Optimal Stopping Strategy for Three Die Rolls

This problem presents a simple optimal stopping game built around rolling a single fair die up to three times, choosing when to lock in a payout. The candidate must reason about whether to accept a known outcome now or continue in the hope of a better future roll, under a hard cap on the number of attempts. The setup is deliberately small and discrete so that the decision tree is fully tractable, making it a common style of screening question for quant research and trading roles where quick, structured reasoning about uncertainty is essential.

Solving it leans heavily on backward induction and comparing immediate payoffs to expected future values. The candidate needs to formalize the future choices into a clean recursive structure and compute the continuation value at each stage. Interviewers watch for a disciplined step-by-step approach rather than ad hoc intuition, correctness of expectations under a uniform discrete distribution, and clarity in translating a verbal game into a quantitative decision rule. Good answers often also comment on how the strategy would generalize beyond three rolls.

What it tests

Optimal stopping problems are governed by the principle of backward induction, where decisions at each stage are made by comparing the immediate reward to the expected value of continuing. The key structure is that at each decision point, the optimal action is to stop if the current offer exceeds the expected value of all future options, and to continue otherwise. This pattern holds because the future is uncertain but statistically predictable, so the rational choice is to maximize expected value at every step. The process is recursive: the value at each stage depends on the optimal decisions in subsequent stages, which are solved first. This approach transforms a seemingly complex sequence of choices into a series of simple comparisons, each grounded in the expected value of what lies ahead.