Statistics and swinginess

[Guy McSucker]

Hello,

It is emerging - as if we didn't know already - that there can be some pretty heavy swings in these shorthanded limit poker games.

When I first took up limit from NL in April, I was horrified at the swings. The standard deviation I observed, around 15BB/100 hands, testified to the increased swings I was seeing. For comparison:

- at NL I was winning around $15/100 with a SD of $60/100. The interesting figure is SD^2/WR which here is $240.

- at limit, decent figures for a player with a basic grasp (which aptly describes my NL game) are probably 2BB/100 with SD of 15. SD^2/WR here is 112.5BB or $675 for $3/6 which is what I was playing and what gives a vaguely comparable win rate.

So that was alarming.

Anyway, I planned my rise through the limits and got lots of advice involving the Party 6-max games. Shorthanded games have always scared me from the variance point of view, so I asked around in this forum to find out typical SD figures. 16BB/100 seemed to be the consensus, and my own figures now agree with that.

Now, 16BB/100 is not much bigger than 15BB/100. I was heartened by this.

However, in practice the swings in 6-max are much bigger than in full ring games. Day in, day out. It is fairly common to drop 40BB and then recover, or win 30BB and then finish losing, in the 6-max games; whereas in the full ring games, this happens, but not very often.

Does anyone with a proper understanding of statistics have an explanation for this?

If you don't have a proper understanding, feel free to chime in as if you do: that's what I am doing in the above.

Guy.

[TazQ]

More hands/hr.

[Schneids]

+ more aggression in all betting levels.

Where as in full ring you might be putting in 3.5BB on all betting streets when you have a small edge, here it's more like 5 or 6BB.

Have an unlucky run here and that negative swing is substantially larger than the same unluckiness that occurs in the more passive full ring games.

Notice this is also the reason why 15/30 full ring games feature the same type of BB roller coasters.

[spider]

[ QUOTE ]
- at NL I was winning around $15/100 with a SD of $60/100. The interesting figure is SD^2/WR which here is $240.


[/ QUOTE ]
If you do a ratio here, wouldn't it be better to use SD than var, since SD is in $ and var is in $^2? Minor point and maybe I don't understand what you are trying to do with the ratio.

As far as swinginess, I don't know the answer but suspect that it has more to do with the psychology of the game than the variance. Because you are headsup more often and in blind stealing/defense situations there is just a lot more bluffing and calling down. That can get frustrating when you lose a bunch of those kinds of coinflips.

[TazQ]

Wouldn't the 15 BB/100 SD or 16 BB/100 number account for the increased aggression on all rounds? Which is why in full games he saw 15 BB/100, and at short he saw 16 BB/100. Or am I wrong here?

[felson]

I think you're right, TazQ. The SD should characterize the size of the swings.

[Schneids]

[ QUOTE ]
Wouldn't the 15 BB/100 SD or 16 BB/100 number account for the increased aggression on all rounds? Which is why in full games he saw 15 BB/100, and at short he saw 16 BB/100. Or am I wrong here?

[/ QUOTE ]

+ worse players so you are more often winning when you're putting in more bets. The fact you're winning a majority of these encounters increases your WR and and helps keep the SD down lower, but still doesn't change the fact there'll be stretches where you're losing every choice encounter.

Basically, SD is derived off of session length and amt won/hr. If I have a 15 minute strech where I win 30BB (all my big encounters hold up), then the next 15 minute I lose 20BB (lady luck swings the other way aggression the aggressive lunatics), then the next I win 5BB, and the last 15 minute I lose 15BB, the equations know this hour of playing no differently than playing full table for an hour and having 15 minute stretches at +5BB, then -1BB, then +6BB, then -10BB. Both are showing up as a break even hour even though one was obviously more volatile than the other.

Unless I'm mistaken about SD and how it's calcululated (which I very well might be since I haven't taken a Stats class in 4 years) our SD comes from deriving average fluctuations/hr, and nothing more. If we made a more specific formula that looked at our sessions more closely we'd fine our short handed SD's are in fact much more varied than full table ones.

[Schneids]

In summary if we were to look at our SD in terms of BB/25 or BB/10 at both games, then multiplied said results by 4 or 10, we'd see a bigger difference between full and short handed.

