Small Edges and Bankroll Growth

[Toffler]

I am relatively new to poker. I am a former BJ player. Until a few years ago, it was thought that a good BJ player should push small edges like splitting TT vs 5 or 6 and doubling down with T v T or T v A when the count is such that the play is correct. A smart fellow, Karl Janacek, showed that such plays actually slow bankroll growth because of increased variance. He showed the plays were only correct when the edge was higher (that is, make the plays at higher counts) and your bankroll would actually double sooner even though you were giving up EV. He even showed that some negative EV plays like insuring a 20 when the count was not quite high enough would result is faster bankroll growth by decreasing variance.

Now to Hold'em poker. I think the same might be true for certain plays in poker. Examples would be going for a gut shot with slightly more than break even odds. Playing small pairs in a loose aggressive game against 5-6 limpers. Playing marginal holding on the button in unraised pots. It might even be incorrect to call a raise or reraise with certain holdings like TT or 99 against certain oppenents even with slightly positive EV. I suspect giving up some small EV in these situations might actually result in faster bankroll growth. Has this been studied with computer sims for poker? Is this concept wrong for poker?

[K C]

I'm no expert in blackjack but assuming a negative expectation, anything that would increase variance would have to increase losses. Then again, maybe you count cards or something and get an edge that way.

With a positive expectation though, you're adding equity to the account each time you take advantage of an edge in the odds. And variance is a zero sum game, as long as you don't have to drop down to a lower playing level. So without that consideration, it is indeed advantageous to push any edge that arises.

Now on the other hand, this is going to depend on your bankroll strategy. The goal for most players is to move up of course. So, the closer the distance is between your levels, the more variance is a concern. And with the more aggressive bankroll management that I teach, which looks to put you up levels much faster than normal, it's not only dropping down that concerns you, it's delays in moving up that you need to worry about as well.

So indeed this may be something you need to look at, depending on your strategy here, and what effect variance has on it. This goes beyond the situations you suggest though, and extends to managing variance in general. You may even want to switch to a different type of game - for instance, from cash games to single table tournaments, if your variance is enough of an issue.

Good insight though. This is pretty advanced stuff for a new player, and if you keep this up, your opponents are going to be paying a big price /images/graemlins/smile.gif

KC
kingcobrapoker.com

[Cheeseweasel]

First of all, thank you very much for your thought-provoking post.

I am also a former blackjack player. I used to use the Uston Advanced Point Count, primarily for its very high betting efficiency. Of course, as you know, betting proficiency is the most important element of playing blackjack, much more important than playing efficiency.

When I play no-limit and pot-limit poker, I calculate the optimum Kelly Criterion percentage of bankroll for bet-sizing. As I'm sure you know, this insures the optimum geometric bankroll growth. However, it has never occured to me to do an analysis of limit poker games.

Are you familiar with Don Schlesinger's "World's Greatest Blackjack Simulation"? I'm sure you know the SCORE. (By the way it was done using Karl Janacek's Statistical Blackjack Analyzer). It seems that a project like that should be done for limit holdem.

I'm sure that you won't find that it's been done already. The great majority of poker "theoreticians" (Sklansky and Malmuth excepted) seem to be more comfortable with hand-waving arguments and vague, ephemeral assertions than their blackjack counterparts.

You have some good examples of plays that may be EV+ but bankroll-draining because of their relatively high variance. I think that the number of marginal poker plays may well be far greater than those in blackjack.

Is this concept wrong for poker? In my humble opinion, a resounding NO. I'm surprised that it hasn't been addressed before now. If I could find a decent poker simulator, I would undertake the project myself.

Best of Luck

[stigmata]

Thought provoking post, although I'm not sure I quite get it yet. Could you explain further, or point me to somewhere I could read more. As far as I understand it, what you are saying is that some minor +EV plays have a major increase on variance. This is fine and understood.

I'm just not sure why this will be detrimental to bankroll growth. E.g. say, for example, I decide to cut out all these minor +EV, large variance plays, but my clone (who plays identical poker) decides to carry on using them. In 6 months time, averaged over all universes:

My bankroll: 10000 +/- 1000
Clones bankroll: 11000 +/- 3000

So my clone is of course doing better, on average. So why are these detrimental to bankroll growth... is it the day-to-day variance that is the problem? E.g. on one give day my clone is at 8000, the next at 14000? So that his bankroll is unstable, with its lowest point being below the lowest point of my bankroll?

Is it this volatility that limits me from moving up stakes quicker? E.g. if I was to sacrafice perhaps 0.25 BB/100 on high-variance plays (basically play a bit weak-tighter), my variance would be reduced such that I would not have as many -50BB downswings. This way I would not need a 500BB bankroll to move up to 10/20, but perhaps 300BB would do?

If this is correct, then it's really how you want to play it. To move up stakes quickly, playing weak-tight might be better. To extract the maximum from your limit, chipping away at every little +EV situation is obviously the way to do it, but you need a bigger bankroll.

Basically adjusting between variance/EV according to your bankroll.


Please correct me if I am wrong, but this is the way I understand it.

[axioma]

one thing i would point out is that bankroll groth is FAR more important in BJ than in poker.

In BJ, the game is exactly the same at high stakes, as it is at lower stakes - this is not at all the case with poker.

therefore the notion to pass up small edges in BJ in order to grow bankroll more smoothly might be profitable in order to assure advancement to higher stakes ASAP, whereas in poker you would want to push those small edges, because you dont have as much riding on your moving up.

just a thought.

[SunOfBeach]

2-tabling 5/10 requires less bank.

at 1BB/hr on each, but with SD(10/20) = 2* SD(5/10), then
2 5/10 tables gives expected hrly rate of 20 with sd = 1.41SD (sqrt of 2), which is obviously less than 2SD.

[stigmata]

OK I think I've confused the point somewhat.... I'm not sure thats what I was getting at. Clearly 5/10 requires smaller BR than 10/20.

I meant could you get away with smaller bankroll at, say, 5/10 if you reduced the number of high variance plays.

Unless your just confusing me even further..... doh

[yoshi_yoshi]

I think the reason this applies more to Blackjack and less to poker is that you have more freedom to choose how much you bet in Blackjack. In fact (I think), if you are using something like the Kelly criterion you always have a mathematically correct amount to bet at each count, depending on your bankroll. So having a steadily increasing bankroll would be useful in BJ because you can bet more in the future.

Contrast this to poker, where, if you have a 300BB bankroll, you are likely to be staying at the exact same stakes for a lengthy period of time, only moving down in cases of very bad luck, and moving up when appropriate. In most of our poker lifetimes, we are only going to be moving throughout 10 or so limits, so the variance issue is not as important as in BJ.

However, I think your theory has more and more relevance to poker as players decrease their necessary bankroll. For example, if a very disciplined player decided to move up to the next level when he has 40BB, and move down when he has 30BB, variance will matter a lot more to him, because he can use the extra bankroll more judiciously than the 300BB player. A 300BB player who wins or loses 20BB can shrug it off as it usually does not affect how much he can bet it the near future, while a 40BB player who wins/loses that much is going to have to change his next bet drastically.

Just my two cents.

[Ed Miller]

Is this concept wrong for poker?

In my opinion, it IS WRONG for poker... at least limit hold 'em. In fact, this concept you've stumbled upon is actually a very popular one among poker "theorists," and it's one of my major pet peeves. Wanna know why it's my pet peeve?

[ QUOTE ]
Playing small pairs in a loose aggressive game against 5-6 limpers.

[/ QUOTE ]

Because a pocket pair after five or six limpers is an incredibly +EV situation, and if you skip it to reduce variance, you are shooting yourself in the foot. Now I know you are new to poker, and you were just throwing out possible examples, but this represents a problem almost everyone who tries to implement this concept has:

They try to use it in completely inappropriate situations. They like the concept.. think it's sexy or what not.. and they try to use it in all sorts of scenarios where it doesn't belong.

In blackjack, you can quantify everything. You can say, "For these three specific plays, lowering your EV actually grows your bankroll faster." There's no temptation to make the play in any other situation.

Not so in poker. Poker isn't blackjack. Simulating doesn't work: The game is too complicated to simulate to any accuracy even remotely near necessary to make these sorts of hair-splitting CV distinctions. At least there's nothing available now that simulates that accurately, and don't hold your breath for something either.

I'm not familiar with the numbers, but I can basically guarantee you that the effect on bankroll growth of splitting versus not splitting tens versus a six at a +6 or +7 count (or whatever the index is) is negligible. That is, you can show mathematically that it ever so slightly grows your bankroll faster to make the -EV play. But if you never made that adjustment, you wouldn't begin to actually appreciate the difference in real bankroll-size terms for centuries, probably.

Now think about what would happen to your bankroll if you liked this "I'm gonna lower my variance by not doubling and splitting in +EV situations" concept and had to figure out the appropriate situations on your own without computer aid. You'd start doing it all over the place. I'm not gonna double soft-15 against a 5 anymore. I always catch a ten when I do that. Bleh.

Maybe I won't split sixes anymore against a three. After all, I don't like having two sixes as hands. That just seems like it's gotta be the right spot to make the adjustment. In fact, I think I'm just gonna stop splitting sixes entirely. Sure it's theoretically +EV, but it just jacks my variance up, and I'd prefer to lower my variance.

Maybe you, personally, wouldn't say that stuff, but a whole lot of people would. I know gamblers. /images/graemlins/tongue.gif

Think about what that would do to your bankroll. SUICIDE. Well, that's what poker players do ALL THE TIME to themselves when they try these "I'm gonna lower my EV and variance" plays. They pick all sorts of inappropriate spots that "seem" right, and they end up playing like douchebags.

99.9% of all limit hold 'em players would make more money, grow their bankroll faster, and have more fun playing if they just forgot about this lowering variance crap and made the highest EV play they could. Because of that, I feel this discussion is totally unhelpful, and I wish it would just die.

EDIT: BTW, you guys who link my posts in discussions when appropriate, please start linking this post. This "let's all lower our variance" nonsense is really beginning to piss me off. /images/graemlins/smile.gif

[colgin]

[ QUOTE ]
BTW, you guys who link my posts in discussions when appropriate, please start linking this post. This "let's all lower our variance" nonsense is really beginning to piss me off. /images/graemlins/laugh.gif

[/ QUOTE ]

Consider it done.

[muck_nutz]

[ QUOTE ]
Not so in poker. Poker isn't blackjack. Simulating doesn't work: The game is too complicated to simulate to any accuracy even remotely near necessary to make these sorts of hair-splitting CV distinctions. At least there's nothing available now that simulates that accurately, and don't hold your breath for something either.

[/ QUOTE ]

You don't think online poker and large hand databases has not made this determination possible or at least increased the accuracy of the estimates to the point that some very low sharpe ratio situations could be determined usefully?

[SlowStroke]

This is an interesting topic. I look at poker from (at least) two points of view.

The micro - decisions about how to play the current hand you have been dealt, call, raise, fold, in no-limit how much to bet or raise.

The macro - decisions about what site to play, what game, what limit, how many multiple tables, full game or short handed and so on.

In the micro view I am always looking for ways to minimize variance if the cost in EV is not too high. In poker there are many ways in which you can trade off one against the other.

For me, if I lower my variance in the micro view I can actually raise my EV in the macro view. Less volatile swings in a game means I'm more relaxed, I'm less likely to tilt, I play better and I can play longer. And the most important reason is that I can play at a much higher limit.

So, less EV per hand can actually be more EV overall.

[Ed Miller]

You don't think online poker and large hand databases has not made this determination possible or at least increased the accuracy of the estimates to the point that some very low sharpe ratio situations could be determined usefully?

I suppose it's possible. You need a damn large database, though... and so much of poker is situational that you'd have to be VERY sure that you aren't introducing systematic error into your results by using data with an unknown bias.

Given what I know about poker "scholorship," such as it is, I think it's quite unlikely that someone will do a good job of identifying these situations accurately in the next five years.

[bobbyi]

You have written out a very lengthy post that only says one thing "Lowering variance by making plays that only slightly lower EV doesn't work because most people who try to do it do it incorrectly". That is not a useful argument. If people do something incorrectly, obviously it isn't going to work. I could just as easily argue "It's impossible to win at poker because the majority of poker players try to win, and fail at it". It just doesn't work that way. The fact that most people don't know how to do something correctly doesn't at all answer the original question of whether it can be done. Are seriously trying to get us to buy the following: "Small pairs against many limpers are very +ev. Folding them to decrease your variance thus means you are trading away a lot of ev. So there must be no way to derease variance without trading off a lot of ev". Yes, folding the pairs in these spots is bad. Yes, the specific example that the OP chose was not a good one. As he says, he is relatively new to poker, so he chose a poor example of a situation where your EV in playing a hand is positive, but very small. Trying to nitpick his individual choice of example (which I agree was a poor one) does nothing to change the fact that there are plenty of marginal situations preflop (not to even mention postflop situations) where a hand can be played for a profit, but only a very small one. When one of those situations arises, folding means that you are trading off a small piece of ev for a reduction in variance. I have worked hard on lowering my variance and it has worked very well for me. It requires only that you have the knowledge and experience to recognize which situations are only very slightly +ev. Can most poker players do that? No, as you say, they cannot. However, that should be expected as most poker players can't even recognize which situations are extremely +ev or -ev, and that is why they make a lot of mistakes and lose a lot of money even though most of them have never heard the word "variance" and wouldn't care about it if they had.

Using the inability of most people to correctly identify marginal situations where folding leads to a decrease in variance with only a neglible effect on variance as a justification for an argument that it is theoretically impossible for these situations to arise and for one to recognize them is exactly the kind of handwaving to which someone else on this thread was referring.

[GrunchCan]

[ QUOTE ]
99.9% of all limit hold 'em players would make more money, grow their bankroll faster, and have more fun playing if they just forgot about this lowering variance crap and made the highest EV play they could.

[/ QUOTE ]

Thanks, Ed. I know a few people in mocros who will appreciate this post.

[twang]

Can someone please explain to me why +EV (no matter size) should never be passed when you are a)playing within bankroll restrictions and b)want to move up in limits ASAP?

/twang

[Ed Miller]

*shrug*

My take on this is like my take on bunching. Bunching exists, and it affects the distribution of cards slightly. But anyone who actually tries to adjust his play based on bunching is almost certainly going to hurt his results.

Another way of saying this is:

The only players better served by worrying about these situations rather than just maximizing their EV are people who already play extremely well. And those people aren't worried about it because they are more worried about how to spend their wheelbarrows full of cash.

I'm speaking to the overwhelming majority of players. "You will do better if you ignore this concept." If you are the exception, more power to you.

[bobbyi]

[ QUOTE ]
My take on this is like my take on bunching. Bunching exists, and it affects the distribution of cards slightly. But anyone who actually tries to adjust his play based on bunching is almost certainly going to hurt his results.


[/ QUOTE ]
I tend to agree with you on bunching (at least in holdem; Mason has said that in games like lowball it can be signficant, which makes sense to me). This is because trying to consider bunching requires the player to take what they are already doing (trying to play winning poker) and add another layer on top of it. So it interferes with the primary task of trying to win. However, I believe that trying to win while keeping your variance acceptable is not any different or harder than merely trying to win.

