Sealed-Bid Auction for a Box of 200 Coin Flips (and an Informed Opponent)

You are one of two participants bidding in a sealed-bid auction for a box. The box pays out based on 200 independent fair coin flips: you receive $1 for each head, so the total payout ranges from $0 to $200. No flip results are shown until after the auction ends. Both bidders submit one sealed bid at the same time; the higher bidder gets the box and pays their bid, while the other pays nothing.

Constraints & Assumptions

• The 200 coins are fair and independent, and each head pays exactly $1.
• This is a one-shot, first-price sealed-bid auction with exactly two bidders; ties are settled by a fair coin toss. If the interviewer does not specify the format, state your assumed format before giving a bid.
• Bids may be any nonnegative dollar amount, not just whole dollars.
• Unless you explicitly argue otherwise, treat both bidders as rational and risk-neutral.
• In Part 2, the information structure is common knowledge: you know the opponent has observed exactly 10 flips, and they know you have observed none.

Clarifying Questions to Ask

• Is the auction first-price or second-price, and are bids sealed and submitted simultaneously?
• Am I facing one rational counterparty, or is the interviewer using a fixed, possibly naive strategy?
• Is this a single auction or a repeated game? Repetition changes what I can learn about the other bidder.
• Should I assume risk neutrality, or do you want me to address risk preferences given the stakes?
• Can I bid any real amount, and how are ties broken?

Part 1

Neither bidder knows anything about the coin flips. What should you bid for the box, and why? Explain your reasoning to the interviewer rather than just stating a number.

Hint — Start from expected value: The dollar payout is a $\text{Binomial}(200, 1/2)$ random variable. Start by computing its mean and standard deviation — a risk-neutral bidder who pays their own bid should never bid above the mean.

Hint — How does competition affect your margin?: Your profit is (value $-$ bid) only in the cases where you win. If both bidders have the same information and reason the same way, what happens when one tries to keep a comfortable margin? Ask whether that margin remains stable once the other bidder can respond.

What This Part Should Cover

Part 2

Now suppose that before bidding, your opponent privately observes the results of 10 of the 200 coins. You see nothing and do not know which outcomes they saw, but you know they have seen 10 flips. How should your bidding strategy change, and why?

Hint — Decompose the opponent's knowledge: If the opponent saw $H$ heads among their 10 coins, determine how their conditional expected value of the box depends on $H$, and how their bid should change as $H$ changes. Then think about which auctions you actually win against a bidder whose bid follows $H$.

Hint — Condition on winning: Compute the box's expected value conditional on your bid winning, not its unconditional mean. Ask what beating an informed opponent reveals about what they probably saw, and incorporate that information into your bid before naming a number.

Clarifying Questions for this Part

• Are the 10 observed coins selected at random, and does the opponent see them before submitting their one sealed bid?
• Is the information asymmetry common knowledge — do they know that I know they have seen 10 flips?

What This Part Should Cover

What a Strong Answer Covers

Follow-up Questions

• What if the opponent had observed all 200 coins before bidding? What is the highest bid you should ever make, and what is your expected profit?
• How does your analysis change in a second-price (Vickrey) auction — does the winner's curse go away?
• What is the most 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?

Overview: This question tests probability and expected-value reasoning, auction and game-theoretic strategic thinking, Bayesian inference under asymmetric information, and quantitative decision-making relevant to a machine-learning-focused data scientist role.