December 02, 2010

Embracing the Variance - Part 1

Over the next few days I will show a series of posts that deal with variance, mostly in LHE tournaments, especially with a very fast increasing blind structure. The inspiration for these posts was a discussion I had at poker lunch with some LHE dudes about 2 weeks ago (further updates via Jesse: here, here and here). The main idea of the discussion is the willingness to play with a positive expectation, while constantly overbetting your bankroll - embracing the variance, so to say.

I will show via simulations, how in essence you should take the more +EV spot as possible ignoring the variance, save for some extreme limiting cases, like the bubble. In some extreme cases, like with very low winrates or with super fast structures, conservative (low variance/slightly winning) play might limp a few more people into the money than aggressive (high variance/higher winrate) play, but overall the latter will win more tourneys, more money, everything!

Conservative play is an old motto of especially live tourney pros. If I had a nickle I've ever heard some "live pro" say on some podcast or some TV table: "I'm gonna fold in this slightly positive EV situation and wait for a better spot"... In fact, when just returning from the aforementioned poker lunch, I was listening to the "Deuce Plays" podcast with David Baker and he was chastising somebody for calling with AJ against 33 (blinds and antes made it a +EV play). He said, he wouldn't even call with AJ if the guy showed up with KQ (3:2 advantage) in this situation. "better spots, better spots... you're out of the tourney... bla bla bla..." So I'll just say it: If you are properly bankrolled for tourneys, that is, you're not taking a once-in-a-lifetime shot at the Main Event, you should take the +EV spot and don't worry about being out of the tourney. And LOL at David Baker not taking the 3:2 spot! Full disclosure: "I do understand ICM and there are certain points where you should fold, this situation was not even close to on of those inflection points".

As I'm producing the graphs for this blog post I'm listening to a podcast RIGHT now, Daniel Negreanu on Grilled Fish saying "Just because something is +EV doesn't mean it's right!", yet he gives no reasoning for it and has all these caveats why he's not shoving in some spots yet shoving in others. The poor hosts try to ask him two or three times and even say "If something is +EV it shouldn't matter how much time there is left in the level, right?" yet he doesn't bite and you are left with wrong reasoning and an angry Bella, because what he's saying is not technically wrong, but all the listeners hear is: "Oh, I should wait for better spots, I will have a greater edge later on, yadda yadda". WTF, spots don't get easier later on, in fact more sharks remain later on, why would I give myself the worst situation later, when I don't have an edge anymore? Anyway, enough about live pros.

Why do we hate variance so much? Why is variance always thought of as a negative thing? - "I like PLO, it's a really interesting game, what I hate is that the variance is so high!". WTF, all the variance does is the broadening of the distribution. For every high loss, you'll have an even higher win, it's not like suddenly, because the game is higher variance that you'll suddenly become a losing player... I think it's psychological, losing hurts more than winning, when your expectation is positive. This is exasperated by happenings around us, for example, in the stock market, periods of high volatility coincide very nicely with bear markets (for the super highest volatility periods since WWII it was the case for 19/20 (!) times). But again, this has psychological reasons, volatility didn't increase, because the Gaussian grew fatter... no, in fact, those periods of general uncertainty don't follow the normal distribution we are used to. In some sense, you could say the market is tilting :P

So if you are absolutely sure of your winrate, you are sure you are a favorite over the field (don't ask me how you know this, you just do), you shouldn't care about variance that much. Sure, you'll have long droughts, but you'll also have periods where you will run hotter than the sun. Just look at great HUHU players, the variance is through the roof, yet their graphs look so smooth, hardly any wiggles... winrate!

But bella, what about going broke? Aren't there all these statements floating around the internet that if you constantly overbet your bankroll you WILL go broke? Yes, but emphasis on the constantly. Sure, the Kelly criterion proved that if you have a 2% edge in a situation, yet bet 4% of your bankroll over and over again, you WILL go broke eventually, no matter what. But there are some cases that go off (you get lucky) and you bet a fixed amount, suddenly that percentage of your bankroll gets lower and lower and you are not overbetting your bankroll anymore. In fact, in LHE, for people with 1000BB bankrolls and healthy winrates, you are betting as little as 1/6th Kelly, which means the best possible results take 6 times as long to reach (that is understandable, since you don't wanna move up and down the whole time and checking your balance constantly) - peace of mind leads to better play which might maximize profits in its own way.

So let me now turn to the first simulation, which is actually more the case for cash games. Overbetting of bankrolls, no increasing blinds. The simulation is presented for 2 starting stacks, bankrolls if you will and 3 different speeds of play. I chose 50BB because of HULA and 20BB, because as we'll see that is sort of what we're striving for as ideal stack in a LHE tourney. Starting parameters are: 10.000 players, 2BB/100 winrates, 15BB/100 SD.

