The key branch is whether to bid in the 3000-6999 range or go up to 7000+.
First, identify the possible bid ranges and the types of sellers who would accept each bid. Then compute the expected profit for a representative bid in each range, considering the conditional probability of the item type given that the seller accepts. Finally, compare the expected profits to find the optimal bid.
Pro tip: In market-making interviews, always clarify whether the probabilities are unconditional or conditional on the seller's acceptance. Here, the 0.7 and 0.3 are unconditional, so you must use Bayes' rule to update after observing a sale.
Determine the minimum bid required for each seller type to sell: defective sellers accept bids ≥3000, genuine sellers accept bids ≥7000. Thus, bids fall into three ranges: below 3000 (no sale), 3000–6999 (only defective sellers accept), and ≥7000 (both types accept).
For a bid b in this range, only defective sellers accept. The item is certainly defective, worth 7000. Expected profit = 7000 - b. This is maximized at the lowest bid, b=3000, giving profit 4000.
For a bid b ≥7000, both seller types accept. Use Bayes' rule to find the probability the item is defective given a sale: P(defective | sale) = 0.7. Expected value = 0.7*7000 + 0.3*10000 = 7900. Expected profit = 7900 - b. This is maximized at the lowest bid, b=7000, giving profit 900.
Compare the maximum expected profits: 4000 (from bidding 3000) vs. 900 (from bidding 7000). The optimal single bid is 3000, yielding an expected profit of 4000.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clearly defining the bidding setting and the signal structure, then apply Bayes' rule to update the prior distribution of the item's value. Recompute the expected value conditional on the signal and derive the optimal bid as the maximum of the posterior expected value minus the bid (or the posterior quantile for a first-price auction).
Pro tip: Explicitly state your assumptions about the signal's informativeness and independence, and discuss how the optimal bid changes with signal precision—this shows you understand the practical implications of partial information.
Specify the prior distribution of the item's value, the auction type (e.g., first-price, second-price), and the signal's conditional distribution given the value. Clarify whether the signal is discrete or continuous and its relationship to the value.
Compute the posterior distribution of the value given the observed signal using Bayes' theorem: P(Value | Signal) ∝ P(Signal | Value) * P(Value).
Calculate the expected value of the item under the posterior distribution, E[Value | Signal]. This is the key quantity for risk-neutral bidding.
For a second-price auction, the optimal bid is the posterior expected value. For a first-price auction, solve for the bid that maximizes expected profit, often involving the posterior distribution's quantiles or solving a differential equation.
Explain how the optimal bid depends on the signal's precision and the prior. Discuss edge cases (e.g., uninformative signal reduces to prior) and practical considerations like risk aversion or competition.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the setup: the item values are now drawn from an exponential distribution with a given rate parameter, and the bidding strategy is based on expected profit maximization. Then, derive the new optimal bid by solving the expected profit integral, and compare it to the discrete case, highlighting how the continuous distribution changes the bid shading and risk profile.
Pro tip: Emphasize that the exponential distribution's memorylessness and skewness lead to a constant absolute bid shading (or a proportional bid if values are scaled), which is a key insight for real-time bidding systems. Also, mention that in practice, you'd validate the model with simulation and consider robustness to parameter misestimation.
Restate the assumptions: item values V ~ Exponential(λ), bids are submitted, and the highest bid wins at the second-highest price (or first-price, depending on context). Confirm the objective: maximize expected profit = (V - bid) * P(win).
For a first-price auction with one bidder, the optimal bid is the expected value conditional on winning, which for exponential is V - 1/λ (if V is the value). For a second-price auction, the optimal bid is your true value, so no shading. Clarify which auction format applies.
In the discrete case, the optimal bid is typically the highest value below your true value (or a shading factor). With exponential, the bid shading becomes a constant absolute amount (1/λ) rather than a percentage, due to the memoryless property.
Discuss how the continuous distribution affects risk: exponential has a long tail, so occasional high values may lead to aggressive bidding. Also, consider how the bid changes with λ: higher λ (lower mean) leads to smaller shading.
Suggest simulation to verify the analytical result, and mention extensions: multiple bidders, unknown λ, or other distributions (e.g., log-normal) for robustness.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.