| Real
Casinos> 2007
- Real Casinos News Archive>
Size Does Count!
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."
|
