Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Will one phone brand take over? Brand loyalty as a Markov chain

AI HL Solid Also in: Statistics, Probability, Finance, Matrices & Markov

Research question

Using survey data on which brand students switched from and to at their last phone upgrade, what long-term market shares does a Markov chain predict, and how sensitive are they to loyalty rates?

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Why it makes a good exploration

Markov chains model any system where people move between states. Surveying real switching behaviour gives you a transition matrix built from your own data.

The mathematics you'll need

Transition matrices from survey data
Matrix powers for future shares
Steady-state vectors and eigenvectors
Sensitivity analysis

Where the data comes from

Survey 60+ people (students and adults) about their previous and current phone brand.

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

A possible outline

Design the survey and define states.
Build the transition matrix.
Predict shares after several upgrades.
Find the steady state.
Reflect on sampling, new brands and changing loyalty.

Pitfalls that cost marks

Too few people in some categories.
Assuming transition probabilities stay constant without comment.
Rows/columns not summing to 1.

Showing personal engagement

Include your own switching story.
Compare with published market shares.
Test how a price change (a change in loyalty) affects the steady state.

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	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	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	Transition matrices from survey data; Matrix powers for future shares

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

Reflection (D)

For this idea, start with the weak points: too few people in some categories; and assuming transition probabilities stay constant without comment. Say how each one affects your answer.

Use of mathematics (E)

HL: transition matrices from survey data; matrix powers for future shares; steady-state vectors and eigenvectors; sensitivity analysis, each explained in your own words and carried out correctly.

How networks & graphs explorations reach the top bands in general: Networks, graphs & algorithms IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Compare the survey of students with a survey of adults and discuss which chain is more stable.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL networks & graphs exploration adds.

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete networks & graphs exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Transition matrices from survey data, matrix powers for future shares, steady-state vectors and eigenvectors and sensitivity analysis.

Where can I get data for this IA?

Survey 60+ people (students and adults) about their previous and current phone brand. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Will one phone brand take over? Brand loyalty as a Markov chain” filled in: the question, the maths and the data sources.

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