Jane Street Quantitative Researcher Interview Question: Optimal Stopping Strategy for Sequential Dice Rolls
The process of optimal stopping is central to both theoretical probability and real-world quantitative finance. A classic problem that tests this concept is the sequential dice roll scenario, commonly posed in interviews for quantitative research positions at elite firms like Jane Street. In this article, we will tackle the question: What is the optimal stopping strategy for sequential dice rolls, and what is the expected payoff if you play optimally? We’ll break down the probability theory, dynamic programming approach, and provide mathematical and code-based solutions, so you can master this elegant interview puzzle.
You are given an opportunity to roll a fair six-sided die (d6) repeatedly. After each roll, you may choose to stop and accept the value shown, or continue rolling. If you choose to continue, you forfeit the current roll’s value forever and must accept the outcome of a future roll (or the process can be allowed to continue indefinitely, or up to a certain maximum number of rolls). The central question is:
This is a textbook example of an optimal stopping problem, a concept with deep applications in finance, statistics, and decision theory.