[Guy McSucker]

[ QUOTE ]

In summary if we were to look at our SD in terms of BB/25 or BB/10 at both games, then multiplied said results by 4 or 10, we'd see a bigger difference between full and short handed.


[/ QUOTE ]

Well, you would, but only because to get a true comparison you'd need to multiply the BB/25 by 2 and the BB/10 by sqrt(10)...

I do hear what you're saying, but I don't think it's the full story. I mean, measuring deviations per 100 hands is still a pretty fine level of granularity, and in any case, I am surprised by both the shorter-term swings - how quickly things change within a 1000 hand period - and the longer term ones. How many good full ring players have 300BB downswings, for instance? There's talk on this forum of several good players having >300BB downswings, which would be almost unheard of in full ring I imagine, and the tiny change in SD (15 to 16) doesn't seem to account for it. Of course neither does the increased number of hands/hr that TazQ mentioned, although that clearly has an impact on how much of a swing you might see in a given period of time.

Guy.

[Guy McSucker]

[ QUOTE ]

Wouldn't the 15 BB/100 SD or 16 BB/100 number account for the increased aggression on all rounds? Which is why in full games he saw 15 BB/100, and at short he saw 16 BB/100. Or am I wrong here?


[/ QUOTE ]

I would have thought so.

Guy.

[Guy McSucker]

[ QUOTE ]

If you do a ratio here, wouldn't it be better to use SD than var, since SD is in $ and var is in $^2? Minor point and maybe I don't understand what you are trying to do with the ratio.


[/ QUOTE ]

I used SD^2/WR since it's the figure that impacts on your required bankroll, according to the guys that know statistics...

(bankroll formula: BR = -ln(r)SD^2/2WR

where r is your accepted risk of ruin.)

So I reckon that's the right figure for measuring swings. Just doing a units calculation you can see it's an amount in dollars rather than a unit-free number, which has got to make it more meaningful.

You'd think.

Guy.

[Grisgra]

I think you'll get a better answer to this not-at-all-easy-to-answer question in Poker Theory.

I think the gist is that while one's SD is a great way of characterizing variance, it's not the full story. If one's swings follow a normal curve, during any given 100 hands (let's say) you are 95% likely to have a result that's within 2SDs of your mean BB/100.

But your deviations may not follow a normal curve. Now, your SD is still fine for calculating your confidence interval around your mean winnings -- because of the central limit theorem, the distribution of the MEAN of most any distribution is normal, even if the distribution of raw values is not.

So in other words: as far as small runs are concerned, the SD may mean different things between ring games and SH games. It MAY. It also may not -- just throwing it out there as a hypothesis.

I, myself, am extremely interested in what the actual distribution of winnings tends to be -- whether it's normal or not. I'm not sure of a good way of figuring this out without data that's equally spaced, i.e., numbers every 100 hands exactly.

[Izverg04]

I have to say I've been under the same impression that you are since I've started playing $10/$20 6-handed.

One explanation is that your observation that the swings in short-handed games don't correspond to the Var/EV numbers is an illusion. Perhaps, when you started playing short-handed your EV dropped by 20%, Var went up by 15%, and these together completely explain the increased swings.

If your observation is indeed real, I think the only plausible explanation would be that the variability of game quality from table to table is much greater in short-handed than in full games. That is the difference in your winrate between a top 10% table and worst 10% table is much greater here than in full-handed games. Since a lot of players here reach the medium-term time scale in 1 week, and there is a large difference in game quality on Monday afternoon and on Friday night, this could explain larger than predicted medium-term swings. What I mean is that if you have long stretches of playing at 1 BB/100 and 3 BB/100, you will have larger swings than predicted by 2 BB/100.

Or again, this might be an illusion. Feeling is not a very reliable thing when statistics are involved.

[Erik W]

Without reading any answer.
The statistics involved is based on a Normal Distribution.

I am 99% certain that that distribution is not correct when it comes to poker as it is not either in financial situations and where much money is lost because of a false presumption.

I am certain that there is fat tails involved (as in finance). That means that the very high and very low outputs is very overrepresentated and that messes up the statistics.

What does it mean to us in poker?
You can still use the standard calculations based on a normal distribution as it is a good approximation of the outcome.

