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Pricing a Mystery Money Box: Expected Value, Adverse Selection, and Competition

Jane StreetQuantitative Trader · Technical Screen
MediumQuantitative TraderTechnical ScreenMachine Learning

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This question evaluates probabilistic reasoning and economic-incentives competencies, including expected-value calculation, risk-neutral decision-making, adverse selection, and surplus division in bargaining contexts.

A company manufactures sealed boxes. Each box contains a cash amount that is an integer number of dollars drawn uniformly at random from $1$ to $100$ (each of the 100 values equally likely). A dealer sells these boxes, and you are negotiating with the dealer over the price of a single box.

The rules of the game change across the parts below. Treat each part as a fresh variation, and in each one reason explicitly about the price you would offer, the expected payoff for each party, and the strategy behind your answer.

Constraints & Assumptions

  • The box's contents $y$ are uniform on the integers ${1, 2, \dots, 100}$, so $E[y] = 50.5$.
  • Both you and the dealer are risk-neutral and maximize expected profit.
  • Offers/prices do not have to be integers.
  • Each part is a one-shot interaction unless the part says otherwise.
  • You may use a continuous-uniform approximation ($y \sim U(0, 100)$) where it simplifies the algebra — just say when you are doing so.

Clarifying Questions to Ask

  • Is the dealer risk-neutral like me, or risk-averse?
  • Who can observe the contents of the box, and at what point in the interaction?
  • Is the price a take-it-or-leave-it offer from me, or is there back-and-forth bargaining?
  • Is this a single trade, or a repeated relationship where reputation matters?
  • Can we make binding agreements before the box's contents are drawn or revealed?

Part 1

The dealer will sell you a box at whatever price you offer. What is the highest price you should be willing to pay for a box? If you buy at price $x$, what is your expected profit?

Part 2

Now suppose the dealer's profit on a sale is the sale price minus the money inside the box — the dealer is personally on the hook for the contents.

  1. Is there any price at which both you and the dealer want to trade? If a dealer did agree to sell at a price attractive to you, what should that tell you?
  2. The company now pays the dealer a $10 bonus for every box sold. What price should you offer now, and why?

Part 3

Now the dealer can open the box and see the money inside before deciding whether to accept your offer. The $10-per-sale bonus still applies. You make a single take-it-or-leave-it offer of $x$.

  1. For a given offer $x$, when does the dealer accept? Derive your expected profit and the dealer's expected profit as functions of $x$.
  2. What offer maximizes your expected profit? How do the resulting payoffs compare to Part 2?
  3. Given how much value this information destroys, what should the two of you do instead?

Part 4

Return to the setting where nobody can see inside the box (dealer still earns price minus contents plus the $10 bonus). Now a second buyer shows up, and the two of you compete for the same box. What happens to the price, and who ends up with the surplus?

Hints

Hint 1

Where to start: For a risk-neutral buyer, the break-even price is the expected value of the contents. What is the mean of a uniform distribution over the integers $1$ through $100$?

Hint 2

Think about total surplus: Add up your profit and the dealer's profit on a trade. In the no-bonus version, what does the sum equal — and what does that imply about mutually beneficial trade?

Hint 3

Splitting the pie: With the bonus, the trade creates a fixed amount of total surplus. The situation is symmetric between the two parties — how should symmetric parties split a fixed surplus, and what price achieves that split?

Hint 4

Acceptance condition: Write down the dealer's profit on a specific box containing $y$ if they sell at price $x$, including the bonus. The dealer accepts exactly when that quantity is positive.

Hint 5

Adverse selection: Once the dealer screens boxes, the boxes you actually receive are no longer uniform over $1$–$100$. Compute the expected contents **conditional on the dealer agreeing to sell**.

Hint 6

The fix: The dealer's peek only redistributes value ex post, but it kills trades ex ante. Is there an agreement the two of you could commit to *before* looking that restores the Part 2 outcome?

Hint 7

Undercutting logic: If the current best offer leaves the winning buyer any positive expected profit, what can the other buyer do? Where does that process stop?

Follow-up questions

  • In Part 2, how does your offer change if the dealer is risk-averse while you remain risk-neutral?
  • In Part 3, generalize the bonus to $B$ dollars. How do the buyer-optimal offer and both expected payoffs scale with $B$?
  • In Part 3, if you interact with the same dealer repeatedly, can the "commit to always trade at a fixed price" agreement be sustained without an enforceable contract? What tempts the dealer to defect, and when?
  • In Part 4, what happens if the two buyers can collude before bidding — and how does the dealer's outcome change?