Parametric Analysis of Von Neumann and Morgenstern's Discrete Poker Model for Deck Sizes S = 2 Through S = 5

We derive optimal strategies for the discrete poker model of von Neumann and Morgenstern for deck sizes S = 2 through S = 5 for all values of p > 1 where p is the ratio of the high and low bids available to the players. Primary results are (1) for S = 2, there are two optimal pure strategies that apply to different values of p; (2) for S = 3, there are two optimal pure strategies for 1 < p ≤ 2, a family of optimal mixed strategies for 2 < p < 4, discontinuous at the endpoints, and an optimal pure strategy for 4 ≤ p; and (3) for S = 4 and S = 5, there are optimal pure strategies for low and high values of p, as well as three or six optimal mixed strategies, respectively, for intermediate values of p, which exhibit multiple discontinuities. In contrast to the previous authors results for the continuous case (where each player is dealt a real number between 0 and 1), for small values of p, players should always bid high, and for sufficiently large values of p, players should bid high only when dealt the maximum value S. In between, the optimal mixed strategies are numerous and behave precariously. We combine empirical results with formal analysis to explain the observed behavior and to indicate what to expect for larger values of S.