When you are trying to play in any +ev situations, no matter how small, as you suggest, what this means is that when you are dealt a hand you must evaluate the situation and decide whether the ev of playing the hand is greater than $0. If so, you play (which may mean calling or raising). If not, you fold. When you are trying to play while keeping your variance low, you do the same. The only differenece is that rather than conditioning on whether your ev is greater than $0, you decide based on whether it is greater than $1 or some other magic number that happens to be greater than $0. So it is exactly the same thinking and reasoning you need to do anyway, you just shift your baseline for playing up slightly. I assert that deciding whether a hand is worth $1 is no easier or harder than deciding if it is worth $0. Does this mean it is easy? Certainly not. But figuring out which marginal hands are worth ever so slightly greater than $0, as you require, isn't easy either. In reality, I don't have an actual magic number in my head like $1. The process of deciding "is this hand good enough for me to play" is basically an intuitive one like it is for someone who will play a hand even if it is only worth $.00001. I'm sure that when you see a starting hand, an actual ev number in dollars does not appear in your head, and the same is true of me. But from a theoretical point of view, this is what happens and I don't think it interferes with poker playing in the same way that worrying about bunching does (which involves adding a new consideration that wasn't there before).

(Trying to keep your variance low could actually be more comlicated than this because your "ev threshold" should vary with how high the standard deviation is associated with the given hand. For suited connectors, you would generally require more ev, for example. But that is only if you want to make things more complicated and a constant threshold still works fairly well. Also, I am only discussing preflop, but making postflop decision is the same: if your ev in continuing is less than some threshold, you fold. Your theshold happens to be $0 while mine is slightly higher. I am also ignoring the metagame imlications of folding because that's a whole other can of worms. Suffice it to say, the ev you should be considering is not only for this hand but should also include how playing will make or lose you money on future hands).

[ QUOTE ]
The only players better served by worrying about these situations rather than just maximizing their EV are people who already play extremely well.

[/ QUOTE ]
I think the average player plays too many hands preflop and looks for too many excuses to call postflop. They are playing slightly -ev situations and rationalizing them as being +ev. Anything that encourages them to fold more in very marginal situations would probably lead to better results.

[That guy]

This subject obviously comes up over and over again and would be a great topic for the online magazine.

At first blush, it seems to me there should be some kind of efficient frontier which balances the EV/variance tradeoff. This frontier assumes you play positive EV poker and just choose the level of risk you are comfortable with... This would result in a frontier (a curve on a graph) where returns would increase at a diminishing rate for a given level of risk increase.

However, at the 'portfolio level' -- your bankroll that is not in use is always sitting in cash. Therefore, if you have 300 BB's as a bankroll and assume you use 30 BB's as a buy-in, you have 270BB's sitting in cash.. The formula for a 'portfolio std dev' is equal to:
(wgt1^2*SD1^2 + wgt2^2*SD2^2 + 2wgt1wgt2SD1SD2P)^1/2

where P = Correlation coefficient
SD= Standard Deviation

Since 90% of you bankroll is in cash, you are left with:
[(wgt1^2)(SD1^2)]^1/2

Since only 10% of your bankroll is exposed to risk and 90% is riskless, a high variance for the 10% is perfectly ok. You should not be passing up +EV situations because of some risk to a small portion of your assets. Even more so, not even 30BBs are at risk on any decision... More like 5-8 BB's...

Net net, the key concern is whether each decision is or is not +EV.

Note that if you are multi-tabling, at least in theory, the correlation coefficient should still be zero.

[maurile]

This makes sense only if you can reduce your variance enough to move up to a higher limit than you'd otherwise be able to without exceeding your desired risk of ruin -- thus increasing your dollar-EV by sacrificing some percentage-EV.

It's easier to do this in blackjack because if you want to bet $125 a hand instead of $115, no problem. But moving up in poker generally involves making a larger percentage jump. Reducing your variance by a bit probably won't let you move from $10/$20 to $15/$30 without substantially increasing your risk of ruin (holding your bankroll constant).

Nonetheless, David Sklansky has sort of written about this topic, although I forget exactly where. He mentions that if you're taking a shot at a bigger game that appears particularly attractive even though you don't really have the bankroll for it, you may want to make certain variance-reducing plays instead of always making EV-maximizing plays. For example, you may want to raise a bettor on your right when you flop a set instead of smooth calling to keep people in behind you, even if smooth calling has the higher EV.

[bobbyi]

[ QUOTE ]
Since only 10% of your bankroll is exposed to risk and 90% is riskless, a high variance for the 10% is perfectly ok. You should not be passing up +EV situations because of some risk to a small portion of your assets.

[/ QUOTE ]
That's some good analysis. When I have time over the weekend, I will come back and examine it more carefully because it is very interesting. You are probably right from a theoretical point of view. However, a big part of the reason I perfer low variance isn't about minimizing the risk to your assets. It's that large downswing are psychologically difficult and can often lead to poor play. In the thread about January downswings, I think there was discussion about how a big downswing can throw someone off their game. Since I main play live (I'm about to finally start playing online as well) and only play ten to twenty hours a week, a big downswing could last me weeks or months and I would find it hard to spend that long with that hanging over me. So I've decided it's safer to sometimes pass up some small edges and marginal hands to try to control my variance even if it means giving up a few dollars per hour in ev. However, from a theoretical point of view, you might be right that someone who manages things well can accept high variance and avoid going bust.

[Cheeseweasel]

tournament

blinds = 50/100
bankroll = t1000

You flop a gutshot. The pot is t2000. T100 to you. Call or fold?