User Uploaded Image


Graph 1: Percentage of players alive versus time in a 10.000 player simulation with 50BB (triangles - dashed) and 20BB (squares - solid) starting stacks over 20 hours. The fast player has reached close to theoretical expectation (marked as black lower limit lines), while the slower one is, understandably still getting there.

Bear with me with the speeds of play stuff, it will be important later for the tourney simulations, just notice that the guy playing slower is still reaching the long run, albeit slower, ldo. By the way, these theoretical curves can be calculated analytically, no need to run simulations, those were just the first step for me to start the program and I thought it woul be nice to show results. See also, Mathematics of Poker: Chapter 22.
Risk of Ruin = exp[-2 * winrate * bankroll / SD^2 ]
Case 50BB starting stack approx. 41.1% die
Case 20BB starting stack approx. 70.1% die

Bringing only 20BB to a live game for an evening if you have a 2BB/100 (easy for live) edge seems like suicide. After all, that is only 800$ for a 20/40 game. But if you stay only for an evening, maybe 6 hours, then you will only go broke ~25% of the time (say you get ~30hands an hour, so half the time of slow case). That's pretty good for having a good time - again, not if you're doing this professionally, but great for having an enjoyable evening.

Talking about results, look at the average BB's of the remaining players. Playing fast with an edge you are really setting yourself up to win big overall. Sure, you'll go broke often, but when you don't, you win. And it's only reaching the long run faster, for the fast player at hour 20, the average bankroll has already reached Kelly level long ago...

User Uploaded Image


Graph 2: Average big bets for remaining players over time. Same parameters as above.

But alas, we're not in cash game regime, we can't play at the same blind level forever in a tourney, it would take too damn long. Next time... first tourney simulation - Rush LHE tourneys, big winrate, fixed skill, fast structures!

Posted By bellatrix at 01:07 AM


Tags: theory tourneys


Schweig posted on December 02, 2010 at 02:56 AM


I disagree with what you're saying about taking every possible +cEV spot in a tournament.

No one has really come close to providing a correct mathematical model for the utility of chips in a tournament with varying skill levels. The closest people use is ICM, which tells us that cEV = $EV far from the money, but it uses the assumption that everyone is of equal skill which distorts the results heavily.

I'm pretty sure that there are diminishing returns on utility in terms of increasing your chip-stack if you have an edge on the field. Sure, there is a point where your edge is nullified because you are too short, but I still believe it's going to be in our best interest not to go bust when we have over 30bbs+ or something, and that can mean passing up some thin edges.

bellatrix posted on December 02, 2010 at 03:27 AM


"I still believe it's going to be in our best interest not to go bust when we have over 30bbs+ or something"

There's a perfect strategy for that - fold until you have less than 30bbs, that way you can NEVER go bust when you have over 30bbs+. Yes, I'm being facetious :)

And yes, there are some very thin EV situations and certain inflection points, ESPECIALLY the bubble, but also others, where you don't take the gamble. That is what I would have loved Daniel Negreanu to explain better, instead of saying "better spots". However, with a 3:2 edge (the AJ vs KQ hand), it's not really close and folding the AJ is a big mistake!

Anyway, I haven't really posted my reasoning and simulations yet. All we have is the cash game situation, just bear with me over the next few days, maybe we'll get to more agreement, for example, if your edge is minimal in the tourney (say less than 2BB/100), you might be better off folding, so yeah, 2% edges, perhaps, 20% edges (3:2) NO! :)

copoka posted on December 16, 2010 at 05:23 AM


"So if you are absolutely sure of your winrate, you are sure you are a favorite over the field (don't ask me how you know this, you just do), you shouldn't care about variance that much."

Bella, don't take it the wrong way, but "absolutely sure of your winrate" is border line silly.
The only way to use "absolutely sure" and "winrate" in one sentence would be "one can be absolutely sure that he can never be absolutely sure about his winrate"

WR is nothing more than suggestive and convenient number and should not be used for any kind of "predictions" or forward looking analyses.
Any simulation made based on assumed WR is basically useless since the assumption made is HUGE and never accurate.

"Sure, you'll have long droughts, but you'll also have periods where you will run hotter than the sun."

That would only be true when you talk about variance produced by the deck of cards.

Yes, there will be periods when you just will not be able to make a hand to save your life and those periods will be followed by periods when you'll flop sets and make flushes at scary rate.
We are talking simple numbers here, and those converge rather fast. Nobody can be lucky or unlucky with simply deck of cards for a long time.

