부산대학교 도서관

This paper analyses the two-player card game Le Her. From the viewpoint of game theory, this is a two-player zero-sum game without perfect information. The Minimax Theorem by von Neumann (1928) guarantees the existence of optimum mixed strategies for all games within this category. This is a very important result on the existence of a ``solution'' for the game, but it does not explain how to find the optimum strategies or the value of the game if one is not able to construct the matrix associated to the two-player zero-sum game in normal form. Once the matrix for the game in normal form is known, the ``solution'' may be obtained by solving an associated linear programming problem. Hence, each particular game requires a specific analysis and the difficulty of analysing it lies in how difficult it is to obtain the matrix. \par The aim of the paper is to find optimum mixed strategies as well as the value of the two-player card game Le Her. The earliest known solution to Le Her was provided by James Waldegrave in 1713. Waldegrave did his study for one standard deck. In 2002, Bejamin and Goldman gave a complete solution to a variant of the matrix two-player card game, with only one suit allowed; they called it the $N$-card version of Le Her. In this paper Vanniasegaram considers a new version for Le Her. His approach allows several suits ($s$) and several denominations ($d$), instead of the standard $13$ denominations. \par In order to find optimal strategies for the two-player card game Le Her, Vanniasegaram distinguishes between the cases $d<7$ and $d\geq 7$. To deal with the latter case, he uses the ``convexity'' of the payoff matrix. This technique was previously introduced by A. T. Benjamin\ and A. J. Goldman [J. Optim. Theory Appl. {\bf 114} (2002), no.~3, 695--704; MR1921173 (2003i:91030)]. \par A formula for the value of the game as well as optimal strategies for the players are given. The main and elegant result states that if eighteen or fewer standard decks are used for playing Le Her, the game is advantageous to Player 1. Otherwise, if nineteen or more standard decks are used, the game is advantageous to Player 2. An asymptotic analysis on the value of the game concludes the paper. This study is done for the cases in which $d$ is fixed and $s$ increases, and $s$ is fixed and $d$ increases.