Winners, losers and variance

[Shandrax]

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I thought you were being sarcastic.

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I deleted the part with people hate to blame it on their lack of skill, so let's say semi-sarcastic. Still the whole idea isn't that far off. If you look a huge sample, you will probably find some extreme cases. It should be possible to be a winning player for 40 years by sheer luck. If we look at the millions of poker players, we might even be able to spot that guy.

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I saw him last year at the WSOP. His name is Jamie Gold.

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Not exactly the guy I had in mind, but Jamie Gold is a perfect example. Out of 6k players the luckiest player won and he did it by an endless streak of getting dealt the nuts or making them along the way. Now this was just one tourney, but it is theoretically possible for this to happen on an entire career. It's of course extremely unlikely, but that doesn't mean it's impossible. All you need is a gigantic sample and the freak events will start coming.

[_D&L_]

Ok i think you raised a fair point.

So lets use a players own wins and losses as his distribution. I'm still thinking variance has to be fairly small regardless of strategy in limit poker, because a "big" win is probably going to fall in the range of 10-15x the small loss.

If after a 1000 hands a player is in a situation where 3-4 big wins aren't going to make him positive, my guess is still its probably more than 95% likely he hasn't been playing with a strategy that is +EV against his opponents. (No i haven't run the numbers to verify, but i still think my guess is in the ballpark).

[Al Mirpuri]

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In limit poker, you should expect to see convergence after a few hundred hands. If after a few hundred hands or maybe 1000 hands, your losing at a limit table, its because your behind in skill not luck.

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Not true. You can lose over a 1000 hands and it proves nothing.

It is also possible for someone not to lose a bet in his life and for someone to never win a bet in his life. Both these extremes can be measured using probability theory.

Sklansky (or was it Malmuth) has written about a player who ran good for years and how it would come about...

[bbartlog]

1000 hands is not enough, even for limit. For no-limit and pot-limit games the variance will be higher, since much of it comes from rare occurences like set-over-set or losing strings of 60/40 matchups for your entire stack.

By way of evidence, look at the results graph for davebreal in this thread: http://forumserver.twoplustwo.com/showfl...e=4#Post9001914

He's a winning player, having won $14K or so over 80,000 hands in one month. But of more interest to the current argument, notice that over the last 40,000 hands in the month he *lost* money (down $2K). Of course you could bring forth other explanations (maybe another pro got his number and decided to try to win money off of him starting in mid-January) but to me it looks like simple variance.

[_D&L_]

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Not true. You can lose over a 1000 hands and it proves nothing.

It is also possible for someone not to lose a bet in his life and for someone to never win a bet in his life. Both these extremes can be measured using probability theory.

Sklansky (or was it Malmuth) has written about a player who ran good for years and how it would come about...



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what's possible and what's probable are two different things. If it could be proven, i would go "all in" everytime on betting that a player who did good for "years" then started doing terrible for a similar length of time was probably the victim of something other than variance.

His strategy could have changed over time, his opponents got tougher, or learned to adapt to him, etc. Its just impossible for me to think that a game whose values are so close together, and where "big wins" are only 20 times small losses, etc, doesn't converge over a fairly short period. I'm extremely skeptical of anyone that thinks you can run great for years just on the basis of luck.

[_D&L_]

I think its a good post/link for the question. Some other things that make it difficult to interpret the data is that the players are posting "all limits." as player bank roll grows, as Dave's did, he may be trying to move up in limits, etc.

I'm not trying to be hypercritical of any contrary data, but really the answer to the question should be mathmatical. I noticed my own win rate varies a lot just depending on the time of day I play (if i play late late at night, i'm usually playing against some of the best players at that limit, if i'm playing during prime time i got a much more varied mix).

I think some of the disagreement is between mathmatical variance, and the non-mathmatical less well-defined question of "can i make a living for X number of years playing at this limit." The latter question can't be answered with math. For instance, US legislation has already knocked out a lot of amateur players. The skill of the average poker player may be increasing over time, etc.

Statistics is best at answering questions looking backwards. Like, was I playing with a +EV strategy in the hands in my database. If the answer is yes, and everything else is held equal - which is an extremely big "if", (e.g. my opponents don't adapt, etc.) then i can expect to keep on winning.

But anyways, good post.

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1000 hands is not enough, even for limit. For no-limit and pot-limit games the variance will be higher, since much of it comes from rare occurences like set-over-set or losing strings of 60/40 matchups for your entire stack.


