POKER MATHEMATICS III: GAME THEORY AND APPLICATIONS
On this page you will find a second installment written by Thomas S. Ferguson. Thomas S. Ferguson is a Professor in the Department of Mathematics and the Department of Statistics at the University of California at Los Angeles.
On the Borel and von Neumann Poker Models
Chris Ferguson, Bright Trading, Westwood, California
Thomas S. Ferguson, University of California, Los Angeles
1. Introduction and Summary.
The study of two-person zero-sum poker models with independent uniform hands goes back to Borel and von Neumann. Borel discusses a form of poker in Chapter 5, “Le jeu de poker” of his 1938 book, Applications aux Jeux des Hazard. Von Neumann presents his analysis of a similar form of poker in the seminal book on game theory — Theory of Games and Economic Behavior by von Neumann and Morgenstern (1944). Section 19 of the book is devoted to certain mathematical models of poker, with both discrete and continuous hands, and with both simultaneous bets and alternating bets. Extensions of the model of Borel may be found in the work of Bellman and Blackwell (1949), Bellman (1952), and Karlin and Restrepo (1957).
In these models, Player I is dealt a random hand X ? [0, 1] where X has a uniform distribution over the interval [0, 1]; the prior probability that X is in any subinterval of [0, 1] is the length of the subinterval. Similarly, Player II independently receives a random hand, Y , according to a uniform distribution on [0, 1]. Throughout the play, both players know the value of their own hand, but not that of the opponent. The structure of the betting in the two models is the same. Each player antes one unit. Player I first decides whether or not to bet. If Player I bets, then Player II decides whether to call or to fold. If Player II folds, Player I wins one unit (the ante) from Player II. If Player II calls, the hands are compared and the player with the higher hand
wins an amount B +1 from the... READ MORE
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