IA idea · Voting, fairness & game theory

How much should you bid in a sealed-bid auction?

AA HLAA SL Ambitious Also in: Calculus, Probability

Research question

In a sealed-bid auction where the highest bid wins and pays its bid, how much below your true value should you bid if others' values are uniformly random, and does a class experiment agree?

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

Optimising expected profit needs probability and differentiation together. A class auction gives real bids to compare with the theory.

The mathematics you'll need

Probability of winning with a given bid
Expected profit (v − b) × P(win)
Maximising with differentiation
Equilibrium bid (n − 1)v/n for uniform values (derived)
Comparing theory with experimental bids

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

Run a class auction for a small prize with randomly assigned private values (with your teacher's agreement).

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 auction and assumptions.
Derive the best bid against one rival.
Generalise to n bidders.
Run the class experiment.
Compare and reflect on risk attitudes.

Pitfalls that cost marks

Assuming others bid their values without comment.
Too few rounds in the experiment.
Not checking the maximum is a maximum.

Showing personal engagement

Run the auction yourself.
Predict how classmates will bid.
Compare with a second-price auction.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Probability of winning with a given bid; Expected profit (v − b) × P(win)
AA HL	Good fit	Probability of winning with a given bid; Expected profit (v − b) × P(win)
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: 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: assuming others bid their values without comment — 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 a second-price auction, where bidding your true value is best, and prove why.

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).