IA idea · Voting, fairness & game theory

Left, right or centre? Game theory for penalty kicks

AA SLAI SLAA HLAI HL Solid Also in: Sport, Probability

Research question

If a kicker's scoring probability depends on which way they kick and which way the keeper dives, what mixed strategies should each use, and do your own penalty data agree with the prediction?

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

Penalties are a textbook example of a mixed-strategy equilibrium, but testing it on data you collect makes it yours.

The mathematics you'll need

Payoff matrices
Expected payoffs
Finding mixed strategies by solving equations (the indifference principle)
Comparing predicted and observed frequencies with a chi-squared test
Sensitivity of the equilibrium to the payoffs

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

Record penalties in training or school matches (kick side, dive side, outcome), or use your own penalty sessions.

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 the game and build a payoff matrix from data.
Find the mixed equilibrium.
Compare with observed choices.
Test whether players are predictable.
Reflect on left- and right-footed players.

Pitfalls that cost marks

Too few penalties to estimate probabilities.
Mixing left- and right-footed kickers without comment.
No test of the prediction.

Showing personal engagement

Take and save the penalties yourself.
Predict your own best strategy.
Advise your team's keeper.

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	Payoff matrices; Expected payoffs
AA HL	Good fit	Payoff matrices; Expected payoffs
AI SL	Good fit	Payoff matrices; Expected payoffs
AI HL	Good fit	Payoff matrices; Expected payoffs

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: too few penalties to estimate probabilities — 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

Add the centre option and solve the 3 × 3 game.

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 skydiver take to reach terminal velocity? (AA 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).