A two-player card game is played with a deck of cards numbered 1-52, which is shuffled and placed face down. Each player draws a card from the top of the deck, then both players reveal their cards and the player whose card has higher rank scores 1 point. Both cards are then discarded. The process is repeated until both decks have been depleted. The winner of the game is the one with the largest number of points.
Suppose you and your opponent can "cheat" at this game. Subjectively, when you reveal a card from your deck, that card is drawn uniformly at random from the set of all cards that haven't been played yet. Instead, before drawing, you and your opponent may each secretly choose to have their card drawn uniformly at random from a restricted subset of cards. You may choose any subset of two or more specific cards that haven't been played yet, and your opponent may do the same. (Neither of you has any information about whether the other person has cheated or which subset they've chosen, and the choice of subset is made before any cards are drawn that round.) The cards you and your opponent draw are guaranteed by the laws of physics to be distinct.
Is there an effective strategy for cheating that improves your chances of winning over playing fairly, or is the best strategy to play fairly? I have tried to tabulate the payoff matrix for simple cases (n=2, n=4 cards) as well as trying to formulate an argument based on the symmetry of the game, but I haven't been able to prove a result either way yet. Any help is appreciated!
See also: I asked an earlier question about a similar game, where a single deck is split into two halves whose contents are known, and the opponent draws cards randomly instead of adversarially. Optimal way to stack deck against uniformly random opponent? Unlike in that question, in this question both players are drawing from the same pool of unplayed cards (rather than both players knowing the contents of their respective halves and drawing from only those halves.)