Texas Hold'em Poker Modeled Successfully by French Physicist

French physicist Clement Sire combined his love of physics and card games and created a model of the poker variant Texas Hold'em that reproduces some properties of Internet poker tournaments, amongst other things, to predict the length of a tournament and determine his ranking simply by assessing the average size of his opponents' fortunes, reports Scientific American this past weekend.

Sire, who is also a champion bridge player and works at the Laboratory of Theoretical Physics, at the University of Toulouse, France, published his work Universal Statistical Properties of Poker Tournaments on arXiv.org. He used real data from online poker tournaments and found that it matched the results of his model.

"What's exceptional about this paper is that Clement somehow took what seems to be a complex and mysterious system and quantified it [with the tools of statistical mechanics] in a very precise way," says Sidney Redner, a physicist at Boston University who works on related problems.

Because of their advanced statistics and analytic math skills, Physicists are frequently asked to study problems other than those we normally associate with physics like atomic particle collision or the big bang theory. They are often invited to work on stock market prediction, weather prediction, catastrophic volcanic or earthquake predictions, and even species migration and extinction patterns.

But unlike the stock market, which is often governed by factors such as politics, war and weather, poker is an especially attractive subject to study because it's one of the few truly isolated systems not affected by external phenomena.

As a result, Sire was able to derive a simplified model of Texas Hold'em to mathematically express several features of the game that experienced poker players would recognize.

In the model, poker hands are represented by a random value between 0 (bad) and 1 (best possible). The "blind,'' or minimum, bet for any table of 10 players gradually increases as the tournament progresses. In any given hand, players can either fold, bet the blind, or go "all-in," as in bet all of their chips.

Sire's model includes functions that reproduce the most basic tasks a poker player must carry out, such as deciding whether to bet strictly on the strength of his or her hand.

Using the model, Sire discovered that there is an optimal value for a player's tendency to go all-in. This value, which he calls q, varies depending on whether a player has few or many chips. He observed that a player who tends to go all-in in a value that deviates from q is going to win less often than a player whose tendency is closer to q.

One feature of Sire's model came directly from his own experience playing in poker tournaments. "I noticed when playing that when I had twice the number of chips as the average," he says, "I was typically in the 10 best people of a 100-person tournament."

Curious, he used data from his model to graph the rankings of players versus the number of chips they held. He found that his anecdotal observations were correct and, also, that his model almost perfectly matched the data he had gathered by visiting two online poker sites and collecting data from 20 no-limit Texas Hold'em tournaments with an initial number of players between 250 to 800.

The physicist also discovered that the maximum number of chips held by the "chip leader," or the player with the most chips at any given time, as well as the total number of chip leaders, are both a function of the number of players who enter a tournament. (Specifically, they're proportional to the logarithm of the initial number of players.)

"This phenomenon has been observed in many different models involving competing agents," Sire notes. "In models of biological evolution, it shows up where you have many species who compete and there is one prominent, or leading, species."

In Internet Texas hold 'em poker tournaments, the minimum bet goes up exponentially over time, which means that it increases by a factor of 10 every hour or two. Tournament organizers do this to ensure that tournaments with 10,000 players don't take 100 times longer to complete as those with only 100 players.

"The increase of the [minimum bet] in tournaments is only to ensure that the number of players decreases sufficiently fast," Sire says. "What's interesting is that organizers must intuitively know this, even though they don't know the math behind it. Essentially they have estimated the rate at which they should increase the blind, but with [my model] they can control very accurately the duration of the tournament."

Sire's ability to reproduce many of the characteristics of a poker tournament indicates that, when taken as a whole, the features of these tournaments are entirely predictable.

But before anyone attempts to use Sire's model to plot a winning strategy, however, they should take heed of Sire's findings.

It turns out that the distribution of the "stack," or fortune, of the chip leaders across tournaments mirrors the pattern found in the distribution of maximum temperatures during every August in history or countless other natural phenomena where physicists have attempted to predict the nature of extreme values.

This pattern, called the Gumbel distribution (a kind of 'leaning and slightly skewed' bell shaped curve), means that the frequency with which chip leaders accrue fortunes of any given size is, in a way, a natural phenomenon that arises as much from the characteristics of the game being played as from the dispositions and abilities of those playing it.

"To have the Gumbel distribution show up here makes sense in hindsight," says Redner, "but it is beautiful to see someone find it in this area for the first time."

Sire closes his article by saying, "It would be interesting to obtain access to the full dynamical evolution of a large sample of real-life poker tournaments, in order to check the predictions of the model concerning the chip leader."

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