Preliminary academic research into poker actually started very early in the computer age. In the book ``Theory of Games and Economic Behavior'' (the founding work of game theory [101]), John von Neumann and Oskar Morgenstern used mathematical models to analyze certain greatly simplified games of ``poker''. Among other things, they demonstrated the fact that bluffing is an absolutely essential component of poker, and that any sound strategy must include bluffing with a certain frequency. While this was interesting, and useful as an example of the application of game theoretic principles, the games studied were too far removed from real poker to be of much practical value.
Other fundamental works into the study of simplified poker were developed by John Nash and Lloyd Shapley [61] and by Samuel Karlin [51, 52]. Collections of related papers on the theory of games are also available [2, 3, 4], as well as an excellent treatise on the analysis of all games [23, 24].
An attempt to adapt these mathematical models to more realistic versions of poker was made by Newman [62], but with only a limited degree of success. More recently, this approach has been revisited and more fully explained by Sakaguchi [71]. Beyond this, there has been little development of the original ideas, probably because they were originally intended as a lesson in the use of game theory, rather than as a serious investigation of poker dynamics.
Consequently, the models which have been developed to date are severely limited with respect to the real game of poker, and are of little use to the practical problem of writing a computer algorithm to play a strong game of poker. Nevertheless, general game theoretic notions can be applied to the practical problem, and the original references may be helpful in directing that method of thought.
There are at least two potentially useful ideas stemming from game theory. The first is the techniques used for determining certain optimal betting strategies. The second is the utilization of optimal bluffing and calling strategies. In both cases, the `pure' solutions to the simplified problems must be adapted to be applicable to the real game, but the underlying principles constitute a solid starting point from which to develop a sound approach.
An optimal betting strategy for pot-limit poker was developed in a paper by William Cutler [32]. Like previous studies, this analysis was based on a simplified poker game with only one betting round and no draw. However, the analysis method is generalized to include games where any number of re-raises are permitted, which is more realistic than the usual no-raise or one-raise scenario. Furthermore, the manner in which the optimal frequencies were computed should still be applicable to a more realistic poker setting, once the effects of multiple betting rounds and the drawing of cards is taken into account.
We now look at two books which undertake a complete game theoretic analysis of real poker games, albeit with limited degrees of success.