Analysis of N-card Le Her

We present a complete solution to a card game with historical origins. Our analysis exploits convexity properties in the payoff matrix, allowing this discrete game to be resolved by continuous methods. In this paper, we analyze a variant of the card-game Le Her, which has a long history in the mathematical literature (cf. Section 18.6 of [7]). The authentic two-player 52-card version is reported and solved by Dresher [3, 4], with some anticipation by R.A. Fisher [6]. Todhunter [10] describes efforts at its solution by N. Bernoulli and Montmort; retrospectively, their lack of the “mixed strategy” concept can be recognized as crucial. The present version, formulated by Karlin ([9], p. 100) who poses the case N = 5 as a problem, involves a single-suit deck of N ≥ 3 cards with respective face-values 1, 2, . . . , N . Let X,Y, Z denote the top three cards, all face down, after a randomizing shuffle of the deck. Cards X and Y are dealt to Players 1 and 2 respectively, leaving Z on top. Each player inspects his or her own card; thus Player 1 knows X but not Y or Z, while Player 2 knows Y but not X or Z. (Our notation will slur the distinction between a random variable and its realization, without real risk of confusion.) Player 1 moves first. He can either keep X, or else elect a swap of cards with Player 2. In the latter case, both players inspect their new cards, and therefore know both X and Y (but not Z). Next, it is Player 2’s turn to move. She can either keep her current card, or else elect to swap that card for Z. That concludes play: the player holding the higher card wins one unit from the opponent. The pure strategies for Player 1 are associated one-to-one with the subsets S of {1, 2, . . . , N}; playing strategy S, Player 1 keeps X if X ∈ S, and otherwise swaps X for Y . Note that if Player 1 swaps, then Player 2 will know that she holds X while Player 1 is holding Y , and will therefore surely keep X if X > Y , and surely swap X for Z if X < Y , the swap “succeeding” iff Z > Y . Thus a pure strategy for Player 2 need specify that player’s action only when Player 1 keeps his card. Those strategies are associated one-to-one with subsets T of {1, 2, . . . , N}; playing strategy T after a “keep” by Player 1, Player 2 keeps Y if Y ∈ T , and otherwise swaps Y for Z. The joint distribution of (X,Y ) is of course given by P (X,Y ) = p = 1 N(N−1) if X 6= Y , P (X,Y ) = 0 if X = Y . Let P (S, T |X,Y ) denote the probability of a win by Player 1 when respective strategies S and T are employed, conditional on (X,Y ); the corresponding unconditional probability is given by