You will need to have a larger cashbase than the statistics suggest because of this phenomena.

This phenomena will be more pronounced in shothanded games than in full handed and that messes up statistics in those games even more.

Erik W

[Benjamin]

[ QUOTE ]
many good full ring players have 300BB downswings, for instance? There's talk on this forum of several good players having >300BB downswings, which would be almost unheard of in full ring I imagine,

[/ QUOTE ]

Several of those who should know have opined that the 15/30 full ring games are swingier than the 10/20 6-max. I've heard of a couple ungodly downstreaks > 300 BB by winning players in that game.

B.

[Grisgra]

This really needs to be figured out once and for all, and I think that with StatKing (which seems to allow deduction total $$ won/lost per every 100 hands) we can do it. Might make a good little article in the TwoPlusTwo magazine. PM me if you have 20k+ hands in Statking and you're willing to take 30 minutes writing down some numbers.

[Schneids]

[ QUOTE ]
[ QUOTE ]
many good full ring players have 300BB downswings, for instance? There's talk on this forum of several good players having >300BB downswings, which would be almost unheard of in full ring I imagine,

[/ QUOTE ]

Several of those who should know have opined that the 15/30 full ring games are swingier than the 10/20 6-max. I've heard of a couple ungodly downstreaks > 300 BB by winning players in that game.

B.

[/ QUOTE ]

Me included. In 10/20 6m despite my very aggro style my longest losing streak has still only been like 220BB, and very rarely do I have streaks that reach 150BB before things have turned around. This, in an extremely large sample size. I do go through 50BB swings like they're twinkies at a fat kid's birthday.

[naphand]

Can you expand a little on the "fat-tails" phenomenon. I understand what you means by this but would like a betteer handle on the mechnics of how it is created, and more details of the implications.

I am currently experiencing a bizarre (and very frustrating) situation where I am winning a huge proportion of my sessions (typically 400-700 hands) but the wins are rarely anything substantial, and every 2 weeks or so I have huge downswing which all but cancels out what has happened before. This could be a fat tail, but I cannot fathom why I am not getting fatties at the other end too. I have closely analysed my play and it has been pretty solid. A few hands went awry but do not represent a significant portion of what I have observed. It is disturbing as I feel this lop-sided distribution may indicate some underlying weakness in my game. It is creating uncertainty, and a familiarity with what could realistically be expected and what constitutes and "acceptable" deviation would be good. I cannot believe that a normal bell-curve type of distribution is normal, from my own experience and from the comments of others here.

Also, it seems that SD/100 is a very small sample size and would result in "granularity" as remarked by someone else. Would not SD over an average session lenght be far more useful. That is, if my session hands average is 500 I should be looking at SD/500 as a more meaningful figure surely? SD/100 is useful as a baseline comparison with other players, or games but, as far as personal relevance goes I would be far more interested to know what a typical session SD would look like.

[Erik W]

The phenomena should exist on both the positive and negative side of "income" so it is weird u just notice it at the downside.

There is really no relevance to dwelve deeper into fat tails and adjust it poker game to it at all more than knowing that u will have larger swings than the statistics we use will show and that we need more cash to be "safe" to those swings.

For a picture of what a fat tail is look at the pictures on this web page.

http://www.bearcave.com/bookrev/misbehavior_of_markets.html

Explanation found on web.
"Tail risk" (fat tails):

"In financial markets, and in other economic and social phenomena, tails are often fatter than the theory predicts. There are more "rare / extreme events" than statistics can show. There is a bigger number of very high and very low values (for example of daily returns) than is theoretically expected. Under / over reactions and herding might explain it.

As those extreme - but infrequent - events, are far apart in time, fat tails might be undetectable inside too short historical series. Then the real shape of the curve does not appear in recent series.



But, however invisible they are in those too short statistics, the risks of extreme prices, returns or volatilities are often higher than a normal distribution law would bring. As a result, investors who are new in the game, who don't have full information on past events or neglect them, or who have a "short memory" (a very widespread bias, see "memory"), tend to get overconfident."

