Question
Alice and Bob trust each other a lot, so they play a game of deterministic poker. Deterministic poker proceeds under the following rules: A standard deck of cards is displayed in the center completely face-up. Alice draws any $5$ cards of her choice from the center, and then Bob does the same. Alice can discard any amount of her $5$ selected cards, in which they will remain out-of-play. If she discards $k$ cards, she can select any $k$ cards from the center to replace them. Bob does the same as the above step. The person with the stronger hand (in the sense of standard poker rules) is the winner. If they have equally strong hands, the game ends in a tie. The cards in the hands of both players are visible at all times. Under optimal play from both players, who has a winning strategy?