Poker, Chance and Skill

The question if poker is a game of skill or a game of chance received a considerable amount of attention mainly because of its potential legal implications. See, for example, [3] and its many references. Most of the material dealing with the subject focuses on legal issues, and only briefly touches the question from a purely scientific point of view. In the present article we address the question as a scientific one. To do so, we provide a detailed analysis of several simplified models of poker, which can be viewed as toy models of Texas Hold’em, the most popular variant of poker. The advantage of considering these simplified models is that unlike the real game, they are simple enough to allow a precise mathematical analysis, and yet there is every reason to believe that this analysis captures many of the main properties of the far more complicated real game, and enables us to estimate the advantage of skilled players over less skilled ones. The analysis suggests that skill plays an important role in poker. As explained in the second half of the article, this fact, together with the Central Limit Theorem, imply that skill is the major component in deciding the results of a long sequence of hands. As the common practice is to play many hands, the conclusion is that poker is predominantly a game of skill. The article is organized as follows. In Section 2 we describe the rules of Texas Hold’em which is probably the most popular poker game played in casinos and card-rooms throughout the world, as well as in online poker sites. Section 3 contains the basic probabilistic information regarding the odds of the main possibilities in the game. In Section 4 we give a detailed analysis of several simplified versions of poker. Section 5 contains a discussion of ∗Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. Email: