IA idea · Voting, fairness & game theory

Cutting a cake fairly when people value it differently

AA SLAA HL Solid Also in: Calculus, Pure maths

Research question

If two or three people value different parts of a cake differently (described by value-density functions), where should the cuts go for 'divide and choose' and for a moving-knife method, and are the results envy-free?

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

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Why it makes a good exploration

Fair division is a real field of mathematics. Describing preferences with functions and finding cuts with definite integrals turns a party problem into calculus.

The mathematics you'll need

Value-density functions on an interval
Definite integrals to value each piece
Solving for cut points
Proportional and envy-free fairness defined and checked
Comparing methods

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

Ask friends to rate parts of a real cake (or a pizza, or a bar of different flavours) and turn their ratings into density functions.

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

Define fairness precisely.
Build value functions from ratings.
Apply divide and choose with integrals.
Apply a three-person method.
Check envy-freeness and reflect.

Pitfalls that cost marks

Fairness not defined precisely.
Value functions with no real basis.
Integrals set up but not interpreted.

Showing personal engagement

Use a real cake and real people.
Find a division someone envies and explain why.
Design your own 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	Value-density functions on an interval; Definite integrals to value each piece
AA HL	Good fit	Value-density functions on an interval; Definite integrals to value each piece
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	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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: fairness not defined precisely — 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

Prove that divide and choose is always proportional for two people, or find an envy-free three-person method.

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 Is a hanging chain a parabola? Comparing catenary and quadratic models (AA 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.

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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).