Are Poker results normally distributed?

[BugsBunny]

Discuss.

Most of us assume that poker results are normally distributed long term. Is this assumption really valid or is the true distribution skewed in some way? Is a true distribution possibly flatter than a normal distribution with fat tails causing the extremes to be more likely than predictions with a normal distribution would expect?

Is there a difference between full games and shorthanded games in this regard? Can the results be different for different players (long term) due to their styles?

I don't know the answers to these questions but would be interested on any insight that anyone may have.

[Siegmund]

Depends what exactly you refer to by "poker results."

If you take any one variable of interest - say, the distribution of the outcome of a single hand - it will have some unique shape that isn't anything remotely like a normal distribution. (A lot of 0s and -1s = hands folded before or on the flop, a modest number of -2s, etc for hands that lost, a small number of +10s and such for hands that won.)

If, however, you mean the change in your bankroll over a large number of hands, yes, that is guaranteed to be approximately normal. A sum of many independent random variables is guaranteed to look more and more like a normal distribution the more things you add up (provided each of the RVs has a finite standard deviation, which it does since your bankroll is finite.)

The more skewed your distribution is, the longer it takes for it to be approximately normal. In limit poker, it takes several hundred hands, because you have many small losses interspersed with relatively large wins every 10th hand or so. There is a rule of thumb that "30 is a large number" for normal distribution purposes -- in poker, you'd better figure on 30 pots won, not 30 hands played, before assuming normality.

At the far extreme are games like video poker, where you will lose money slowly for thousands of hands before hitting the big royal flush jackpot. "In the long run" your win rate at video poker is normally distributed too - but the long run is millions of hands rather than just a thousand as in poker.

[DrPublo]

If you're really interested, pick a limit (Say Party 3/6), datamine it for several hundred thousand hands (millions even?), and examine the BB/100 of players for whom you have over X number of hands logged. I would not be surprised to see some sort of normal distribution.

The Doc

[BugsBunny]

I wouldn't be surprised either /images/graemlins/smile.gif I also wouldn't be totally shocked to see a slightly flattened distribution with fat tails to at least some degree.
BUT
the more I think about it the more Siegmunds answer makes sense. And I suspect that what looks like fat tails to some people is actually caused by short term results with a high SD and, possibly, a true earn that's lower than they think it is. We also have the fact that large swings are guaranteed to happen with large enough sample sizes, and people are playing lots of hands nowadays. Combine the fact that individuals are playing 100's of thoosands of hands over the course of a few months with the fact that lots of individuals are doing this and it's not surprising that some of them are hitting massive downswings. We're talking outliers here, but outliers are bound to appear. It would be stranger if we didn't have outliers with all the hands that are being played by the cumulative player base.

Anyway thanks for the feedback people. If anyone has anything else to contribute please do.

[Derek in NYC]

Shouldnt it depend on what you mean by "results"? Certainly starting hand distribution should be normal, as it is a random event. However if by "results" you mean win rate or something, that dependent variable is effected by non-random independent variables, namely each player's individual post-flop decisionmaking skills. A LAG, LAP, TAG, or TAP's decisionmaking, which is non-random, would greatly drive results and the distribution of those results. Furthermore, I think the post-flop decisionmaking variable (if you want to call it that), is not static. It changes over time--whether as a result of improvement, tilt, etc.

[BugsBunny]

[ QUOTE ]
... win rate or something, that dependent variable is effected by non-random independent variables, namely each player's individual post-flop decisionmaking skills. A LAG, LAP, TAG, or TAP's decisionmaking, which is non-random, would greatly drive results and the distribution of those results. Furthermore, I think the post-flop decisionmaking variable (if you want to call it that), is not static. It changes over time--whether as a result of improvement, tilt, etc.

[/ QUOTE ]
Just to clarify by results I do mean earn, win rate (whatever you want to call it) (which I think would have been obvious from the context of the original question)

It's true that the decision making process is non-random, otherwise we couldn't turn a profit in the first place. It's also true that a persons style (and development as a player) will effect certain variables of the ultimate results.

However I'm not sure if it effects the actual form of the distribution (long term). The true earn will be effected, the SD will be effected, and both of these may well change over time (hopefully for the better in our case) effecting the actual distribution values but not necessarily it's form. And the form of the distribution is the question here.

[That guy]

So this thread is really trying to get at:

When returns are not 'normal' -- measures such as standard deviation will most likely understimate the true risk exposure to your bankroll.

In the financial markets, 'fat tails' are often caused by investor psychology. If a participant (investor or poker player) exhibits behavioral biases (risk seeking behavior increases as losses mount --- 'to get back to even'), then results may not be normally distributed because the behavioral aspects are screwing up the natural randomness of upswings and downswings. Said differently, when someone goes on tilt and starts overplaying hands, their results will tend to have a 'fatter tail' on the downside than a normal distribution would suggest. If this is the case, larger negative overall bankroll moves would occur more frequently than a measure like Standard Deviation would suggest.

I would argue that this is probably the most important factor in poker -- avoiding tilt. Dan Harrington said that the difference between him and others is not when they are winning, often he is no better than others in this situation --- but when they are losing, Harrington feels he is a much better player...

So if you run the data for a wide sample of players, I would hypothesize that the distribution will exhibit some fat tail characteristics -- because many poker players will go on tilt and become 'risk seekers' when losing. This would suggest that the data is not normally distributed. But I believe this is because of the behavioral aspects of the players, not the game of poker.

I have read that the demise of many good players was by having one really bad day after X good sessions. ie, they win for 5 days and give it all back on day 6... This would suggest a 'fat tail' on the downside.

Setting a stop-loss seems to make a ton of sense. You avoid that one terrible session. You are going to hit streaks of bad sessions (I had 9 losing sessions in a row in January) but you need to minimize the magnitude of the loss in each session.

To me, this is one of the most interesting parts of being a poker player: the mental challenge of dealing with the swings... Given most gamblers propensity for obsessive behavior, you can see how easy it would be for your average gambler to go into a tailspin when confronted with one terrible (but ordinary) downswing ==> ie, the downside tail gets fatter and fatter...

[Pokerscott]

Bankroll and ROI statistical results reported here (http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=1570111&page=6&view=colla psed&sb=5&o=14&fpart=1)

The analysis used a Monte Carlo approach that did NOT rely on a normality assumption. However, the distributions do rely on a stationary assumption (meaning your 1st/2nd/3rd probabilities are constant). You could easily add a random element to your 1st/2nd/3rd probabilities if that floats your boat although unless it is drastic the results will not change much.

If you look at the results closely, you will see that after 50 or so SnGs the ROI and bankroll distributions look very close to normally distributed.

Pokerscott

[Piers]

No

But you can get some extreemly useful results very simply if you assume that it is.

I believe the law of large numbers applys to poker results as well.