Suppose a standard deck of playing cards is shuffled and dealt into one card piles on a table.

Each player flips cards one by one.

The players compare the values of the cards (J=11, Q=12, K=13).

If a player has a card higher than the other player's, the difference represents money in dollars the player with the lowest card owes.

The revealed cards are removed from the game.

Any player can end the game after a round where cards have been flipped.

If both players are rational and count the cards that have been revealed, when would it be rational to stop the game if:

* Both want to maximize dollars earned

* Both want to lose as least as possible.

Using "+2" to mean P1 has earned a net of 2 dollars, and "-2" to mean P1 owes a net of 2 dollars.

ex.

Game begins.

P1 flips 7. P2 flips 9. Net: -2

P1 flips 4. P2 flips 1. Net: +1

...

Solution Ideas:

I do assume both players are greedy and will not cash out as soon as any amount is earned (ends at first draw), but it certainly is a practical strategy. (say the first round, P1 flips K and P2 flips 1)

Try with a smaller set of cards to try and find a pattern. (ex. 4 aces, 4 twos, 4 threes)

Calculate the expected value the game would earn and as soon as the value is reached for either player, end the game.

Try with replacement first as an estimate (cards wouldn't be removed from the game, but would be reshuffled)

Simulate the experiment through programming where draws are exhausted and identify the step for each experiment where the absolute value was the maximum for the entire game.

Ideas are appreciated. A full solution is not required.