Choose optimal posted price under adverse selection
Company: Imc
Role: Data Scientist
Category: Machine Learning
Difficulty: medium
Interview Round: Onsite
You are negotiating to buy an item whose true quality is unknown to you.
- With probability **0.7**, the item is **defective** and would be worth **$7,000** to you.
- With probability **0.3**, the item is **good** and would be worth **$10,000** to you.
- You make a single take-it-or-leave-it offer price **p**.
- The seller will accept if and only if:
- If the item is defective: **p ≥ 3,000**
- If the item is good: **p ≥ 7,000**
Assume the seller knows the quality; you only know the prior probabilities.
1) As a function of **p**, compute:
- the probability the offer is accepted,
- your expected profit **E[value − p]** (ex ante, i.e., before knowing whether the offer is accepted).
2) Find the offer price **p** that maximizes your ex-ante expected profit.
3) Compute the posterior probability the item is good given that the seller accepts your offer, **P(good \| accept)**, for the key price regions.
4) (Follow-up using exponential distribution) Suppose that if the item is defective, the time-to-failure **T** is exponentially distributed: \(T \sim \text{Exp}(\lambda)\), and if the item is good it never fails in your time horizon. For an offer price **p** in each acceptance region, compute \(\Pr(T \le t \mid \text{accept})\).
Quick Answer: This question evaluates probabilistic reasoning, Bayesian updating, expected-value optimization under asymmetric information, and basic mechanism-design intuition within the Machine Learning domain for Data Scientist roles.