranked vote in your class or club on a real decision (with your teacher's permission).

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

Collect ranked ballots.
Apply each method and record the winners.
Find the Condorcet winner if one exists.
Simulate random ballots to estimate how often methods disagree.
Reflect on which method suits your decision.

Pitfalls that cost marks

Methods described vaguely, so results can't be checked.
Non-anonymous ballots.
Assuming random ballots are realistic without comment.

Showing personal engagement

Use a decision your class really has to make.
Predict which method will be the odd one out.
Design ballots that make all four methods disagree.

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 voting method defined precisely with a worked example; Pairwise comparisons and Condorcet winners
AA HL	Fits, but add an HL technique	Each voting method defined precisely with a worked example; Pairwise comparisons and Condorcet winners
AI SL	Good fit	Each voting method defined precisely with a worked example; Pairwise comparisons and Condorcet winners
AI HL	Fits, but add an HL technique	Each voting method defined precisely with a worked example; Pairwise comparisons and Condorcet winners

Level: Accessible. A good first extended piece of maths, with room to go deeper. 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: methods described vaguely, so results can't be checked — 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

Measure how often each method elects the Condorcet winner when one exists.

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).