# Price a Sequential Four-Box Search Game Four boxes contain one \$100 prize. Each opening costs the same fee `x`; you choose an unopened box and may stop after any failure. Find the largest `x` for which starting optimally has nonnegative expected value. Then, after deriving that break-even fee, decide whether a player who paid that fee for the first opening and failed should continue. ### Constraints & Assumptions - The prize is equally likely in each box. - Each additional opening costs `x`. ### Clarifying Questions to Ask - Should the continuation decision compare only future conditional value and future costs? - Does finding the prize end the game immediately? ### What a Strong Answer Covers - Backward induction, stopping choices, the break-even fee, and sunk-cost reasoning. ### Follow-up Questions - How does the answer generalize to `m` boxes? - What if unopened boxes have unequal probabilities?