Easy call, right? The pot odds are nearly twice that required for a call. But let's look at what Mr. Kelly has to say.

let
f* = the fraction of your bankroll that that you should bet to maximize your expected value
A = pot odds
p = win probability

f* = e/A where e = (A+1)p-1>0

Let's plug in some numbers.

e = (20+1)*(4/47)-1 = .79

f* = .79/20 = .04

Therefore, your optimal bet is t1000*.04 = 40. Easy fold.

There has been no correction made to f* due to the variability of the payoff so this is extremely conservative.

All of this "let's stop hating variance and learn to embrace it" nonsense is starting to piss me off.

[muck_nutz]

[ QUOTE ]
I suppose it's possible. You need a damn large database, though... and so much of poker is situational that you'd have to be VERY sure that you aren't introducing systematic error into your results by using data with an unknown bias.

[/ QUOTE ]

I agree with you on this part.

[ QUOTE ]
Given what I know about poker "scholorship," such as it is, I think it's quite unlikely that someone will do a good job of identifying these situations accurately in the next five years.

[/ QUOTE ]

I'm not so sure here. There are large databases in the world _and_ the amount of money in poker has increased dramatically in the last few years bringing smart minds to the table that wouldn't have been interested in the past.

[deepdowntruth]

Variance is not a problem if you are adequately bankrolled for your blind level and are in a ring game.

Variance is a problem in a tournament. In Sklansky's advanced tournament book he asserts it is probably best to choose a lower EV option if that option has a better chance to win. He's probably right about that.

[Ed Miller]

This is both irrelevant to the discussion and gravely in error.

[rmr1976]

I'm going to bookmark this for the next time I have a poker debate with a friend of mine, who in spite of his intelligence, plays from a weak/tight mindset.

Example: I asked him how he would play AA on the button with a flop of KQJ rainbow.

Pretend that you are in a game with a bunch of huge maniacs, who throw chips into the pot as if they had an unlimitted supply. Assume the betting was capped pre-flop and capped post-flop, and that it will be capped on the next 2 rounds. There are 5 or more people on the flop and you expect no one to fold.

Suffice it to say--this is going to be the mother of all pots.

Pot size at river: 60 BB ((5 players * 4SB * 2 rounds)/2) + (5*4*2)

Price to call thru river: 8 BB
Odds 60 to 8 or 7.5 to 1

My friend thinks folding is appropriate, because it is likely you are drawing to only 2 outs. I think it is essential to stay until the river.

Is it possible that my friend is correct, and that a fold is proper here?

[Ed Miller]

My friend thinks folding is appropriate, because it is likely you are drawing to only 2 outs. I think it is essential to stay until the river.

Is it possible that my friend is correct, and that a fold is proper here?

If you assume your opponents have truly random hands, and that they will always cap the betting and never fold, no matter what they have, then folding your aces is truly ridiculous.

Furthermore, with such simple (albeit unrealistic) assumptions, this becomes a math problem. You can calculate the EV of continuing versus folding exactly. So your friend doesn't have to take my word for it. He can do the math for it and find the answer himself.

BTW, I can't imagine why anyone would describe AA on a KQJ board as having two outs.

EDIT: BTW, go to http://www.pokerstove.com/ for software that performs these calculations.

[rusellmj]

[ QUOTE ]
*shrug*

The only players better served by worrying about these situations rather than just maximizing their EV are people who already play extremely well. And those people aren't worried about it because they are more worried about how to spend their wheelbarrows full of cash.


[/ QUOTE ]
Funniest thing I've read this week. Great thread. Thanks for your contributions Ed.

Russ

[rmr1976]

Ed Miller wrote:
"BTW, I can't imagine why anyone would describe AA on a KQJ board as having two outs."

He would argue that all this action suggests your opponents have something better than a random hand, ie. KK, KQ, KJ, JT, or something like that.

Your odds of improvement are much lower with all of those people in the hand, and he feels the probability of winning is much less than what you would expect.

So he would dispute the idea that your aces are up against random hands.

[Ed Miller]

[ QUOTE ]
Pretend that you are in a game with a bunch of huge maniacs, who throw chips into the pot as if they had an unlimitted supply. Assume the betting was capped pre-flop and capped post-flop, and that it will be capped on the next 2 rounds. There are 5 or more people on the flop and you expect no one to fold.

[/ QUOTE ]

To me, this characterization says "random hands." You can't let him just go around changing the rules on ya... /images/graemlins/tongue.gif

[ACW]

This sounds the wrong way round to me. I think that reducing macro variance can be important, but that reducing micro variance never is.

[O Doyle Rules]

[ QUOTE ]

In the micro view I am always looking for ways to minimize variance if the cost in EV is not too high. In poker there are many ways in which you can trade off one against the other.

For me, if I lower my variance in the micro view I can actually raise my EV in the macro view. Less volatile swings in a game means I'm more relaxed, I'm less likely to tilt, I play better and I can play longer. And the most important reason is that I can play at a much higher limit.

So, less EV per hand can actually be more EV overall.

[/ QUOTE ]

I think this is a valid point. For certain individuals, a lower variance style of play perhaps could be +EV just for them due to their emotional makeup and their ability or inability to handle large bankroll swings.

