Compute optimal stopping in a die-rolling game

Think about three sequential die-rolling games using a fair six-sided die (faces 1–6). After each roll you can either stop and collect the number showing as dollars, or keep going. If you haven't stopped by the final permitted roll, you are forced to accept that last roll's value.

Game A: At most three rolls, with no fees. Determine the optimal stopping rule and the exact expected payout.

Game B: At most three rolls; deciding to go on after roll 1 or roll 2 costs $1 before the next roll (every continuation costs $1). Find the optimal stopping thresholds and exact expected payout.

Game C: At most three rolls; the second and third rolls each cost $1 to make (that is, you always pay $1 to take roll 2 if you continue after roll 1, and another $1 to take roll 3 if you continue after roll 2). Find the optimal policy and expected payoff.

Bonus: Extend Game B to an n-sided die with faces 1 through n and a continuation cost c; derive the stopping threshold(s) in terms of n and c.

Overview: This problem assesses a candidate's command of optimal stopping theory, expected value calculation, and finite-horizon decisions under uncertainty, including the ability to recognize threshold-based policies and apply backward induction, which are relevant to data scientist roles.