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I acknowledge enough in my post w.r.t that no limit games will have more variance, and thus convergence will be slower. Still even after 100 hands, there will exist some -BB threshold that if your below, we can say with 95% certainty that you are a losing player. Granted the -BB threshold will be very large w/ only 100 hands, but we can always calculate a threshold for any size sample, save maybe a sample of size 1.

[CopTHIS]

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Explain what i am missing? Especially w.r.t to limit poker.

Lets define convergence as knowing whether you're strategy is a +EV or -EV strategy within say 95% certainty, given the skill of your opponents. How many hands, and how far negative do you have to be to know your strategy is losing?

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I don't generally follow limit, but in NL you only have to look at the graphs people post to see the variance you actually expect. It's very difficult to model something like this so if there is a huge amount of actual real-life data then why not consider that. It's not uncommon for NL players to have 20/30 buyin swings, or 25k break even runs. Then you can win 12 buyins in 500 hands. I mean, if you actually watched someone play only 200 hands you could probably get a fair idea about their standard - but even then they could be playing a specific way against certain opponents, or they could just off their game for some reason - but if you just look at the basic net profit/loss you need a decent number of hands to be able to have anything like reasonable confidence in the level they are at.

[Al Mirpuri]

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Not true. You can lose over a 1000 hands and it proves nothing.

It is also possible for someone not to lose a bet in his life and for someone to never win a bet in his life. Both these extremes can be measured using probability theory.

Sklansky (or was it Malmuth) has written about a player who ran good for years and how it would come about...



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what's possible and what's probable are two different things. If it could be proven, i would go "all in" everytime on betting that a player who did good for "years" then started doing terrible for a similar length of time was probably the victim of something other than variance.

His strategy could have changed over time, his opponents got tougher, or learned to adapt to him, etc. Its just impossible for me to think that a game whose values are so close together, and where "big wins" are only 20 times small losses, etc, doesn't converge over a fairly short period. I'm extremely skeptical of anyone that thinks you can run great for years just on the basis of luck.

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You are being sceptical of Sklansky because he wrote it and it is in print somewhere but I have read over twenty two plus two texts so I cannot pinpoint it.

[bbartlog]

I agree that outside factors introduce non-mathematical variance which makes the empirical data questionable. Personally I'd like to be able to graph my BB/100 in pokertracker as a function of how long I've been playing (this session), since I'm pretty sure it would show a gradual decline as I got tired.

The thing is though that a purely mathematical analysis is not so easy either, simply because of the odd (non-normal) distributions of expected returns on a hand. Not only that, but returns are not indepedent (if you are dealt six premium hands in a row, you will be getting a lot more action for each hand on average than you would if they were dealt to you over a period of 50 hands). And of course in conceding the importance of extrinsic factors we are sort of admitting that the mathematics of variance isn't all that relevant to a practical player.

For what it's worth, my brother plays professionally and he agrees that time of day is very important in terms of the quality of opposition you face.

[pzhon]

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I acknowledge enough in my post w.r.t that no limit games will have more variance, and thus convergence will be slower.

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This is a common misconception of those who haven't played much. It is contradicted by a tremendous amount of evidence from winning players. The variance in NL is larger when expressed in big blinds, but no serious player compares a NL game with a big blind of $1 with a limit game with a big blind of $1. For typical low to medium stakes limit and NL games, an expert player has a much greater advantage in relation to the variance in the NL games. That means the expert will hit the long run much more rapidly, and that an expert who wins the same amount per hour in each variant will have a lower variance in NL.

This is one factor that leads people to wonder whether you are being serious. Another is that you are grossly underestimating the amount of time it takes to hit the long run in a limit game. You are off by more than an order of magnitude.

Typical standard deviations in online limit games are about 15 BB/100 hands in full tables, or 17 BB/100 in 6-max games. Win rates of solid winners vary by games, but 2 BB/100 is a good win rate in many games. After 10,000 hands, a solid winner with these stats has a rough 95% confidence interval of 200 BB +- 300 BB (2*100+-15*2*sqrt(100)). It doesn't take hundreds of hands to hit the long run in limit; it takes tens of thousands.

For comparison, a winning NL player may have a win rate of 10 PTBB/100 (16 big blinds/100) with a standard deviation of 40 PTBB/100.