Distribution Issues

[ben mo]

After reading Malmuth's Gambling Theory and other topics, I had an unsettled feeling that some of the major premises of the book might be wrong -- specifically, I wondered if many outcomes in poker aren't normally distributed.

My first example of this TYPE of fallacy is particularly obvious in StatKing, which presumes that win RATES are normally distributed, which is not the case at all. If I have a win rate of $85/hr in a 20-40 holdem game, with a 95% confidence interval of +-80, that would suggest that my actual win rate is somewhere between 5 and 165, which is patently false. My actual win rate is probably somewhere between $0 and $50. Maybe over an infinite number of trials a person's winrate is normally distributed around 0, but even that I'm skeptical of. In order to calculate a reasonable range for an actual winrate, given certain data, we need a reasonable model for how winrates are distributed. I haven't seen one yet.

My second example is more quibbling. I haven't seen any proof that short-term results are normally distributed, as Malmuth presumes when calculating e.v.'s etc. My first concern was that in limit holdem, you seem much more likely to win a lot of money in the short term than to lose a lot of money. Sure, it's still possible to calculate an e.v. if this is the case, but some things that are potentially valuable aren't being accounted for: for example, if someone LOSES 100 big bets or more, I think the calculation of whether they are a winning player should be quite different than if someone WINS 100 big bets -- specifically, I think someone who has a huge loss is more likely to be a losing player than a person who has a big win is likely to be a winner.

Also, the normal distribution really seems to fall apart in Pot or No limit games. Let's say someone wins at a rate of 10BB/hr in a pot-limit game with an hourly SD of 100BB (I have found this reasonable). This would predict that a -1000BB hour would be a 10 SD Event (basically impossible), but that is clearly not the case in pot-limit, where even the best players can have massive swings. One might be tempted to say that this is due to a miscalculation of ones SD, but I don't think so. Pot limit holdem, in my experience, is one of the highest CV games around -- low variance relative to winrates, but it also has the highest short term fluctuations. Perhaps getting the unit right would help solve this problem -- like maybe calculating over 10 hour blocks instead of 1 hour blocks approaches normalizatoin -- but I don't know.

Any help figuring this stuff out would be appreciated.

ben

[rigoletto]

I'm not sure I understand your post, so I have a couple of questions:

If I have a win rate of $85/hr in a 20-40 holdem game, with a 95% confidence interval of +-80, that would suggest that my actual win rate is somewhere between 5 and 165, which is patently false. My actual win rate is probably somewhere between $0 and $50.

How do you arrive at the at the conclusion that $0-$50 is the true winrate? If you don't think your Statking result are accurate, it could be because you dont have enough data.

My first concern was that in limit holdem, you seem much more likely to win a lot of money in the short term than to lose a lot of money.

How du you arrive at this conclusion???

I think someone who has a huge loss is more likely to be a losing player than a person who has a big win is likely to be a winner

What makes you think this?

[SoBeDude]

Maybe over an infinite number of trials a person's winrate is normally distributed around 0, but even that I'm skeptical of. In order to calculate a reasonable range for an actual winrate, given certain data, we need a reasonable model for how winrates are distributed. I haven't seen one yet.

I think this is the key of what he is asking. His point is that there he does not believe win-rate normalizes around 0.

The assumption is that two numbers equal but opposite around 0 should mean the same, but opposite thing. In other words, a loss of 100 has the same weigthing as a win of 100 (but in the other direction, or course). He does not believe this is so. I tend to agree with him.

This means that the calculation of a program like stat king, which uses this assumption, is possibly providing inaccurate information.

I do not have the math abilities to help answer his question though.

-Scott

[Nottom]

Maybe I'm not understanding what you are trying to say, but you are correct when you say your results aren't distributed around 0. They should be distributed around your avg. win, in your case $85. If you say that your wins are between 0/hr and $50/hr and StatKing is saying you are averaging $85/hr then something is wrong somewhere since this just makes no sense.

As for the statement that a losing player is more likely to lose 100BB than a winning player, well of course they are, but this doesn't and shouldn't affect the calculation at all only the results. Whether someone is a winning player or not is the simplest calculation you can do, is their total winnings > 0. Short term luck will affect this result over the short term, but over the long term losers will lose money and winners will win it, individual sessions are irrelevant.

As for no/pot limit SD, I don't really have a lot of experience there, but would think that even a good player would see a much higher SD than his sb/bb limit counterpart, but the potential winnings can more than make up for it.

Just some thoughts from a n00b.

[togilvie]

You should look into the Central Limit Theorem to give you significantly more confidence in these issues. It's one of the key theorems in applied statistics, the proof of which forms the basis for Mason's assumptions.

It's stated as follows:
The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-Normal.

Is it a simplification? Absolutely. But it's a great approximator that allows one to do sophisticated analysis. I expect (but am unwilling to do the legwork /forums/images/icons/wink.gif ) that you will find that the CLT addresses your issues after working them through.