[naphand]

Very good link, thanks for taking the time. One particular paragraph caught my attention, reproduced here for the benefit of the Forum:

[ QUOTE ]
...Mandelbrot mentions that there are periods where the market acts like a calm sea. In this case we might expect that the distributions will also be relatively well behaved and do not stray that far from a Gaussian curve. In other cases Mandelbrot describes the storms that beset markets. In these regimes, when the market is in the throws of a bubble or after a crash, we might expect that the distribution of return will have "fat tails", reflecting extreme behavior.

[/ QUOTE ]

My italics. This almost sounds like the comparison between a table full of loose-passives and a tables of LAGs, and the interchanges that occur when players go on tilt. I think you are certainly onto something here. Clearly though, the math would be complicated but this kind of exercise helps to illustrate the overall behaviours which may be of considerable in understanding why people frequently experience extremes.

If this is to be looked into further, but without becoming a too-academic exercise, I for one would like to see some sessional results for players; total BB wins/losses by session, compare SD/100 and SD/session (or sessional average?) and look at AF/V$IP/WR. Graphical illustration could be very revealing as a histogram of $win/loss frequency. I doubt we will make any great breakthroughs but a familiarity with the kind of fluctutations that occur would go a long way to mentally prepare people for what is likely, and possibly characterise the behaviours sufficiently well to produce some kind of expectancy. I know Mason has produced mathematical calculations for determining variance as largest loss and how many hours to experience this, but these were only partially complete as they relied entirely on SD and WR, and assumed a normal distribution (I presume).

As for fat tails, today I got one towards the positive end at last (+55 BB), nothing amazing but something decent at least. The only problem was, it was playng full ring $2/$4 games on Party... /images/graemlins/ooo.gif. Makes me wonder if I am not more suited to this, I found playing 3-tables of full-ring incredibly relaxing compared to 6-max. I think 4-tables would be easy and so little needed as far as reads are concerned. Bizarre.

[Izverg04]

Let's assume that our skill and the skill of the overall pool of players does not change over time. Then given sufficient number of hands, the distribution of the win of this number of hands becomes normal according to the central limit theorem, and the standard risk-of-ruin calculations can be used.

Negative streaks of size R typically occur over the number of hands N=R/winrate. This means N=2-10k hands for significant negative streaks. We want to examine our observation based on "feeling" that the distribution of results over N=2-10k hands is not yet normal, that is the central limit theorem has not kicked in yet.

Each hand we enter has a distribution of results associated with it. This distribution is not normal, but looks very close to normal. If every hand had this exact distribution of results, then the average over N hands would become normal very quickly. Certainly by N=100 hands it would become indistinguishable from normal.

We know that every position has a different distribution of results. However, this variation has a period of 6, and this completely washes away also very quickly.

So the only plausible explanation is variation of underlying result distribution that is comparable to N=1000. If we play sessions of N=500 hands, and the winrate and variance vary significantly from session to session, this would possibly not wash away by the time N=2-10k hands.

[naphand]

I am not entirely clear what you are trying to say here, if it is not more than saying that the table conditions for sessions sizes we play affect the distibution of results more significantly than can be reasonably expected from looking at hand/position distribution alone?

[ QUOTE ]
Let's assume that our skill and the skill of the overall pool of players does not change over time.

[/ QUOTE ]

We really cannot make this assumption. I do not disagree that we are forced to adopt the position that our own play and that of other players remains reasonably constant, however this does nothing to address the issue of fluctuating table conditions. This may be much more important than previously thought. Obviously mandelbrot/chaos theory has gone a long way the proving the so-called "butterfly-effect", and it may be that he arrival of just one new player at a table, even if this players seems "normal", may be enough to spiral the table from a "settled" state to a much more "chaotic" state, akin to the market variations quoted in my previous post.

I think it is very useful to calculate baseline figures, assuming a normal distribution, and Kiddo has posted elsewhere (and I have repeated) the calculations Mason has used for this. They are useful, and eye opening, but do not take into acccount table conditions. We simply cannot assume the distribution for a 6-max table will be the same as for a full-ring game, or that $5/$10 is the same as $10/$20. We know this is different, the conditions are different. What we need to do is try to get to grips with how this affects our sessions. We do not yet have any statistical basis, apart from the rather ad hoc figures posted on the forum, for characterising the distribution or comparing to a normal distribution. This would be a logical next step, but would require some sizeable data. I would be happy to collate such a sample, though my statistical/mathematical ability will not stretch to provide the kind of rigour necessary to formalise anything produced, I think collating sessional information could be very interesting.

