There has not been a puzzle here in a while, so…

Alice is playing a card game with her nephew, Bob, using a standard deck of playing cards. Alice starts with the 26 red cards in her hand, and Bob starts with the 26 black cards in his hand. In each of 26 rounds (or tricks), Alice and Bob simultaneously select and play one card from their remaining hand. The player with the higher-ranked card takes the trick; cards of the same rank are discarded with neither player taking the trick. After all cards have been played, Alice wins (loses) one point for every trick won (lost).

At this point, by symmetry, this seems like a fair game. To skew things slightly in her nephew’s favor, Alice suggests a wrinkle: aces are low… except for one of Bob’s aces, the ace of spades, which is high, ranking higher than all other cards.

1. What are Alice’s and Bob’s optimal strategies and corresponding expected score in this game?
2. Suppose that instead of a point for every trick, the player with the most tricks wins the game (with a score of +1, or 0 if both players take the same number of tricks). What are Alice’s and Bob’s optimal strategies and expected score?
3. How does this generalize to arbitrary initial hands? That is, suppose Alice and Bob start with known card ranks and , respectively (with all cards known to both players).