SIZE DOES COUNT!
13 April 2007
French researcher combines physics and games to
study poker player behaviour
News reports of a French study of player behavioural
patterns in the popular poker variant Texas Hold 'Em
(see previous InfoPowa report) have been fleshed out in
some interesting Easter holiday reading from The
Scientific American this week.
The article reviews the work of Clément Sire, a
statistical physicist and champion bridge player working
at the Laboratory of Theoretical Physics, University of
Toulouse, France. Combining his love of physics and
games, he has created a model of Texas Hold 'em that
enables him to do everything from predicting the length
of a tournament to figuring out his ranking simply by
assessing the average size of his opponents' fortunes.
"Physicists," Sire explains, "are now more than ever
involved in the study of complex systems that do not
belong to the traditional realm of their science." He
published his work "Universal Statistical Properties of
Poker Tournaments" on arXiv.org recently, having 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 Clément
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.
Poker is an especially attractive subject, because it's
one of the few truly isolated systems. Unlike, say, the
stock market, which is often governed by factors such as
politics, war and weather, poker tournaments are not
affected by external phenomena. As a result, even Sire's
simplified model of Texas Hold 'em appears to
mathematically express many features of the game that
experienced players would recognise.
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.
Clement 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. But any player, whose
average tendency to bet the farm deviates from q, is
going to win less often than a player whose tendency to
go all-in is closer to q, he says.
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 from online poker
tournaments.
The researcher 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 an order of magnitude (a factor of
10) every hour or two. Tournament organisers 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 quickly," Sire says. "What's interesting is
that organisers 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. 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, 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."
Online Casino News courtesy of InfoPowa
More news here.
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