I know my climb through the limits (online play) from .50/100 thru 3/6 was a steady profitable one by playing closer to a Lee Jone's style of play (look for a reason to fold) and this was for a large sample of hands.(>350k hands)

My results since switching to a SSH style of play has been much more volatile, and less profitable. (approx. 150k hands)

Assuming that I employed both styles of play to the same level of correctness, (I know there is no way I was playing either style 100% correctly) I am starting to feel that "just for me" (and maybe certain other players) that a lower variance style of play might in fact be better for their "peace of mind" when playing poker and therefore more profitable for them longterm.

[alThor]

[ QUOTE ]
Until a few years ago, it was thought that a good BJ player should push small edges ... A smart fellow, Karl Janacek, showed that such plays actually slow bankroll growth because of increased variance.

[/ QUOTE ]

Just for the record, these ideas (Kelly) were known over 50 years ago. If you want bj-specific references, start with Joel Friedman's risk-aversion article, which was published in 1980 (a couple years after KJ was born).

alThor

[Paul2432]

[ QUOTE ]
Nonetheless, David Sklansky has sort of written about this topic, although I forget exactly where.

[/ QUOTE ]

Take a look at this post from RGP (http://groups-beta.google.com/group/rec.gambling.poker/msg/83b10ea7501c04da) . Sklansky says:

[ QUOTE ]
Some of the big players have an incorrect opinion because when I do step up to 300-600, I do so because I feel that the game is good enough that a very tight, sub optimum strategy, will still result in a high hourly rate while maintaining a low SD.

[/ QUOTE ]

Paul

[2ndGoat]

Argument 1) Individually Positive EV plays can slow geometric bankroll growth.
Argument 2) Altering your play to reduce variance is a can of worms.

I agree with both. My one suggestion for reconciliation: when you can't make up your mind for a particular action, consider another factor to try to tip the scales. And if you still can't make up your mind, consider another... when you exhaust all EV-minded decision criteria and you're STILL exactly on the fence, take the lower-variance route (then go back later and figure out another relevant EV-minded factor). Now you're not intentionally sacrificing any EV, but you're reducing your bankroll requirements.

Of course, whatever benefit you derive from this strategy better be labelled "epsilon."

2ndGoat

[ACW]

I wonder if the root of this discussion is the splitting of hairs between the arithmetic and geometric means?

EV is the weighted arithmetic mean of the possible outcomes. In BJ, you can keep each bet as a fixed percentage of bankroll, and in such a case it is the geometric mean that matters. It's not hard to construct scenarios where the arithmetic mean is positive, but the geometric mean is less than one.

In poker, if you are playing within your bankroll, and changing stakes rarely (compared with the number of hands you play), the arithmetic mean is what matters. The less well bankrolled you are, the bigger the relative importance of the geometric mean, but unless you take your entire bankroll to a big bet table, your results will always more closely follow the arithmetic mean than the geometric mean.

[Toffler]

Janacek's work involved more than Kelly. You can find his paper, I think it's titled "The Theory of Optimal Betting Spreads", on the internet.

Toffler

[Cheeseweasel]

"The Theory of Optimal Betting Spreads" was written by Brett Harris.

[BradL]

Fantastic post.

-Brad

[SinCityGuy]

[ QUOTE ]
I'm not familiar with the numbers, but I can basically guarantee you that the effect on bankroll growth of splitting versus not splitting tens versus a six at a +6 or +7 count (or whatever the index is) is negligible.

[/ QUOTE ]

Actually, it's probably negative EV when you factor in the probability of getting "the tap on the shoulder" and getting 86'ed after you split your tens vs. the dealer's six on a high count. /images/graemlins/wink.gif

[TheHenchmen]

Excellent question. I must say that you've got the mindset of a winner, and kudo's on studying the game and taking the correct approach.
The answer to this question really depends on your tolerance for variance. I personally do not like the downswings of pushing small edges, even though the EV will show in the long run. For me, the current downswings effect my play in a small, yet noticable way. The effects on my play easily counter the slight increase in EV. But that's just me.
You on the other hand, might not mind the variance in bankroll. If such is the case, go ahead and shove every edge with all your might. I simply prefer to get my money in the middle with a larger, more reliable edge.
Anyways... Manage your bankroll by whatever means you are comfortable, continue to think analitically about the game, and good luck to you.

[O Doyle Rules]

[ QUOTE ]

The answer to this question really depends on your tolerance for variance.

[/ QUOTE ]

Bingo!

[Paul2432]

[ QUOTE ]
[ QUOTE ]

The answer to this question really depends on your tolerance for variance.

[/ QUOTE ]

Bingo!

[/ QUOTE ]

Well this was the original question:

[ QUOTE ]
I suspect giving up some small EV in these situations might actually result in faster bankroll growth. Has this been studied with computer sims for poker? Is this concept wrong for poker?

[/ QUOTE ]

You might as well say, well it really depends on your desire to make money. If you don't really care about making money then go ahead and don't push small edges.

I think this thread has shown that pushing small edges DOES NOT inhibit bankroll growth and that was the question.

If player falls apart after a couple of bad beats, then that player has bigger problems than missing out on a small edge here and there.

Paul

[O Doyle Rules]

Hey Paul,

You are correct and there is no doubt that by pushing any edge no matter how small will increase one's bankroll for the long haul. I wish I had went back and re-read the original question before replying in the fashion I did.

