Choose a Numbered Deck Against an Optimal Repeated Guesser

Build a deck using any number of copies of cards labeled 1 through 5. An opponent knows its composition. Each round, the opponent names a number i and draws a uniformly random card. Whether the guess is right or wrong, the opponent pays you i. If correct, the game ends. If wrong, the card is returned, the deck is reshuffled, and another round begins. The opponent minimizes expected total payment. How should you choose the deck to maximize that minimum expected payment? ### Constraints & Assumptions The deck is finite and nonempty, sampling is with replacement after an incorrect guess, and the opponent can choose a new guess each round. There is no fee for deck size and no limit on the number of rounds. The payment on the successful round is included. ### Clarifying Questions Is a wrong guess paid? Is the card returned? Can the opponent change guesses using previous observations? Are zero copies of a label allowed? ### What a Strong Answer Covers Express the opponent's value in terms of label probabilities, justify why adaptive guesses cannot beat the optimal stationary choice, and find an attainable maximizing deck composition. ### Follow-up Questions What happens if one label is absent? Why is a uniform deck not optimal here? How does the answer generalize to positive payments associated with more labels?