IA idea · Voting, fairness & game theory

How likely is a voting paradox? Rock–paper–scissors preferences

AA HLAI HL Ambitious Also in: Probability, Simulation

Research question

With three candidates and voters whose rankings are random, what is the probability that no candidate beats both others head to head (a Condorcet cycle), and how does it change with the number of voters?

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 paradox can be found exactly for small numbers of voters by counting, and estimated by simulation for large numbers, so theory and simulation check each other.

The mathematics you'll need

Counting rankings and voter profiles
Exact probability for three voters by enumeration
Simulation for larger numbers of voters
Convergence of the probability as voters increase
Confidence intervals for simulated probabilities

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.

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Confidence intervals for a mean. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

No data needed; optionally compare with ranked ballots you collect.

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 Condorcet cycles with an example.
Calculate the exact probability for three voters.
Simulate for larger numbers.
Describe how the probability behaves.
Reflect on whether real preferences are random.

Pitfalls that cost marks

Counting errors in the exact case; check by listing.
Too few simulations to see the trend.
Ignoring ties for even numbers of voters.

Showing personal engagement

Find a real cycle in your class's preferences.
Predict the probability before calculating.
Add a fourth candidate.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Counting rankings and voter profiles; Exact probability for three voters by enumeration
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Counting rankings and voter profiles; Exact probability for three voters by enumeration

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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: counting errors in the exact case; check by listing — 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

Model voters with a shared left–right scale (single-peaked preferences) and show cycles disappear.

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 How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) 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).