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? ```hint 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 $100$? ``` #### What This Part Should Cover - Correct expected value of the discrete uniform distribution. - Framing the decision as expected profit relative to a break-even price. - Cleanly stating expected profit as a function of the offer $x$. ### 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? ```hint 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 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? ``` #### What This Part Should Cover - Computing what the sum of your profit and the dealer's profit equals in the no-bonus game, and reasoning through what that implies about whether a mutually agreeable price can exist. - Identifying that the \$10 bonus is the entire source of surplus. - A symmetry/fairness argument that pins down a specific price and the resulting expected profit for each side. ### 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? ```hint 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 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 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? ``` #### What This Part Should Cover - The correct acceptance rule and the conditional distribution of contents given a trade. - Expected-payoff expressions for both parties as functions of $x$, and the buyer-optimal offer. - Quantifying the collapse in payoffs relative to Part 2 and naming the phenomenon (adverse selection). - A commitment/coordination strategy that restores the full surplus, and what makes it credible or fragile. ### 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? ```hint 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? ``` #### What This Part Should Cover - Bertrand-style competition logic: bidding continues while positive expected profit remains. - The limiting price and each buyer's expected profit at that price. - Who captures the \$10 surplus once buyers compete. ### What a Strong Answer Covers Across all parts, the interviewer is testing whether you can track one economic object — the \$10 of surplus per trade — through changing information and competition structures: - Fluent expected-value computation and clean payoff algebra in every variant. - Identifying in each part where surplus comes from, who captures it, and why (bargaining symmetry, private information, competition). - Recognizing adverse selection: private information can destroy total surplus, not just redistribute it. - Understanding commitment as a remedy — agreeing not to use information can make both parties better off. - Explicitly stating assumptions (risk neutrality, tie-breaking on indifference, continuous approximation) rather than silently relying on them. ### 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?