First, variance is the standard deviation of your win rate. If you don't understand what those terms are, don't comment on variance.

There are three things that I can imagine reducing variance without sacrificing your win rate.

1) Finding something outside the game structure to base your decisions on. Decisions based on player actions follow the reasoning, "ok, X% of the time when my opponent does this he's holding {X} range." Other kinds of decisions are "Ok, when my opponent's hand shakes, he's holding the nuts." They are much lower variance decisions because they generally involve less drastic percentages implicitly within the reasoning. Variance is not eliminated entirely because you won't always get this kind of information, and also there's some small fraction of the time that your opponent is really blinking because he's got something in his eye. Generally however, finding these things leads to less variance along with a higher winrate.

2) Playing more tables at a lower buy-in rather than one table at a higher buy-in. Diversification reduces the effects of non-systemic risk. It also makes the larger swings that are due to systemic risk occurr less frequently. However, there is not a great amount of non-systemic risk for a person who table selects judiciously, and the benefits to systemic risk reduction will not show up in the Standard Deviation calculation given a large enough sample. But generally, your winrate will suffer less when you sit at a table that has too many regs, and you'll be less likely to hit a really dry run of cards in the interval that's typically relevant to your SD. Also, variance as measured by SD of $/hand will decrease (although $/(hand*amount of tables) will be the same).

3) Most importantly, playing more consistently. If sometimes you decide to get it all in as a 55% favorite and other times you fold, you're increasing the variance of your variance, which is really just absorbed in your variance as a mean. Bottom line, you don't want to constantly be second-guessing your play style throughout the interval that you're using to calculate your variance. This involves that delicate balance of confidence and humility that is one of poker's most fundamental skills. So if you master consistent play, you will go a long way to reducing your variance. My personal advice is to only evaluate your play strategies outside the game, and do it while looking at objective data about your performance.

Kind of ironic that your definition is completely wrong - the standard deviation is the square root of the variance. That's pretty important.

Variance is the mean of the squared deviations.

Your theories also have some errors.

My poker software uses an SD for the correlcation of my winnings data set. If other peoples' uses the actual mathematical variance, that's fine. I actually wasn't aware that "variance" was the same term of art both in math and im poker. The concept doesn't change much just by squaring it, anyway. If there's mathematical problems with my assumptions above, please discuss on the merits.

A different line of reasoning that I was thinking along is that not only would most changes to decrease variance also decrease edge, they'd also be exploitable by a thinking opponent, right?

I'm not a statistician, but isn't it true that the thinner Hero's edge is on any particular decision, the more variance he'll experience if he makes that decision 1000 times in a row? If Hero is pushing when he knows his equity is exactly 50.1% he's more likely to lose money than if he demands an edge of 75% to push; therefore, he's more likely to experience large swings from expectation in the 50.1% scenario than the 75% scenario, right?

Is it possible for a good poker player to decrease his variance while holding his edge constant, playing the same game, without fundamentally improving as a poker player (by making less FTOP bad decisions)?

Yes. The normal swings are bigger on 50/50 bets than on 90/10 bets. You are taking less risk as a 90% favorite, thus you have less variance (which measures risk).

To estimate a normally distributed variance, just multiply edge * (1-edge) * sample size. On a 50/50 bet 1000 times, "normal" random variance is 250 with SD of 15.8. On a 90/10 bet 1000 times, "normal" random variance is 90 with SD of 9.5. The SD is the bounds on the range of how much you expect to be + or - the mean in ~68% of the samples. 2SD is the bounds on the range of how much you expect to be + or - the mean in ~95% of samples.

If you fold everything but Aces, you will have very low variance, but also very low winrate. No risk no gain.

Not neccessarily because you have to factor in how much money the player lost waiting for a 75% edge. For the most part you are correct. But as a blanket statement, someone waiting for a 75% edge isnt less likely to lose as a person willing to press any edge. There are too many other factors involved to determine if someone is more likely to win/lose. There could very well be a game with the blinds/antes/rake at such an amount that waiting for a big edge is not going to work.

blah blah, what I mean is someone nut peddling could very well have a graph that steadily goes down with small peaks that dont overcome the previous slope.

I think a better way to word it would be given one session, one buyin, or a very short timeframe, a person pushing any/all edges is more likely to lose.

lol.

The only way to "reduce" variance is to rig the game.

Don't worry about variance, worry about your skills.

This is obviously not true. You can reduce your variance to 0 if you want to. I'm not saying the result will be good but it's possible.

It often IS possible to reduce your variance SOMEWHAT without a large reduction to win rate, and when you can do it, it's beneficial because you can play higher stakes with less chance of going broke.

Let's say you play in a rebuy/add-on MTT rolled for 300 $100 buy-ins.

Let's assume your expectation in a $300 buy-in MTT (that ISN'T a rebuy/add-on) is identical.

Would playing the rebuy/add-on reduce variance?