Just one more thing, why N=100 and N=1,000? What about other values of N? How do we determine what value of N we should be working with, this would appear to be critical to determiing when we can expect results to begin to normalise (when we say normal, are we talking guassian or fat-tails?).

[pfkaok]

[ QUOTE ]
"In financial markets, and in other economic and social phenomena, tails are often fatter than the theory predicts. There are more "rare / extreme events" than statistics can show. There is a bigger number of very high and very low values (for example of daily returns) than is theoretically expected. Under / over reactions and herding might explain it.

[/ QUOTE ]

This could easily be applied to poker, and particularly the 6max games, as with fewer people, everyone is more familiar with how each person at the table is doing. The herding effect is definately there, since when people see you losing, even weak opponents become tougher and play back at you in spots that they normally wouldn't. And when you've won a few big pots and shown down some monsters, people subconciously get scared of you, and will fold more, and play more predictibly against you... which will both increase your EV, and reduce your swings. This would be an extremely dificult thing to prove, as a lot of it is on the subconscious level, and you'll never actually know if your opponent would have played any given hand differently if you had won 50BB in the last hour.

I know that I noticed this when I used to play a lot of 3-6 in B & M casinos, but once I started reading more, and studying more about poker I always assumed that I Must have just had selective memory, and/or had an insufficient sample size. I'm still certain that my sample size was increadibly insufficient, but now I believe that there might have been more merit to my thought than I had realized? Maybe it has to do with the fact that most people who play cards truly believe in rushes and strings of luck (including bad luck, "when it rains it pours" and such), I don't know. But when I see things like this, and really think about it, it makes me believe more that the right kind of stop-loss money management system might have more value than simply alleviating tilt. Of course it won't turn a bad player into a winner, but it might add to your profits by avoiding slightly +EV, high swing situations... like when you've quickly lost 25 + BB at a table and weak to average players begin taking shots at you in pots that they might just let you have if they didn't think you were "tilting" or frustrated. And when you switch from a table like that to one where you have a clean slate, you maximize your chances to be in a hugely profitable, lower variance situation... like when you run well right off the bat, and everyone thinks you're on a rush, so people would readily surrender pots in spots that if they thought you were "tilting", they'd get tricky and play back at you.

I've noticed this more than ever playing 6max though, b/c if people see me win a few pots with good hands, after betting them the whole way. Then they see me call somebody down, and take a pot down with A high, then its perfect. If people think that you're always going to call down "without a pair!@?! You're the worst player ever" (as they type in the chat box sometimes)... then they are going to play so straightfoward against you, and when they bet or raise you know they've got something. If you could somehow always be in a game where your opponents thought you were "on a rush", and "always call down", I would love to see the kind of BB/100 you'd put up... I'm guessing it'd be more than any world-class player could put up in a normal game.

[Benjamin]

pfaok wrote:[ QUOTE ]
the right kind of stop-loss money management system might have more value than simply alleviating tilt. Of course it won't turn a bad player into a winner, but it might add to your profits by avoiding slightly +EV, high swing situations... like when you've quickly lost 25 + BB at a table and weak to average players begin taking shots at you in pots that they might just let you have if they didn't think you were "tilting" or frustrated. And when you switch from a table like that to one where you have a clean slate, you maximize your chances to be in a hugely profitable, lower variance situation... like when you run well right off the bat, and everyone thinks you're on a rush, so people would readily surrender pots in spots that if they thought you were "tilting", they'd get tricky and play back at you.

[/ QUOTE ]

I think your thoughts on the advantages of running well have merit, but I think that one can also take advantage of people who are loosening up and taking shots at you because they sense you are weak. You just have to stop bluffing and value bet like mad. I've frequently had tables where I've lost substantially more than 25 BBs, and then come storming back for a win.

If your opponents already tend to be loose aggressive, as they are at 10/20 and 15/30, then if they go more in that direction because they sense weakness, that is good for you. If your opponents tend toward loose passive and weak postflop, then you don't want them to move toward aggression, but you can still take advantage of their looseness.

B.