Value a Die Game with Optional Continuation

You repeatedly roll a fair six-sided die. You always collect the number rolled. If the roll is 1 or 2, the game ends immediately. If the roll is 3, 4, 5, or 6, you may either stop with your accumulated winnings or continue rolling. Assuming risk-neutral preferences and no discounting, what is the fair price of the game and what is the optimal stopping policy? Derive the value rather than estimating it by simulation. ### Constraints & Assumptions - Winnings accumulate across rolls. - The decision to continue is made after collecting a 3, 4, 5, or 6. - There is no fixed horizon, transaction cost, or loss on a later roll. - “Fair price” means expected monetary value to a risk-neutral player. ### Clarifying Questions to Ask - Are the amounts accumulated or does only the final roll pay? - Is stopping allowed after a 1 or 2, or is termination forced? - Does continuing ever forfeit prior winnings? - Is the player risk neutral? ```hint Write a continuation-value equation After any optional stopping point, the future game has the same value as it did before the first roll. ``` ### What a Strong Answer Covers - A Bellman or first-step equation for the continuation value. - Justification that continuing after 3 through 6 is optimal. - The numerical expected value and a distinction between fair value and personal willingness to pay. - A check that the expected game length is finite. ### Follow-up Questions - How would a per-roll fee change the stopping decision? - What if a later roll of 1 or 2 erased all accumulated winnings? - How would risk aversion affect the maximum entry price?