Price a Sequential Four-Box Search Game
Company: Worldquant
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
# Price a Sequential Four-Box Search Game
Four boxes contain one \$100 prize. Each opening costs the same fee `x`; you choose an unopened box and may stop after any failure. Find the largest `x` for which starting optimally has nonnegative expected value. Then, after deriving that break-even fee, decide whether a player who paid that fee for the first opening and failed should continue.
### Constraints & Assumptions
- The prize is equally likely in each box.
- Each additional opening costs `x`.
### Clarifying Questions to Ask
- Should the continuation decision compare only future conditional value and future costs?
- Does finding the prize end the game immediately?
### What a Strong Answer Covers
- Backward induction, stopping choices, the break-even fee, and sunk-cost reasoning.
### Follow-up Questions
- How does the answer generalize to `m` boxes?
- What if unopened boxes have unequal probabilities?
Overview: Four boxes contain one \$100 prize. Explain the assumptions and derivation clearly, check edge cases, and show how the result changes when those assumptions no longer hold.
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