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The physics of poker tournaments

par Clément Sire - 19 février 2007


Physicists are now more then ever involved in the study of complex systems which do not belong to the traditional realm of their science. Finance (“options” theory,...), human networks (Internet, airports,...), the dynamics of biological evolution and in general of competitive “agents” are just a few examples of problems recently addressed by statistical physicists.

Texas hold’em poker tournaments have become amazingly popular throughout the world. Although a priori governed by purely human laws (bluff, prudence, aggressiveness...), poker tournaments can be accurately described by a simple model recently introduced by Clément Sire, a physicist at LPT (link to the article). Note that the famous mathematicians Émile Borel (the founder of modern integral, and one of the very first theorists of the game of bridge) and John von Neumann (at the origin of modern game theory, among other major contributions) were among the first scientists to get interested in poker, looking for the best strategy (including bluff) in a simple version of head-to-head poker.

One of the nice aspects of a poker tournament lies in the obvious fact that it is one of the very few truly isolated human system, which is not affected by any external phenomenon (unlike the stock market, for instance).

Definition of the poker tournament model

Initially, the players (up to 10000 in real tournaments) sit at tables accepting up to ten players, and receive the same amount of chips. At each table, the player next to the dealer has to post the blind bet. Then, each player receives his hand, whose value is a random number between 0 and 1. Depending on the value of their hand and their individual strategy, players can fold, bet the amount of the “blind” (“call”), or go “all-in”, hence betting all their chips (but then risking to be called). The player with the highest hand wins the pot. Players with no chips left are eliminated.

The model retains the two main aspects of real poker tournaments :

  • The minimal bet is the blind, which grows exponentially with time. In real Internet tournaments, the blind grows gradually (40$, 60$, 100$, 150$, 200$, 300$, 400$...), and is hence multiplied by a factor 10 every 1-2 hour. The growth of the blind sets the pace of the tournament.
  • Most of the deals end up with a player winning a small multiple of the blind. However, during certain deals, two or more players can aggressively raise each other, so that they finally bet a large fraction, if not all, of their chips. In the model, players with a very good hand have a finite probability q>0 to go “all-in”, in order to mimic this effect.

The solution of the model shows that there exists an optimal value for q : If the players go all-in too often, the game is dominated by all-in processes and the average number of chips per player can get rapidly large compared to the blind. The first player to go all-in is acting foolishly and takes the risk of being eliminated just to win the (negligible) blind. Inversely, if q is too small, players (especially those with a declining number of chips) would be foolish not to make the most of the opportunity to double their chips by going all-in. One expects that real poker players would, on average, self-adjust their q near its optimal value. In a more realistic version of the model (work in process), the optimal value of q is not constant and is found to depend on the stacks of all the players at the table. It reproduces the well known fact that players with a small stack go all-in more often than rich players (but are also often called, unfortunately for them).

Some results of the poker model

The figure below shows the probability distribution f(X) to find a surviving player with the fraction X of the average number of chips per player. The full black lines and green circles correspond to actual Internet poker tournaments data, and the thin dotted lines to numerical simulations of the model. The blue dashed lines are the result of the mathematical solution of the model. Note that very similar results are obtained averaging over the five main events of the World Poker Tour 2006 season (10000$ buy-in) or over many Internet tournaments (22-55$ buy-in), with a final result independent of the skill level of the tournament. The maximum of the distribution corresponds to players holding around 55% of the average stack.

We also plot the distribution cumulative sum F(X) which represents the percentage of players owning less chips than a player with X chips (in units of the average stack). For instance, a player owning twice the average stack (X=2) precedes 90% of the other players, whereas a player with only half the average stack (X=1/2) is ahead of 25% of other players. Hence the model permits to evaluate the current ranking of a player, which can be useful for real players, when the live ranking of the tournament is not available to them.

Interestingly, these distributions do not depend on time or on the number of surviving players (which must not be too small though). Note that the model includes no adjustable parameter.

A poker tournament is a toy model of an economy (money exchange, greed... ; the increase of the blind exactly acts like inflation) and the chip distribution hence plays a similar role as the wealth or salary distribution in a real economy, which up to now, cannot be predicted !

Other results of this work concern the “chip leader”, i.e. the player currently possessing the largest amount of chips. One finds that the probability distribution of the stack of the chip leader is universal, and is given by the so called Gumbel distribution, well documented in probability theory. In the figure below, the dashed lines correspond to the Gumbel distribution, and other lines/symbols to numerical simulations of the model. Note that other extreme events arising in very diverse contexts obey the same kind of statistical laws (like, for instance, the maximum summer temperature in Paris).

Moreover, the average duration of a tournament, the average maximum of the number of chips of the chip leader (in units of the average stack), and the total number of different chip leaders all grow proportionally to the logarithm of the initial number of players. The latter result is similarly observed in several biological evolution models.

Finally, this model also makes the connection between the “science of poker” and the persistence problem widely studied in physics, as well as some recent models of evolution in biology, and extreme value statistics in probability theory.