So, although number of AA dealt, sets flopped or straights we make on the river can not effect difference in skill for a long time, there is another thing that can, is and will.

There are monies involved.
Difference in action we give in unlucky situations and action we get when we run good is when luck in poker plays its major role.
Now we are getting into the dark territory of unbalanced coolers, as I call it, and one can be stuck there for life.

First, those things are fairly rare comparing to the number of hands dealt so they converge much much slower in "calendar terms" so to speak. Nasty thing is that although they are rare, the effect on your earn is huge.

Second, even if we are lucky enough to see them balancing out, deck is still stacked against us due to measures we have to take to survive in this game.

Lets say we are raped by sick coolers at 10/20.
Our edge, albeit a decent one, over competition is simply not enough to overcome a huge damage made by the deck and we are losing ungodly number of big bets.

To protect our BR we must go down to 5/10 and then we finally start running good and at some point "good" coolers at 5/10 balance out with "bad" coolers at 10/20.
We make all those BB back and then some, off course, thanks to our superior skills.

The problem is, all that "and then some" is not enough to get us back in dollar terms. We were lucky to see coolers converged, but we still down money, and the more ugly cycles like that we have the more money we lose. It will work the same way when moving up the stakes.

Third. There is so much human and/or social factors involved in giving and getting action in poker, that the argument can be made that bell curve simply cant govern this process for the same reasons it is useless in any other field where human nature runs wild, like stock market, for example.
So what convergence are we talking about in a first place? Where would that come from?

"Just look at great HUHU players, the variance is through the roof, yet their graphs look so smooth, hardly any wiggles... winrate!"

I'd say you are looking at graphs of skilled poker players who where lucky enough positive cooler balance for a long time.
The proportion of skills vs luck is unknown.
Prove me wrong.

Oh, btw, there is a pretty good HU poker player named Matt Hawrilenko. Have you seen his graph lately?

There is another one. Very talented and very naive.
Talented enough to break the game of HULNE down to the science.
Naive enough to declare in one of his blogs that "making 300k playing poker is a child's play"
His name is is Bryce Paradis.
Take a look how much he made over past 12 month.

Schweig posted on May 02, 2011 at 07:28 AM


I never realised I got a response to my post here until now (due the fact that you don't get e-mails for blog responses).

Anyway, I read your follow up blog posts which were interesting and I'll respond directly to them on the actual posts, but I'll first reply to what you said here and expand on what I was saying. (Mostly ignoring the other guy who responded, he seems to be talking crap and I couldn't be bothered to read it.)

But yeah, I agree that passing up AJ when you know your opponent has KQ is awful in almost any situation.

And yeah, my comment about best interest to not go bust of course doesn't mean that it overrides our interest to actually win pots, so we are not passing up decent edges. I still stand it's not a case of +cEV is +EV even far off from the money, and that the inflection points for what is considered a thin edge can vary dramatically depending on a lot of factors mostly to do with what we think the utility of our chips or 'tournament life' actually is.

For example, imagine a situation where we have 3 players on our left who we have just covered and defend very rarely in blinds to 2-2.5x opens, and only 3-bet shoving AJ+, KQs, ATs, 88+, have the fold box ticked otherwise, trying to move up the pay ladder, and close to never make moves. This may seem extreme but you used to be able to find this situation or similar in most large field, $20-50 tourneys on FTP with <100 players left, and it probably exists sometimes in live tourneys when you have a weak table.

Now given you have something like 30bb and they have something like 20bb deep, it's very profitable for you to open a ton in the CO and BTN vs these players, but you should raise/fold even strong hands like AQ which will have the correct odds to call a shove, just because if you lose the all-in your stack loses a ton of its utility because you lose the ability to steal that easily when you no longer have them covered (also, although doubling gives you slight extra leverage it probably improves your winrate in this situation very slightly.)

Meanwhile, if you maintain this situation you can actually chip up without ever risking your utility by just raise/folding a bunch until someone adjusts (and they usually didn't.)

Now consider a very different situation where a very troublesome, somewhat loose, aggressive guy moves into your direct left with 40bb. Now your situation doesn't look so great, your utility is much less than it was before and although you aren't necessarily looking for situations to double up (for leverage and the extra chips are worth more because they are in play vs this guy as its worth more utility here), you are no longer shying away from them when they are +cEV. Now you're def not raise/folding AQ and not passing up spots where you have 45% with dead money.

Anyway, I hope I clarified my position better and I'm going to make some comments on those graphs you posted over at Part 2.

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komal posted on June 16, 2018 at 06:49 AM


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