# Optimize a Two-Box Place-or-Take Game A one-player game has two opaque boxes and lasts exactly 100 turns. On a place turn, a third party adds one unit of money to one uniformly random box. On a take turn, one uniformly random box is emptied and its contents are paid to you. You do not observe the contents of either box or the amount collected until the game ends. Determine an optimal policy and the expected payoff. ### Constraints & Assumptions - Every random box choice is independent and has probability one half for each box. - Both boxes start empty. - A take action leaves the chosen box empty and does not reveal which box was chosen. - A policy may depend on the turn number and your previous actions, but not on hidden outcomes. ### Clarifying Questions to Ask - Does the player learn which box was selected by either action? - Must all 100 turns be used, or may the player stop early? - Is utility linear in final money, so only expected payoff matters? ```hint Use symmetry When the boxes are observationally identical, track the expected total or a symmetric value function. ``` ```hint Work backward At each remaining-turn state, compare the expected continuation after place with the continuation after take. ``` ### What a Strong Answer Covers - A state based on sufficient expected box contents rather than unobservable histories. - A backward-induction recurrence comparing place and take actions. - A justified optimal policy and a reproducible numerical expected value. - An explanation of why hidden payouts do not create an exploitable observation strategy. ### Follow-up Questions - How would observing the amount taken change the state needed by the policy? - What changes if take lets you choose a box rather than selecting one randomly?