However, one's emotional control and state of mind I would venture to say impacts alot of player's results. All I am saying is that for a number of players who fit in this category, a short term, lower variance style of play might help their longterm "poker mental well being" and thus make them more profitable for the longterm.

Are they extracting the maximum amount that is mathematically possible from the game? Certainly not.

Are they playing the best style for them that perhaps leads to the best longterm results for them? I think it can be argued.

[Guruman]

Here's my view from way over here:

Attempting to lower variance based on dropping minimally +EV plays first requires a very accurate analysis of EV.

Analysis of EV in poker is by nature less exact than in Blackjack because of the fact that poker opponents are not restrained to a specific behavior the way that Black Jack dealers are. As an example, you can gain EV in poker by causing opponents to incorrectly fold with well-placed bets. There is no correlation to this in BJ since the dealer is always going to play his cards the same way. This distinction drastically simplifies BJ EV calculation when compared to poker.

Calculating EV in poker accurately enough to drop minimal +EV hands to reduce variance can often include factors outside of the cards themselves (eg position, opp player tendencies, implied odds, etc) that may have a degree of subjectivity to them.

The sum effect of all of this is that it can become difficult to drop marginal +EV plays at first because you first must become very consistent at determining which plays actually are positive or negative EV. Once you've masted that, then you have the tools to adjust your play to taste. Until then, it's probably best to go by the book. /images/graemlins/smile.gif

[maurile]

[ QUOTE ]
Attempting to lower variance based on dropping minimally +EV plays first requires a very accurate analysis of EV.

[/ QUOTE ]
Because if you're a little bit off, you might accidentally increase your EV instead of decreasing it, which would be disasterous. /images/graemlins/wink.gif

[TStoneMBD]

[ QUOTE ]
Since only 10% of your bankroll is exposed to risk and 90% is riskless, a high variance for the 10% is perfectly ok.

[/ QUOTE ]

i too have thought about this several times, but from a more generalizing standpoint and never put it into mathematical analysis. im not certain if your theory is correct or not, id like it if an authority on bankroll management could intervene.

ed miller:

when i started reading this thread, my immediate impression was that it was entirely incorrect for you to state that lowering variance in sacrifice of ev is wrong. if for instance, you are in a no limit cash game with a buyin of 10k, and your bankroll is only 10k, should you push allin on a 51/49 coinflip? of course not. granted, you should not have bought into the game in the first place, but that is besides the point. even a novice of poker should be able to understand and utilize this concept.

however, after you replied to bobbyi, stating that novices end up taking lowering variance too far, and should simply look for ev situations, you redeemed yourself. novices would be far better off if they simply did not consider variance except when approaching the most obvious of scenarios such as the one i mentioned previously. however, i know that people will abide by your request in posting your thread when the word variance comes up, but id truly wish they did not as you created a universal concept that is not fundamentally correct when applying it to all universal situations in a vacuum.

[bobbyi]

[ QUOTE ]
you first must become very consistent at determining which plays actually are positive or negative EV... Until then, it's probably best to go by the book. /images/graemlins/smile.gif

[/ QUOTE ]
What book? You're saying that there is a book that teaches us how to play poker without being required to consistently determine which plays are actually positive or negative EV? If so, that book sucks.

[Zim]

nt

[That guy]

im not certain if your theory is correct or not, id like it if an authority on bankroll management could intervene.

Me too. While they are at it, let me elaborate...

This is just straightforward math. The point is not to prove anything revolutionary, it is just to apply common finance/statistical logic and apply it to poker to 'prove' what most already know -- play small relative to your bankroll and push clear +EV situtations. I agree that many situations arise that are really close in terms of +EV, 0EV or slightly -EV. Only true experts can differentiate these and they are going to get some wrong simply because there is error, often significant, whenever you are estimating.

With that in mind...

Take 2 situations:
Case A: You use your entire bankroll of 30 BB's
Case B: You use 10% of a 300 BB bankroll

Assumptions:
You are playing online and have a SD (in BB's) of 16/hr
(If SD = 16BBs, then a 16 BB bankroll will be equal to 1 SD
We are saying 30BBs, so 16/30 = 53.33%)

As I stated before, the way to think about this is to think about the big picture, your overall bankroll variance -- not just your session variance. Of course, if you are playing with your entire bankroll, then your session variance equals your overall bankroll variance

Case A:

30BB Bankroll (100%)
[(100%^2)(.5333^2)]^1/2
Reduces to [(.5333)^2]^1/2 = .5333
So overall SD = 53.3%

If you are using 300BBs, using the same math but not to be redundant you end up getting 10% of 53.3% = 5.33% (90% of your bankroll has a standard deviation of zero).

OK, now the point... (assume you can play 65 hands/hour)

If you are engaging in a 53% Std Dev practice (per hour), you are taking a wild wild risk.

In fact, using standard statistical measures, you will be bankrupt ~2.15% of the time in the first hour (after 65 hands). You will be bankrupt ~16% of the time in a little under 5 hours (325 hands).

If you only use 10% of your bankroll, you have 104 hours (~6700 hands) to demonstrate your skill. Even still, you will be bankrupt 2.15% of the time just due to variance. You will be bankrupt 16% of the time after 16,055 hands due to variance...

This all assumes you are not sure if you are a winning player or not (your 'win rate' is assumed to be 0).

