Inspired by A game with 52 cards
Alice and Bob play the following game with two (identical) standard decks of $52$ cards.
- First Alice secretly arranges one deck of $52$ cards in a long row on the table. All cards are face-down, and Bob has no knowledge of the positions of any of the cards.
- Then the game goes through several rounds. In every round, Bob first pays $1$ Euro to Alice. Then Bob arranges the second deck of cards in a long row on the table, parallel to Alice's row and with all the cards face-up. Then Alice tells Bob all the face-up cards in his row that agree with the corresponding face-down cards in her row. Then the next round starts.
- The game ends, as soon as Bob knows the positions of all the face-down cards in Alice's row. Bob then receives $x$ Euros from Alice.
The original problem asked for the minimum risk-free wager. Instead:
What is the minimum payout $x$ for Bob to break even on average?