Clearly style is the biggest factor in determining your variance as many have already pointed out but with regards to your original question of using skill to reduce variance, yes, great plays can both increase and decrease variance but if you separate out skill from style with regards to variance, i.e. bluffing more as variance due to style while bluffing more effectively as variance due to skill than I think it creates lower variance overall.

Start by looking at closing action calls/bluffs/value bets.

If your calls are 100% correct your swings for any particular hand will be between the largest pot won - your own investment to the largest investment lost when you made a correct fold. If your 50/50 on your calls, than the swings will be increased by the amount of your bad calls so anything in-between will reduce variance proportionally.

It's true that there are plenty of situations where improvement will necessitate increased variance. i.e. minimum variance is never calling so adding calls will both improve play and variance. Also because of pot odds, you can have more bad calls than good while still being net positive on the call decisions, though not necessarily over all, so improving your good to bad call ratio will also increase both winnings and variance before passing the 50/50 point will reverse this, since we are talking about great play to reduce variance, than we are mostly talking about improving the good/bad ratio overall.

For bluffs it's much the same except the act of bluffing itself adds variance but I'd classify that as more a style factor than skill factor. So again having more successful bluffs than failures improves results while reducing variance for great play.

For good to great value bets you actually increase variance by adding more thin value bets but I still think, great decision making in general for closing action will result on lower variance.

Prior to ending action, I think part of great poker also lends itself more towards lower variance while keeping within the bounds of the high variance style.

bluffing/betting to improve your balance does induce higher variance but again that's more about style while skill is more about improving successful bluffs to failed bluffs ratio when selecting bluffs to balance with. Also knowing when to give up on a bluff can also be a huge factor in reducing variance.

Betting increases variance but is part of style. Improving betting lines, pot control, bet sizing are the skill aspect and overall improving these lowers variance.

calling for correct pot odds/implied odds goes both ways and is a bit of a wash when thinking of alternatives of folding/raising.

Note most of these points are assuming deep stack play which I would think would be the default format when discussing great play in general. I'm sure an online player could provide a better answer for 100bb poker than I could.

As far as pre-flop play goes, it's not that big a factor in deep stack play other than bet sizing will have some impact on later betsizing which is important for variance.

On the whole I'd say that great skill for deep stack play will reduce variance within a wider variance framework that is more determined by style however reducing variance should never be the intended goal other than for pot control.

Calling or so big money in marginal situations will increase the fluctuations more or less, depending on how big ones edge usually is when the big money goes in. Other aggressive play produce all kinds of things.

Small ball keeps the pots smaller and becomes optional in situations where the profit difference between the options is small. Small ball doesnt include calling where raising or so isnt profitable, like call is often profitable but a raise isnt.

I don't think anyone answered the question. The answer is "yes". If by "good" you mean the ability to read your opponents. To see this, imagine that you know your opponent's hole cards. If you do, your win rate will be higher and your variance lower than any champion player who doesn't.

If I understand your question correctly player A plays $100 rebuys and always has one rebuy and 1 addon so it costs $300. Player B plays $300 shootout and has exactly the same roi as player A.

If I assume player B has the same finish distribution as A and hence the same roi with the same payout structure then the variance will be exactly the same.
I think even if A sometimes plays with 2 or 4, 5 or 6 re-buys but overall has the same average BI ie, $300 and the same roi the variance will be the same.

The small changes for reduced rake in re-buys will have a small effect but with the same roi (B's finish dist would be slightly higher to overcome the extra rake), it will still be about the same.
I would think variance would be very very slightly lower for the re-buy player due to paying a reduced rake and hence having a slightly lower finish distribution (hence v. slightly lower variance) to keep in line with the same roi.

Variance of roi in tournaments is dominated by the payout structure and the finish distribution you have and the cost of the tournie.

Assuming this answers the question "Can You Be Good Enough To Greatly Reduce Variance?"
I am not totally convinced of this if this player always attempted to win the max possible and not adjust to avoid high variance spots, ie, taking every +ev spot.

Variance would be 0.0 if all losses and also 0.0 if you could win every single hand always for the same amounts. Somewhere in between these extremes there will be a maximum variance. Can this knowing player get so far past the maximum variance point they get back to a greatly reduced variance compared to a moderate winner?

It would likely depend on the type of game such as HU or full ring etc.

Although this player 'knows' the opponent's card they can't win all the hands, would still get rivered on some big pots. A good few preflop folds, lots of smaller losing pots and a plenty big winners but this still will have a reasonable sized variance to it along with the fantastic win rate. It maybe would be quite high as the pot sizes are going to fluctuate quite a lot and with a high frequency.

