Quant puzzles, solved step by step

Puzzles: 100
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018I will draw a card from a shuffled deck. You may pay Rs 6 to play a bet that pays Rs 10 if the card is red. Before deciding, you may pay to be told the card's colour. What is the most you should pay for that information?Market making, betting and sizingCoreOptiverChicago · 2025
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What is the information worth?
Show the worked solution
Rs 2. Without information the bet is worth 0.5 x 10 - 6 = -1, so you decline and your value is 0. With the colour known, you play on red and make 4, and skip black and make 0, which averages 2. Information is worth the improvement in your best decision: 2 - 0 = 2. If it would not change what you do, it is worth nothing.
How do you value a piece of information?
Suppose a weather forecast costs money and you are deciding whether to carry an umbrella. If you would carry it anyway, the forecast is worthless to you; it is valuable only if some answer would change what you do. The value of information is the expected value of your best decision with it, minus the expected value of your best decision without it. Work out both decision trees separately and subtract. Never value information by the size of the payout it relates to.
Blind, the bet has an expected value of minus 1 so you decline and get 0; told the colour first, you play only on red and make 4 half the time, an average of 2, so the information is worth Rs 2.
Why is it not worth Rs 4 or Rs 5?
Rs 4 is what you make when the card is red, but it is red only half the time. Rs 5 is half the payout, which ignores the Rs 6 you pay to play. The information saves you from the losing half of the bet and lets you keep the winning half, and that is worth half of Rs 4, which is Rs 2. Pay more than Rs 2 and you would do better declining the offer of information and declining the bet.
The relationship
$\text{VOI} = E\big[\max(\text{payoff}, 0)\big] - \max\big(E[\text{payoff}], 0\big) = \tfrac12\max(4,0) + \tfrac12\max(-6,0) - \max(-1,0) = 2$
payoff 10 - 6 = 4 on red, -6 on black
E[max(payoff, 0)] your value when you can choose after seeing the colour
max(E[payoff], 0) your value when you must choose blind
What it says in wordsInformation is worth the gap between deciding after you know and deciding before.
When is information worth the most?
Vary the price of the bet. At a price of 5 you are exactly indifferent blind, and the information is worth 2.50, its maximum; at a price of 0 you would always play, and it is worth 0. Information is valuable when you are close to indifferent and the decision could go either way. The formula also has the shape of an option payoff: knowing first lets you exercise only when it pays, which is why traders talk about paying for optionality and paying for information in the same breath.
Where candidates lose it
The trap is answering with the size of the win, Rs 4, or half the payout, Rs 5. Both value the information by the bet it is about, not by the decision it improves.
The second loss is forgetting that without information you would decline. Candidates who compare with playing blind, at -1, get 3. The comparison is always with your best action without the information, which here is to walk away.
What the interviewer asks next
What is the information worth if the bet costs Rs 3?
What if the information is only 80% reliable?
You can pay to see one card of a two-card hand before betting. How do you decide what that is worth?
Asked at Optiver, Quantitative Research, Chicago, 2025 (Wall Street Oasis): Valuing information, taking directional bets when not plus EV.
089Make me a market on the number of disposable nappies used in the UK in one day. Build the estimate from stated assumptions and choose a width you would actually trade on.Market making, betting and sizingCoreDRWLondon · 2025
Try it first
If each of four inputs could be about 10 to 25% off in either direction, how uncertain is the product?
Show the worked solution
About 9.4 million a day, and I would open at 8 bid, 11 offered, in millions. Assume about 700,000 births a year, 2.5 years in nappies, six changes a day and 90% disposable: 9.45 million. Multiplying the low and high ends of each input gives 5.5 to 15.0 million, so a quote of 8 at 11 is tight enough to trade and still honest about the uncertainty.
How do you build the estimate so the interviewer can follow it?
Chain it through things you can reason about. Children in nappies are roughly births a year times the years each child spends in them. Assume about 700,000 births a year, a round number worth checking against the latest official statistics, and 2.5 years in nappies: about 1.75 million children. Each child uses about six a day on average, more as a newborn and fewer as a toddler, and assume 90% of families use disposables: 1.75 million x 6 x 0.9 = 9.45 million a day. Say each assumption out loud and give it a range as you go.
Multiplying the four central assumptions gives 9.45 million nappies a day, but multiplying the four lows and the four highs gives 5.5 to 15.0 million, so the honest uncertainty is roughly a factor of 1.6 either side, and a quote of 8 at 11 million sits inside it.
The relationship
$N = B \times Y \times c \times d = 700{,}000 \times 2.5 \times 6 \times 0.9 \approx 9.45\text{ million}$
B births a year, an assumption
Y years a child spends in nappies
c changes a day
d share of families using disposables
What it says in wordsBuild the count from quantities you can defend one at a time, and multiply.
Where should the width of the market come from?
From the ranges, multiplied. A shopkeeper who is unsure of both price and quantity is more unsure of revenue than of either. Put a low and a high on every input and multiply the lows together and the highs together: here 5.5 million to 15.0 million, about a factor of 1.6 either side of the centre. Centre the quote near the middle on a multiplicative scale, the geometric mean of the ends, 9.1 million, which sits close to the central estimate.
Then choose the width you will actually trade. A market as wide as the whole range, 5.5 at 15, is useless: nobody trades against it and it tells the interviewer you have no view. Quote tighter, 8 at 11, and move it as they trade: if they keep buying at 11, raise both sides, because their trades carry information. Say the scope questions too: does the count include adult incontinence products, and a school-age child in night-time pants? Those can move the answer more than any of the four inputs.
Where candidates lose it
The first loss is giving one number, or a market whose width is a round guess such as plus or minus a million, with no link to the assumptions. The interviewer wants to see where the width came from.
The second loss is the opposite: a market so wide it is safe and worthless. Show the full range, then quote a tighter two-way price and explain how you would move it when they trade.
What the interviewer asks next
I buy 5 lots at your offer. Where is your market now?
What single piece of data would you buy to narrow the range most, and why?
How would you size the market if the settlement were a count of nappies sold rather than used?
Asked at DRW, Trading, London, 2025 (Wall Street Oasis): Make me a market on the amount of diapers used in the UK daily

payoff	10 - 6 = 4 on red, -6 on black
E[max(payoff, 0)]	your value when you can choose after seeing the colour
max(E[payoff], 0)	your value when you must choose blind

B	births a year, an assumption
Y	years a child spends in nappies
c	changes a day
d	share of families using disposables

Try it first

How do you value a piece of information?

Why is it not worth Rs 4 or Rs 5?

When is information worth the most?

Where candidates lose it

What the interviewer asks next

Try it first

How do you build the estimate so the interviewer can follow it?

Where should the width of the market come from?

Where candidates lose it

What the interviewer asks next