POKER MATHEMATICS II: HEAD TO HEAD POKER

On this page you will find a paper 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.
This paper is by no means light reading and is not for the cranially challenged. It is however, a very interesting read and i wish to pass it on to my valuable visitors! Enjoy!
UNIFORM(0,1) TWO-PERSON POKER MODELS

Chris Ferguson, TiltWare, Los Angeles
Tom Ferguson, Mathematics, UCLA
Ce'phas Gawargy, Mathematics, Universitie Paris 1

Section 1. Introduction and Summary.
The study of two-person zero-sum poker models with independent uniform (0,1) hands goes back to Borel and von Neumann. Borel discusses a model 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 model of poker in the seminal book on game theory

— Theory of Games and Economic Behavior by von Neumann and Morgenstern (1944). Most subsequent work on these models has been to extend the Borel model to allow several rounds of betting or more bet sizes. The von Neumann model, though more closely tied to actual play, is harder to treat mathematically. In this paper we solve several extensions to the von Neumann model. See Ferguson and Ferguson (2003) for a discussion and comparison of these two models.

1.1 The Model of von Neumann. In the von Neumann model, Players I and II both contribute an ante of 1 unit into the pot, so that the initial pot size is 2. Then they receive independent uniform(0,1) hands, x for Player I and y for Player II. Player I acts first either by checking (in which case, the hands are immediately compared and the higher hand wins the pot) or by betting a prescribed amount B > 0 (putting that amount into the pot). If Player I bets, then Player II acts by either folding (and conceding the pot to Player I) or calling (and putting B into the pot). If Player II calls the bet of Player I, the hands are compared and the player with the higher hand wins the entire pot. That is, if x > y then Player I wins the pot; if x < y then Player II wins the pot. We do not have to consider the case x = y since this occurs with probability 0.

The solution may be described as follows. Player I has a unique optimal strategy of the form for some numbers a and b with a < b: bet if x < a or if x > b, and check otherwise. Although there are many optimal strategies for Player II, and von Neumann finds all of them, there is a unique admissible one. (A strategy is admissible if no other strategy gives a better expected pay off against one strategy of the opponent without giving a worse expected pay off against another strategy of the opponent.) It has the simple form for some number c: call if y > c, and fold if y < c. The optimal values of a, b and c in terms of B are  
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POKER MATHEMATICS II: HEAD TO HEAD POKER
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