Case 003Market-making gamesHard

You make a market in a coin-flip contract paying Rs 100 on heads. One trade in four is the interviewer, who knows the outcome and trades only when it helps; the rest is balanced. What is the narrowest breakeven spread, and what does quoting 45 at 55 cost over 40 trades?

Citadel SecuritiesLondon · 2026

The situation

In Tulsivan Markets' trading round you are the market maker. Each round a fresh fair coin is flipped out of sight, and a contract pays Rs 100 if it lands heads and nothing if tails. You post a bid and an offer around 50 and must trade one contract with whoever arrives.

One arrival in four is the interviewer, who has seen the coin: they buy at your offer when it is heads and sell at your bid when it is tails. The other three arrivals are uninformed, equally likely to buy or sell. You play 40 rounds.

Your task

Find the narrowest symmetric spread around 50 that breaks even, and your expected P&L over 40 rounds if you quote 45 at 55.

Quick check

What is the narrowest breakeven quote?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The narrowest breakeven quote is 37.5 at 62.5; quoting 45 at 55 loses about Rs 300 over 40 rounds. Each uninformed trade earns the half-spread and each informed trade loses 50 minus it, so the expectation per trade is the half-spread minus 12.5. At 45 at 55 that is minus 7.5 a trade. The breakeven offer of 62.5 is exactly the coin's value given that someone chose to buy.

Step 1Who are you actually trading with?

Split the arrivals before touching numbers. A shopkeeper who sells umbrellas at a fixed price does fine with ordinary customers, but a customer who has already seen the weather forecast buys only on the days it will rain. The uninformed three quarters pay you the half-spread; the informed quarter trades only when your price is wrong, and then it is wrong by 50 minus the half-spread. This is adverse selectionThe tendency of a quote to be hit mainly by traders who know it is mispriced, so the fills a market maker gets are worse than average., and the spread exists to pay for it.

The relationship
E[P&L per trade]=0.75 h  −  0.25 (50−h)=h−12.5E[\text{P\&L per trade}] = 0.75\,h \; - \; 0.25\,(50 - h) = h - 12.5
hhalf-spread around 50, so you quote 50 - h at 50 + h
0.75share of arrivals that are uninformed
50 - hyour loss when the informed trader lifts your offer on heads or hits your bid on tails
What it says in wordsEach unit of half-spread is earned on every arrival in the end, and the informed trader costs a fixed 12.5 a trade, so the half-spread must be at least 12.5.
Per-trade profit for the quoter against the half-spread-15-10-5+5+10+150breakeven: half-spread 12.5quote 37.5 at 62.545 at 55: -7.5 a tradegain on uninformed, 0.75hloss to informed, 0.25(50 - h)net, h - 12.50510152025Half-spread h around 50, Rs
The quoter's expected profit per trade is the half-spread minus 12.5: the uninformed gain rises with the spread, the informed loss shrinks with it, and the net crosses zero at a half-spread of 12.5, so 45 at 55 loses 7.5 a trade.
Step 2What does 45 at 55 cost over the game?

Plug in h = 5. Expected profit is 5 minus 12.5, minus 7.5 a trade, and over 40 rounds that is minus Rs 300. Break the total into its two parts, because that is the sentence the interviewer is waiting for: about 30 uninformed trades earn Rs 150, and about 10 informed trades lose Rs 45 each, Rs 450. A tight spread feels competitive and looks generous to the uninformed, but the informed quarter drains three times what the spread earns.

Forty trades at 45 at 55: who you trade with decides the result, Rs30 uninformed trades x +5+15010 informed trades x -45-450Net over 40 trades-300The spread earns 5 a trade; one informed trade in four costs 45.
Quoting 45 at 55 for 40 rounds, about 30 uninformed trades earn Rs 150 and about 10 informed trades lose Rs 450, an expected loss of about Rs 300.
Step 3Why is 62.5 the right offer and not just a breakeven?

Ask what you learn when someone buys. An uninformed buyer arrives with probability 0.375 whatever the coin shows; the informed trader adds 0.25 only on heads. So the chance of heads given a buy is 0.625 divided by 1.0, which is 0.625, and the contract is worth Rs 62.5 to you at the moment you are lifted. That is the logic of the Glosten and Milgrom model: a fair offer is the expected value conditional on being bought from, and a fair bid is the value conditional on being sold to, 37.5 here. The breakeven arithmetic and the conditional-value arithmetic land on the same numbers because they describe the same thing.

Close with what you would do in the game. Quote 37 at 63 or so, slightly outside breakeven, and watch the flow. If the informed share looks smaller than one in four, tighten; if buys cluster, the next quote should move up. The limit of the model is that it assumes you know the informed share, and in a real game estimating it from the flow is most of the job.

Where candidates lose it

The usual loss is quoting tight to win flow, as if every counterparty were random. The interviewer designed the game so that the informed quarter is the whole story, and a 10-wide market pays them 45 a time.

The second is computing the loss to the informed trader as 50 rather than 50 minus the half-spread. Your quote is still collected on their trade, so the half-spread offsets part of the damage, and missing it gives a breakeven of 16.7 instead of 12.5.

What the interviewer asks next

  • The informed share rises to one in three. What is the breakeven quote now?
  • Two informed traders arrive in a row, both buying. Where should your next quote sit?
  • You are allowed to refuse one trade in ten. Which would you refuse, and how much does it save?
  • How does this change if the coin pays Rs 100 on heads and Rs 20 on tails?

Asked at Citadel Securities, Quantitative Research, London, 2026 (Wall Street Oasis): 3rd I got rejected it was different brainteasers and trading game

Company names and figures are illustrative.