← Jane Street Interview Insights
Expected value is $100, standard deviation around $7.
First, recognize that the box's expected value is $100 (200 flips * $0.50 per flip). In a first-price sealed-bid auction with one other bidder, your optimal bid depends on your belief about the opponent's bid. Assuming the opponent is rational and also values the box at $100, you should bid below $100 to ensure positive expected profit, but high enough to win. A common strategy is to bid a fraction of the expected value, such as $50, but you should justify based on game-theoretic reasoning.
Pro tip: In first-price auctions, bidding your true value guarantees zero profit even if you win. Instead, shade your bid to balance winning probability and profit margin. Mention that if the opponent is naive and bids $100, you can bid $99 and profit $1, but if they are sophisticated, you need to consider mixed strategies.
Calculate the expected value of the box: 200 flips * $0.50 per head = $100. This is the maximum you should be willing to pay.
Recognize it's a first-price sealed-bid auction: highest bidder wins and pays their bid. Your profit is (value - bid) if you win.
Assume the opponent is rational and also values the box at $100. In symmetric equilibrium, both bid a fraction of the value. For two bidders with uniform values, optimal bid is half the value, so $50.
Based on the equilibrium, bid $50. This maximizes expected profit given the opponent's likely bid. If you believe the opponent will bid differently, adjust accordingly.
Explain that bidding $50 balances winning probability and profit. Bidding higher increases chance of winning but reduces profit; bidding lower increases profit but reduces chance of winning.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Recognize that your opponent's private information creates an adverse selection problem, so you must shade your bid downward to avoid overpaying. Quantify the expected value of their information advantage and adjust your bidding strategy to account for the winner's curse.
Pro tip: Frame the problem in terms of information asymmetry and the winner's curse—this shows you understand market microstructure and can apply game theory to real-world pricing decisions.
State that your opponent has a private signal from seeing 10 flips, which gives them an edge. You must assume they will only bid aggressively when their information suggests a favorable outcome.
Estimate how much their 10 observations can shift the expected value of the remaining 190 flips. For example, if they see more heads, the overall proportion of heads is likely higher, so the coin may be biased.
Since you only win the auction when your opponent bids lower (likely because their private information is unfavorable), your expected value conditional on winning is lower than the unconditional expected value. Shade your bid downward accordingly.
Use Bayesian updating to compute the expected value given that you win. The optimal bid is the expected value conditional on winning, which may be significantly below the unconditional expected value.
If the auction format allows, you might bid more conservatively or even pass if the shading required is too severe. Alternatively, you could try to infer their information from their bidding behavior, but that's risky.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Recognize that if your opponent knows the exact sequence of 200 flips, they have perfect information and will only accept a bid if it guarantees them a profit. Therefore, you must bid at or below the minimum possible value of the game to avoid adverse selection. Compute the minimum sum of 200 fair coin flips (all tails = 0) and bid accordingly, yielding an expected profit of 0.
Pro tip: In adversarial settings, always consider the worst-case scenario for you, not the average. Here, the opponent's perfect information means you must price at the lower bound to avoid losing money.
The opponent knows the exact sequence of 200 flips, so they will only accept your bid if it is favorable to them. You must assume they will exploit any overbid.
The sum of 200 fair coin flips (0 for tails, 1 for heads) ranges from 0 to 200. The minimum possible sum is 0 (all tails).
To ensure you never lose money, you must bid no more than the minimum possible value, which is 0. Any positive bid could be accepted only when the true sum is less than your bid, causing a loss.
If you bid 0, the opponent will accept only if the sum is 0 (since they are indifferent at 0, but assume they accept if it's at least 0). Your profit is 0 - 0 = 0. If they reject, profit is 0. So expected profit is 0.
Bidding negative would guarantee a loss if accepted, so the optimal bid is exactly 0, yielding expected profit 0.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Second-price makes truthful bidding dominant for a symmetric uninformed bidder, but the winner's curse doesn't fully vanish when there's asymmetric information.
First, clarify the distinction between first-price and second-price auctions and how bidding strategy changes. Then, analyze whether the winner's curse persists in a second-price auction, considering rational bidding and information asymmetry. Finally, discuss implications for data science and pricing decisions.
Pro tip: Emphasize that in a second-price auction, bidding your true value is a dominant strategy, but the winner's curse can still occur if bidders are uncertain about the item's value and fail to account for the information revealed by winning.
Briefly explain first-price (winner pays their bid) and second-price (winner pays the second-highest bid) auctions, highlighting the strategic differences.
State that in a second-price auction, bidding your true valuation is a weakly dominant strategy, unlike in first-price where you shade your bid.
Discuss whether the winner's curse persists: even with truthful bidding, if bidders have common value uncertainty, winning reveals that others had lower estimates, which can lead to overpayment relative to the true value.
Connect to data science: in practice, models must account for selection bias and information asymmetry; the winner's curse may manifest as overestimating the value of a won auction or ad placement.
Summarize that while second-price auctions simplify bidding, the winner's curse does not necessarily disappear; it depends on whether bidders rationally incorporate the information from winning.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Framed as a value of information question.
Recognize this as a classic expected value problem where you pay a fixed amount upfront for the right to observe 10 fair coin flips, and then presumably bet on the outcomes. Calculate the expected profit per flip by determining the optimal bet size and direction given your information advantage, then multiply by 10 and discount for risk and the upfront payment. The maximum you'd pay is the expected value of the information minus any risk premium or costs.
Pro tip: Frame your answer in terms of expected value and risk-adjusted returns, and explicitly state your assumptions about the betting structure (e.g., even-money bets, ability to bet any amount). This shows you can handle ambiguity and think like a trader.
Ask clarifying questions: Is it a fair coin? Can you bet on each flip? What are the betting rules (even money, bet size limits)? Are you obligated to bet? This ensures you understand the problem before calculating.
For a fair coin, if you can bet on the outcome after observing the flip, you have a guaranteed win. With even-money bets, the expected profit per flip is the amount you bet. If you can bet any amount, the value is theoretically infinite, so assume a bet size limit or that you can only bet a fixed amount.
Multiply the expected profit per flip by 10 to get the total expected value from observing all flips. If there's a bet size limit, use that; otherwise, assume a reasonable limit or that you can bet your entire bankroll each time (leading to exponential growth).
Since the payment is upfront and the returns are uncertain (if the coin is not fair or if you can't bet), apply a discount for risk. The maximum you'd pay is the risk-adjusted expected value, which might be less than the theoretical expected value.
Give a concrete number or range based on your assumptions, and explain how it would change with different parameters. Emphasize that the answer depends on the betting structure and risk tolerance.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.