A great problem to have - "A 500bb/100 w-r but just look at this variance!"

you for sure can, however it will generally make your game more passive and tighter. Phil hellmuth would be the best answer due to his tournament results. However on the other end, phil Ivey who is ultra aggressive has 10 bracelets aswell..
top tournament players these days play a very aggressive game and tend to greater swings...
its just like deciding...do you want to min cash every tournament? or do you want to take those marginal spots and go for the win?

to be the most profitable, you'd take the highest variance route because you will take every .00001+bb EV spot even if it meant committing 5BI's of your standard stakes...so it really comes down to whats more important to you. Variance is player dependent and can be controlled to a decent extent..
However to answer your question, being incredibly great would lead you to not attempt to reduce variance, instead take the optimal play so the idea to put them together is contradictory.

Although not necessarily true, I believe its generally true that better players play a significantly higher variance form of poker. Building pots against bad players and exerting pressure on them costs variance. But if you don't know what you're doing and don't have the skills necessary to recognize when a 3bet with some garbage suited hand is marginal instead of plain awful, you're just throwing money away. So players generally play it safe to begin with and then start adding more marginal plays (usually bluffs) to their arsenal as they improve.

Actually I did an action analysis of variance within the framework of style and tried to determine how variance would fluctuate when both right/wrong, bluffs, thin value bets on closing action and within the context of applying "good practice" of pot control, balance on earlier streets.

post #36

variance caused by skill, while ignoring variance created by style would both increase/decrease depending on how it was applied but it looked like a net lower variance when comparing them quantitatively.

That sounds almost like it could be considered a Fundamental Theorem of Poker or something...

I’ve been having a debate over in the MSMTT section of this site that involves, to some extent, this question. Since many of you seem to have a solid handle on this whole variance/EV question, I’m hoping you can shed some light.

My knowledge of statistics derives from a single class I took some 30 odd years ago (!) in college, so forgive me if my questions seem basic.
My understanding of EV is that, given a large enough sample size, our actual realized EV should approach theoritical EV. It seems logical that variance can greatly affect the outcome of any single trial, so we need a large number of trials to overcome these effects. Getting our chips in when we are favored with +EV situations is what constitutes skill over the long run in poker, if I'm not mistaken.

I may have mistakenly believed that a large enough number of trials was necessary to realize EV.

I was told that, when considering a +EV play in poker, that one trial is the same as 500 or 5000 or 500,000. That a +EV play is +EV regardless of how many trials we expect to run. Is this accurate? To use an extreme example, if we had a 50.1 edge vs 49.9, wouldn't any single trial have a wider scale of variation than if we were 90-10 favorites? And given that fact (at least I think it’s a fact, please correct me if I’m wrong), then if we KNOW we are not going to have enough trials to reach a statistically significant sample size, then a thin edge +EV play that realizes it’s potential over a large enough sample is no longer +EV in a single trial.

Help please, I am definitely a lost puppy here. I always believed that sample size was paramount when considering statistical outcomes, and the concept of a one trial +EV play doesn't seem intuitively correct to me.

I seriously doubt it based on all of the HEM/PT database stats I've seen. The discussion about always folding (zero variance) or clairvoyance (presumably lower but still > 0 variance) are so far out of the scope of reasonable play that I don't think they are very meaningful.

Edit: Someone could look at the UB account statistics in PT to see what clairvoyant variance may look like.

I think that some of your confusion may lie in the fact that when we make our decisions at the table it's made for optimizing our profits or minimizing our losses if necessary for every decision. We can make calculations based on our best guess of all of the opponents hands find an average point of Expected Value to gauge what the best decision is. So while we are playing based on this single number, it will never be the result for a single hand wich is either we will win/lose/chop the amount in the pot but if we imagine playing the same hand the same way many times, it will eventually average out to be very close to that point.

However while we can calculate our EV to a single point, we need to be prepared for the fact that we could get unlucky over the short term due to the variance involved with all possible outcomes.

So during the hand we play to optimize which involves precision in our average EV of all possible outcomes. Off the table we prepare for the worst which involves the variance of all possible outcomes that will probably only be realized after many thousands of hands to any degree of certainty.

For online games, it's such a numbers game with an enormous player base that you will be limited with your control of variance and mostly should only be concerned with EV and let variance be handled only through BR management.

If you play more live, especially at non destination casinos where you can better learn the player habits and tendencies of the regulars, you can control variance to a much larger degree. Playing to reduce variance can also be a larger concern live for the following reasons.

For live play, good players can expect much higher win rates than on-line but must make huge sacrifices on volume so it's less reasonable to simply play more hands.

To optimize you play with smaller margins for your bankroll to get the most from the 20-30 hands/hour and larger win-rates can help with that with a much higher winning/losing session ratio, but it's also reasonable to sometimes sacrifice some small amount of EV to reduce your variance to make up for the lack of numbers that will often never see convergence of variance.

What evidence do you have that you can "control variance to a much larger degree" in live play other than just asserting it?

Playstyle difference, Live tells

Playstyle difference in that you can afford to play a much safer (tighter) game at the casino since players are much looser