IA idea · Voting, fairness & game theory

Who should pay what for a shared taxi? The Shapley value

AA SLAI SLAA HL Solid Also in: Finance

Research question

When friends share a taxi that drops each of them off in turn, how should the fare be split fairly, and how does the Shapley value compare with splitting equally or by distance?

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

A real everyday argument with a precise mathematical answer. Calculating the Shapley value by averaging over orders is accessible and surprising.

The mathematics you'll need

Characteristic function: cost of each group's journey
Averaging marginal costs over all orders (permutations)
Comparing with equal and proportional splits
Properties of fairness checked
Generalising to four people

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 real fares or a taxi app's estimates for each combination of drop-offs (record dates), or a fare rule from your local taxi tariff.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.

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

Get the cost of every possible shared journey.
Calculate the Shapley value.
Compare with simpler splits.
Check fairness properties.
Reflect on what 'fair' means to your friends.

Pitfalls that cost marks

Missing journeys for some groups.
Order counting errors.
No discussion of what fairness means.

Showing personal engagement

Use a real journey with your friends.
Ask friends which split they think is fair first.
Find a case where equal splitting is very unfair.

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	Characteristic function: cost of each group's journey; Averaging marginal costs over all orders (permutations)
AA HL	Good fit	Characteristic function: cost of each group's journey; Averaging marginal costs over all orders (permutations)
AI SL	Good fit	Characteristic function: cost of each group's journey; Averaging marginal costs over all orders (permutations)
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: missing journeys for some groups — 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

Compare with the nucleolus or another method, or split costs for a shared flat.

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).