You are one of two bidders in a sealed-bid auction for a box. The box's payout is determined by flipping 200 fair coins: it pays **\$1 for every coin that lands heads**, so the payout is somewhere between \$0 and \$200. None of the flip outcomes are revealed until after the auction. Both bidders simultaneously submit a single sealed bid; the higher bid wins the box and pays their own bid, and the loser pays nothing. ### Constraints & Assumptions - The 200 coins are independent and fair, and the payout is exactly \$1 per head. - First-price sealed-bid auction, one shot, exactly two bidders; ties are broken by a fair coin flip. If the interviewer leaves the format open, state the format you are assuming before you commit to a number. - Bids can be any non-negative dollar amount (not restricted to whole dollars). - Assume both bidders are rational and risk-neutral unless you explicitly argue for something else. - In Part 2, the information structure is common knowledge: you know the opponent has seen exactly 10 flips, and they know you have seen none. ### Clarifying Questions to Ask - Is this a first-price or second-price auction, and are the bids sealed and simultaneous? - Am I bidding against one rational counterparty, or against the interviewer playing some fixed (possibly naive) strategy? - Is the auction one-shot or repeated? Repeated play changes what I can learn about my opponent. - Should I treat myself as risk-neutral, or do you want me to discuss risk preferences given the stakes? - Can I bid any real amount, and how are ties resolved? ### Part 1 Neither bidder has any information about the flips. How much do you bid for the box, and why? Walk the interviewer through your reasoning, not just the number. ```hint Anchor on expected value The payout in dollars is a $\text{Binomial}(200, 1/2)$ random variable. Compute its mean and standard deviation first — a risk-neutral bidder who pays their own bid should never go above the mean. ``` ```hint What does competition do to your margin? Your profit is (value $-$ bid) only in the states where you win. If both bidders share identical information and reasoning, what happens when one of you tries to preserve a healthy margin? Ask yourself whether that margin is a stable strategy once your opponent can react to it. ``` #### What This Part Should Cover - Correct expected value (and spread) of the box's payout, computed quickly. - Framing the bid as a trade-off between the probability of winning and the margin conditional on winning. - Reasoning about what symmetric competition between two identically-informed, rational bidders implies about how much margin either one can sustainably keep — carried through to a conclusion, not just asserted. - Explicitly stating assumptions (auction format, opponent rationality, risk neutrality) before naming a bid. ### Part 2 Now suppose that before bidding, your opponent secretly observed the outcomes of 10 of the 200 coins. You do not get to see anything, and you do not know what they saw — but you know they have seen 10 flips. How does your bidding strategy change, and why? ```hint Decompose the opponent's knowledge If the opponent saw $H$ heads among their 10 coins, work out how their conditional expected value of the box depends on $H$, and how their bid should move as $H$ changes. Given that, think about which auctions you actually end up winning against a bidder whose bid tracks $H$. ``` ```hint Condition on winning Evaluate the box's expected value conditional on the event that your bid wins — not its unconditional mean. Ask what winning against an informed opponent tells you about what they likely saw, and price that information into your bid before naming a number. ``` #### Clarifying Questions for this Part - Are the 10 observed coins chosen at random, and does the opponent see them before submitting their single sealed bid? - Is the information asymmetry common knowledge — do they know that I know they have seen 10 flips? #### What This Part Should Cover - The adverse-selection mechanism: winning against a better-informed opponent is itself informative, so you must value the box conditional on the event of winning rather than using its unconditional mean (the winner's curse). - A concrete model connecting what the opponent observed to their conditional valuation and their bidding behavior. - A clear, reasoned argument for how the uninformed bidder's strategy should shift relative to Part 1 — the direction and the mechanism, worked through rather than an asserted number. - Who captures the surplus in this auction, and why the information itself is what's valuable. ### What a Strong Answer Covers Across both parts, the interviewer is listening for: - A clean separation between "what is the box worth" (valuation) and "what should I bid" (strategy against a specific opponent). - Consistent game-theoretic reasoning — treating the opponent as a best-responder by default, while noting how the answer changes against a naive opponent. - Fast, correct quantitative work: means, standard deviations, and conditional expectations done in your head or on paper without stalling. - Stating assumptions up front and adapting gracefully when the interviewer perturbs the setup. ### Follow-up Questions - What if the opponent had observed all 200 coins before bidding? What is the most you should ever bid, and what is your expected profit? - How does your analysis change in a second-price (Vickrey) auction — does the winner's curse disappear? - What is the maximum amount you would pay, before the auction, for the right to observe 10 coins yourself? - Suppose the payout were $1{,}000$ per head instead of \$1. Does risk aversion change your bid, and how would you quantify that?