Math and Poker

Two plus two has another thread on that perennial question: How many hands do I need to play to know whether or not I’m a winning player?

How many hands to know…

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if you’re a winning or losing player? 5,000? 10,000? ie. how long does it take for the bad beats/luck to even out enough so that you have an accurate idea of how you’re doing at poker?

The responses are the standard ones. Confidence intervals. Poker has large standard deviations. You need large samples. Blah, blah, blah.

None of the responses address the fundamental problem with trying to use classical confidence intervals to determine whether your win rate is “significantly” bigger than zero — non-stationarity. Your results stream doesn’t come from a constant distribution, the underlying mean and variance of the process changes.

The bigger the sample you use the more likely you’re dealing with a mixture of distributions that is extreme.

Bigger samples don’t give you more reliable estimates of your win rate. Classical confidence interval analysis requires an assumption of a sample stream of independent, identically distributed observations.

The answer is to look at smaller samples, not larger samples.

Pick one hand a day. Not randomly, but pick the one that involved the most money, the biggest one hand swing you had that day. Won or lost doesn’t matter. Then look at that hand in great detail. Did you make any mistakes? Should you have made that pot bigger? Should you have put less money in that pot? What prior information did you have about the opponent hand distribution? Was the result consistent with that? etc, etc.

That kind of analysis, a hand a day for 30 days, will give you tons more information about whether or not you’re a winning player than confidence intervals derived from the results of a million hands.

And it’s a statistically sound approach while confidence intervals are not statistically sound when you are not sampling from a constant distribution.

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Good luck

A lecture on intuition by Daniel Kahneman.

long island price fixed dinner test

As I Please makes an observation about the distinction between value bets and bluffs on tghe river

I take the position that a player should bet for value on the river when there is enough likelihood of being called by a player with a worse hand, but that bluffing should be reserved for one’s very worst hands, the ones that have no chance of winning a showdown. Game theory tells us that the size of the range of losing hands that should be bluff-bet relates to the size of the range of the hands that should be bet for value the way the bet size relates to the pot size. And there should be a wide range of hands with which a player checks and calls a bet, and a range where the right move is to check-fold.

This is a distinction that matters on the river but does not matter when there are still more cards to come. Before the river it doesn’t even make sense to think about hands in terms of a region of the number line, there is no unique rankings of hand values.

I mentioned earlier that Good Math, Bad Math was doing a series of posts on game theory.l

In a recent post he briefly discusses zero sum games.  He gives home game poker as an example of a zero sum game.  He talks about non-zero sum games as those where potential wins can come from outside the game, some external source of value.  That means you can have a win which has some source other than a loss of an opposing player.

Raked poker is a different kind of non-zero sum game.  Rather than an external source of value it has an external value sink.  You can lose money without it becoming a win for another player.

I’ve never really thought of it much before but it seems like that might have an effect on equilibriums.  I think it’s pretty much common wisdom that poker can be treated as if it’s a zero sum game because the rake is just a constant drain.  I’m not so sure though.  If your model treats the game as just an individual decision then that’s probably true.  But how about a more dynamic model where a sequence of decisions is modeled more explicitely?

If you’re making the standard game theory assumption that all opponents behave in a way to achieve equilibrium then you have a game with negative expected value and where the expected value gets worse for long games.  That just seems like it has to have an effect on optimal equilibriums.

Good Math, Bad Math has a short introduction to game theory. I think he’s planning some followup posts. Read the comments also.

Michael Trick links to a cartoon on the traveling salesman problem.

It illustrates an important consideration for poker players, not just operations research analysts or traveling salesmen.

Don’t let yourself get bogged down in detailed computation of exact optimal solutions. Take a step back and look again. Sometimes a real easy route lies in another direction.

The program for INFORMS SouthWest Regional Conference in College Station April 18, 19 is available.

 My talk is scheduled for noon on April 19.

I might try to live blog the conference.

The Journal of the American Statistical Association had a book review of Curve Ball: Baseball, Statistics, and the Role of Chance in the Game (from 2001)

Micheal Trick has a post about a presentation he attended on the heads up match between PokerBot and a couple of pros a few months back.  I made a comment on his blog.