There is a companion article which discusses this project’s role in decentralized community and citizen science at ArkFab.

You can find the current paper at Scribd here or download it here.

A while back, I got the idea to investigate how the entropy of a poker tournament evolves with time. In thermodynamics, entropy is a measure of how ‘spread out’ energy is amongst the states available to it. When the energy in a system is concentrated in one place (like a hot cup of coffee in a cold room), the entropy of the system is low. When the energy is spread out (a few hours later, both the room and the coffee are the same temperature) the entropy of the system is high.  Although originally defined for distributions of physical energy, entropy can be defined more generally to study arbitrary distributions – for example the distribution of capital, in the form of chips, between players in a poker tournament.

Just by looking at the formal structure of the game, you can tell some things about how entropy behaves. For example, it is formally required that entropy falls to zero with time. On the one hand, this is a fancy way of saying, ‘one person will eventually win the tournament'; on the other hand, it is interesting to consider that this is the exact opposite of what happens in the physical, thermodynamic world. The entropy of a closed thermodynamic system necessarily increases with time: hot coffee in a cold room will cool down, but warm coffee in a warm room will never heat up. However, the entropy of a closed poker table necessarily decreases. It has a second law of thermodynamics that runs in the opposite direction from ours.

Beyond a bottom-up analytical approach, I wanted to see how real-life tournaments behave. Although online gaming has generated a wealth of data, accessing the data is difficult, and I could find only one other paper which investigated the phenomenon. This was ‘Universal statistical properties of poker tournaments’ by Clement Sire (Sire 2007). The author notes that most of the game-theoretic work on poker has been on largely restricted to optimal betting strategies in head-to-head tournaments. Sire builds a relatively simple model of player behavior: a player bets according to a simple evaluation of their hand and the table, and goes all-in if their hand is evaluated to be above a certain quality. This model predicts that tournaments will have certain statistical properties; this prediction is born out in real-life tournaments.

I was only able to get two suitable datasets, so it’s hard to draw solid conclusions about what is going on. However,  there are interesting observations to be made. Here’s a visualisation of one tournament:

Tournament fortunes and entropy for one set of data. The top graph shows the holdings of each player; the bottom graph shows the entropy of the tournament as a function of time (green). The red lines are the upper and lower bounds for the tournament; this is a function of the number of players, which is in turn a function of time.

For one thing, the entropy remains close to its theoretical maximum value, generally ~90% of the absolute maximum. In the tournament pictured, entropy appears to increase to a maximum, and then slowly decline, before the loss of a player abruptly changes the distribution of chips (the sudden changes in the stair-step of the max/min entropy.)  Furthermore, when the tournament entropy is normalized by its maximum entropy, there is a significant upwards trend (p = 0.012). Over the course of the tournament, the entropy increases towards its theoretical maximum. Additionally, it is interesting to me that, in between the losses of players, entropy appears to increase, reach a maximum, and then decrease again before collapsing. (It’s more clear in this image) I interpret this as the redistribution of the winnings of the leaving player (eg, of cyan to black and then to the rest of the table in hands 1-25) followed by a concentration of chips which eventually pushes a player out (yellow vs. the rest, hands 25-40).

However, none of these observations held in the second tournament. One possibility is that, because the second tournament was faster paced, players were eliminated much faster, and these frequent perturbations are obscuring the pattern. On the other hand, it’s entirely possible that the first tournament was a fluke. The only way to resolve this question is with more data!

One reason I am interested in this question has to do with a series of papers written by Arto Annilla from the University of Helsinki. He’s shown that protein folding, genomics, abiogenesis, ecological succession – pretty much every aspect of nature – is not merely constrained by the second law of thermodynamics, but a direct consequence of it. Most relevant to this project, I think, is his analysis of economies and ecosystems. The ultimate goal of each, he argues, is not only to increase entropy to a maximum, but to do so as fast as possible. To the extent that poker tournaments can be thought of as a toy model of an economy, they may provide empirical insights into thermoeconomics. Of course, we’ve already seen that tournament entropy is formally constrained to decrease with time, though it would be interesting to see the behavior of a tournament which is not driven by a rising minimum bet, as these are. The first tournament may show an upward trend after perturbation from quasiequilibrium conditions, and the relative entropy may show a tournament-scale increase. (Or, it may not. Argh! Why oh why must n=2?!)

Dr. Annilla’s discussion of economics also applies to ecosystems, and I often think about tournaments in ecological terms, with the chips representing environmental resources which various species (players) compete for. Indeed, the top graph in the above image is modelled after a common visual representation of relative abundance of bacterial species in microbiomes:

The relative abundances of (and hence, the distribution of resources across) different bacterial taxa on a leaf. The horizontal axis is time; each vertical bar is a snapshot of the bacterial community on the leaves of a cottonwood tree, at points in time over the course of a year. clix for suace11! '

In this context it is interesting to me how often, at least in the slow tournament, there is a continuous decline in a dying population until the player is forced to go all in and loses (E.g. yellow and green in the first image.) This strongly reminds me of species whose population is slowly reduced until it’s pushed over a critical threshold, and goes extinct.

I have always been interested in the tendency of multiple interacting systems, sometimes trivially simple ones, can create a supersystem with complex behavior. Cellular automata, the nervous system, and ant colonies are all examples. Might a poker tournament be viewed as a collection of automata, interacting along thermodynamic lines? (Sire 2007) notes,

Although a priori governed by human laws (bluff, prudence, aggressiveness…), we shall find that some of their interesting properties can be quantitatively described.

What sort of self-organizing properties might a collection of interacting automata have when equipped with poker strategies, even very simple ones such as those used by Sire?

There’s a lot of interesting questions left to explore (for example, is it more meaningful to normalize entropy by division, or to add the change in entropy caused by player dropout? The latter could be physically interpretted as the entropy of a closed system including both players and explayers who take away a certain amount of entropy when they leave the table.) I’ve had some questions raised about the validity and meaning of the distribution I used to calculate the entropy (though I think that I did it right). I’ve also gotten to have fun conversations with people about interesting problems. And I’ve had sweet theme songs to back me up.

If you are interested and want to hear more or have ideas, please comment! And definitely, if you or someone you know have tournament histories and would like to contribute them, please contact me at ThermoPoker(at)gmail.com.

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Clément Sire (2007). Universal statistical properties of poker tournaments J. Stat. Mech. (2007) P08013 arXiv: physics/0703122v3

Annila, Arto (2009). Economies Evolve by Energy Dispersal Entropy, 11 (4), 606-633 DOI: 10.3390/e11040606

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