## Tuesday, January 5, 2010

### Playing Poker for a Living: Who are you playing against?

Because poker is played against other players instead of against the house you don't need to be phenomenally good at poker in order to have a positive expectation, you only need to be better than someone you are playing against. As the existence of casinos demonstrates many people are willing to lose money on average in order to have a chance to win some money. The fact is that a lot of people find it fun to gamble and even more people find it fun to fantasize about winning at gambling. So very good poker players simply end up taking money from players who are less good.

Online poker is the best way to try and make a living at poker. The cost of playing online poker is smaller than the cost of playing poker at a physical casino, also many players who do not have access to a casino are willing to play online. Online activity isn't bound to just one physical place and so it is different times for different players giving more uniform behavior over time.

There are a large number of people who play poker occasionally for fun and a few people who play poker constantly, some for a living. I would be willing to bet that the distribution of the number of players who play with a certain frequency roughly follows a power law. N(f) = a*f^k for some choice of constants a and k where N(f) is the number of players who play with the frequency f. I expect this to hold true mostly for relatively small frequencies. Obviously there are many more low frequency players than high frequency players which means that here we expect k to be negative.

Lets take a moment here to consider the question of what this sort of distribution would mean for who you are likely to be playing when you log in to a poker site. Higher frequency players play more often but there are a lot more lower frequency players. If the power law distribution holds (and there is some reason to think that it would) then a player which plays with a frequency f would have a chance of being picked with probability density proportional to f*N(f) since the probability of a particular player being picked is the combination of the probability of that player being on which is f and the number of players with that frequency. Normalizing this density we get

P(f) = (k+2)*f^(k+1).

taking this a step further we calculate the average frequency of play as

integral f*P(f) df = (k+2)/(k+3) = faverage

This of course isn't very helpful unless we have some idea of the value of k. As we already said k must be negative to make lower frequencies more common than higher frequencies. But this means that there is a singularity at f=0 and examination of the probability density that we just came up with suggests to us that if we want to keep the total number of people online at a given time finite we need a k which is greater than -1.

for the limiting values of 0 and -1 we get an average play frequency of 2/3 and 1/2 respectively. The middle values have values in between these. for the middle of the road k = -1/2 we get 3/5 rather obviously no player can be playing with frequency 1 and it is doubtful that the average player is playing 2/3 of their day since that would mean they just take out 8 hours to sleep and thats it. Even the low estimate of 1/2 is too high. Qualitatively though the point is that high frequency players are on more often than low frequency players. If we take these frequencies to be a fraction of the highest normal play frequency then these numbers make more sense. The highest realistic value for the top player frequency is probably about 1/2 which would correspond to playing 12 hours of poker every day. A regular 40 hour work week would correspond to a frequency of 5/21. Using this as a base line I would say the highest tier of poker players probably have a playing frequency of around 1/3. This gives much more realistic figures for the average frequency of around 2/9, 1/5, and 1/6 for values of k 0, -1/2, and -1 respectively.

At any rate this suggests that most people you play will be people who play a very significant fraction of their time. This however doesn't take into account the fact that people do not play at all times of the day evenly. Taking this into account will skew the distribution back towards low frequency players during peak times and towards high frequency players at off times.