[That guy]

continued...

the issue with 'pushing small edges' is important in the degree to which it impacts your overall bankroll variance.

If you are playing with a big bankroll, using the assumptions in my previous post... some incremental variance will produce a very small incremental overall bankroll variance. It is so diluted that you shouldn't worry about it. As Ed Miller and Sklansky say, if you are playing short, then variance is concerning and you need to not push small +EV plays as your incremental variance is not being diluted relative to a large bankroll.

This is all common wisdom among good players but just like stock market diversification -- I bet a decent % of the public ignores it...

Mason says in Poker Essays 2 that variance will increase as you rise in stakes because there will be more aggressive players at the table. In this case, SD will rise and all the assumptions above could be understated.

[Cheeseweasel]

Please show me the error of my ways.

[Ed Miller]

[ QUOTE ]
In my opinion, it IS WRONG for poker... at least limit hold 'em.

[/ QUOTE ]

when i started reading this thread, my immediate impression was that it was entirely incorrect for you to state that lowering variance in sacrifice of ev is wrong. if for instance, you are in a no limit cash game with a buyin of 10k, and your bankroll is only 10k, should you push allin on a 51/49 coinflip? of course not. granted, you should not have bought into the game in the first place, but that is besides the point. even a novice of poker should be able to understand and utilize this concept.

Notice I said LIMIT hold 'em. I would say no such thing about no limit, particularly large stack no limit.

And yes, I am saying something that is only 98% true and ignoring the 2% case. I do that a fair bit.. it's a pedagogical tool I use often... and one I happen to like.

I consider myself a teacher first, TStone, not a theorist. I teach novice and intermediate limit hold 'em cash game players to whoop up on small and medium stakes games. And I believe that the overwhelming majority of players whom I am trying to teach would be better off if they pretended variance didn't have anything to do with any decision they make at the table.

[pzhon]

The main problem with applying the ideas of Kelly to tournament play is that the value of tournament chips is roughly linear. The equations you used assume the value is logarithmic, which is ridiculous. That would mean it is never worth taking a risk that might mean you bust out.

There are other problems with that example:
<ul type="square">
The poker situation is implausible. Usually you are not sure of your outs and that you will not get a chance to draw again on the river.

You computed the optimal fraction of your bankroll to call on a gutshot with these pot odds, but that does not directly tell you whether you are better off calling the actual bet or not. With a logarithmic utility function, the optimal bet size when getting pot odds of 20:1 may be T39, but you should call a bet of T93. T100 isn't a clear fold.

The "optimality" of T39 is a misnomer. You would rather see a bet of T20 or a check than the optimal T39.

The variability of the payoff means you may callect more bets after you hit. That helps you, even though it is additional variance, so you have underestimated the amount you should call. [/list]
I'm all for using the Kelly Criterion for rational risk aversion. Try applying it to the context under discussion, a limit ring game where you have a bankroll of about 300 BB. For example, what is the standard deviation on a typical marginal hand with which you limp? Call it 3BB. In that case, the Kelly Criterion recommends that you play those hands that gain at least .5*(3BB)^2/300BB = 0.015 BB.

[bobbyi]

[ QUOTE ]
And yes, I am saying something that is only 98% true and ignoring the 2% case. I do that a fair bit.. it's a pedagogical tool I use often... and one I happen to like.

I consider myself a teacher first, TStone, not a theorist.

[/ QUOTE ]
Oh, sorry, I must be lost. I had been looking for the poker theory forum. I seem to have stumbled onto the pedagogy forum instead. Can you please direct me to the forum where people try to gain a higher understanding of poker by discussing actual theory with all of its subtle nuances rather than ignoring the exceptions that don't fit conveniently into their world view so that they can make the game appear simple for novices? Thanks.

[TheHenchmen]

Well, Paul. Let me just say that I am FAR from falling apart at the poker table. Nobody wins everyday, and I can handle the swings. I should've been more clear in stating that small edges arise in a multitude of hands (depending on how they're played), and should you push every edge, you are going to end up losing several pots, just to win that one pot every once in a while that will net you a profit.
I also stated incorrectly that it affects my play. Well it does, but only indirectly, as I will explain. If you are in a lot of pots pushing small edges, we have to examine our level 3 thinking. Example: I've been in a lot of pots, with a lot of marginal hands, and I've lost several of them. My oppenents are currently seeing me as 'loose-aggressive,' so I will get paid off on my good hands.
While this may be the proper image and play for several of you, my playing style is better suited by an image that will allow me to pick up the blinds, and push people off of the best hand once in a while.
All in all, my play is only affected by the image I am projecting, thus my response to situations must be changed. I simply prefer a tougher image... that of a player who you must always wonder, 'is he going to show me the nuts?' Let me be clear. NOT a player who you already know, 'he's going to show me the nuts.' I want opponents to second-guess themselves and make mistakes, so I do push some medium edges on occasion. It's all about mixing up your play, and I do. Just not enough that my opponents label me as a 'Wildman.'
I hope I've restated my opinion in a manner that can be better understood.

[scotty34]

[ QUOTE ]
My friend thinks folding is appropriate, because it is likely you are drawing to only 2 outs.

[/ QUOTE ]

Maybe I'm missing something, but are the four T's making you the nut straight not considered outs?