500th post thesis - the cost of marginal preflop decisions

[rzk]

introduction.

i've been thinking about preflop recently and more specifically how much i may be losing due to making the wrong decisions in marginal spots. so in my 500th post i will calculate how much a smart player with 250000 hands in his database (for a given type of game conditions such as stakes, rake, softness of the game), who bases his preflop decisions on his own previous results, is losing on average due to making the wrong decisions because of an insufficient sample size. even if you don't follow the math, the conclusion is worth your attention.

before i start, a couple of points:

- you may have a different number of hands in your database. if, in fact, you have a 1000000 hands instead of 250000, you will need to divide my final estimate for the loss by sqrt(1000000/250000)=sqrt(4)=2

- you may think the calculation doesn't even apply to you because you don't base your preflop decisions on your own database but instead follow stox's or somebody else's recommendations. (in fact, most people probably belong to this category). however, stox himself bases his recommendation on his own data, so the calculation certainly applies to him and, by extention, to you.



step 1. single marginal spot

i will now start the calculation with a specific example. suppose you've played 250000 hands and all this time you've been opening KJo utg as a default. suppose you look at your data and see that KJo has been break-even. that means you actually don't know if it's profitable to play it. how much does this lack of knowledge cost you?

1/6 of the time you are utg. KJo is 0.9% of all your starting hands, so you are faced with this decision 0.9/100*1/6=3/2000 of the time. the standard deviation for a hand that sees the flop is about 3bb/hand, so the actual winrate with KJo is anywhere between -3bb/hand*sqrt(250000*3/2000) and +3bb/hand*sqrt(250000*3/2000) with 68% confidence. calculating the square root, we have:

winrate is between -0.11bb/hand and 0.11bb/hand (68% confidence)
winrate is between -0.22bb/hand and 0.22bb/hand (95% confidence)

from your data, there's a 50% chance that KJo is unprofitable, so with a 50% probability you will be making a mistake if you continue to play this hand. how much will this mistake cost you? roughly speaking, anywhere from 0 to 0.22bb/hand, so roughly 0.11bb/hand. since there's a 50% chance you are making this mistake, the lack of precise knowledge about KJo is costing you about 0.05bb/hand on average.

how much does this affect your overall winrate? since you end up in this situation 3/2000 of the time, if you play 100 hands you will face it 100*3/2000=0.15 times on average so you'll lose 0.05bb/hand*0.15hands=0.0075bb.

so your winrate suffers by 0.0075bb/100 from your lack of knowledge/data.

this looks like great news, since the effect on your winrate is essentially negligible. but it's not that simple - the problem is that there are lots of similar marginal decisions and your mistakes in them add up.



step 2. all marginal spots.

let's now estimate how much you are losing on _all_ of your marginal preflop decisions.

at a 6max table there are 6 positions you can be in and for each position there are several marginal unsuited hands: typically Axo, Kxo, Qxo, Jxo (could be more or less depending on the position). so you have roughly 24 marginal spots with unsuited hands and your total average loss on them is:

24*0.0075bb/100 = 0.18bb/100

similarly, there are marginal suited hands and marginal pocket pairs. a suited hand is dealt 3 times less often than an unsuited one. if you go through the same calculation for, say, K9s, instead of KJo you'll see that your average loss will be only sqrt(3) times smaller than for KJo (and not 3 times smaller as one could expect). so your loss for a marginal suited hand is:

0.0075bb/100 *1/sqrt(3) = 0.0043bb/100

and similarly, for a marginal pocket pair:

0.0075bb/100 *1/sqrt(2) = 0.0053bb/100

so, assuming there are 30 marginal spots (corresponding to 6 positions and Axs,Kxs,Qxs,Jxs,Txs) for suited hands and 6 marginal spots for pocket pairs, the total loss would be:

24*0.0075+30*0.0043+6*0.0053 = 0.34bb/100

now this is a pretty sizeable chunk of change!



step 3. additional considerations.

this, btw, is a conservative estimate for a couple of reasons:

- for example, i considered a Axo hand in, say, the co as a single marginal spot. however in reality it consists of several marginal spots: what to do with no players in front, against a raise by a lagtag or lp or tag or lag, against a limp by lag or lp, agaist several limpers, etc. etc. this actually increases are estimate for the errors. roughly speaking, if a marginal spot actually consists of N distinct marginal spots, each of which happens with a probability of 1/N, the estimate for the loss should be increased by a factor of sqrt(N). on the other hand, if you, for example, knew that opening KJo utg has the same ev as opening KTo in the hj (which might not be such a bad of an assumption), you could combine these two marginal spots and treat them as one. combining N marginal spots into one decreases our estimate by sqrt(N). this, however, is dangerous, as we can never be sure which marginal spots can be successfully combined. overall, a conservative estimate for the increase in the number of marginal spots due to all of the above would be 4, so the loss should be increased by at least a factor of 2.

- i assumed that only one hand of, say, a Kxo type is marginal for a given position. however, in many cases there may be 2 or more such marginal hands. what i mean is: there are successful players who play a 26/17 style and there are similarly successful players who play a 35/25 style. they have widely different preflop standards, often spanning 2 or more pips for a given situation. for example, some players only open K9o+ otb while others additionally open with K5o,K6o,K7o, and K8o. these hands can all be called marginal. the effect of this could be as much as a factor 1.5 increase in our estimate for the loss.



conclusions.

overall these two additional considerations increase our estimate by as much as a factor of 3 compared to our previous estimate of 0.34bb/100 and we are led to an interesting conclusion:

marginal preflop spots are costing you 0.34bb/100 by a conservative estimate and could in fact cost as much as 1bb/100!

remember, this doesn't assume you made some kind of mistakes in analyzing your data - this is your average loss assuming you analyzed all the marginal spots as best you could from your data. i think it's safe to say that 99.9% of decent players have never analyzed the data in as much detail as this calculation assumes, so a typical decent player loses even more due to incorrect preflop play!

what conclusions can we make from this:

- even the simplest street - preflop - still contains lots of opportunities for significantly increasing your winrate.

- one should not be content to just base your decisions on your own, or stox's, or any other expert's data. more careful methods of analysis are needed.

[mute]

Maybe a dumb question, but the .34BB/100 assumes that you happen to be wrong about all these marginal spots, right?

Still some food for thought. I've always been an advocate for the "preflop, w/e"-attitude, when people are discussing raising A7o in HJ and so on.

Ofcourse there's some shania aspects to be considered as well, which are pretty much impossible to quantify.

[rzk]

[ QUOTE ]
Maybe a dumb question, but the .34BB/100 assumes that you happen to be wrong about all these marginal spots, right?


[/ QUOTE ]

no, it actually assumes you are wrong in 50% of these spots.

[mute]

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from your data, there's a 50% chance that KJo is unprofitable, so with a 50% probability you will be making a mistake if you continue to play this hand. how much will this mistake cost you? roughly speaking, anywhere from 0 to 0.22bb/hand, so roughly 0.11bb/hand. since there's a 50% chance you are making this mistake, the lack of precise knowledge about KJo is costing you about 0.05bb/hand on average.


[/ QUOTE ]

Doesn't this say, that there's a 50% chance, that KJo is unprofitable and if you're wrong then it costs you .055BB/100 on average.

I mean if KJo is profitable then we would make the same .055BB/100 mistake by folding, wouldn't we?

So there's 4 scenarios isn't there? Fold correctly/incorrectly and raise correctly/incorrectly.

I might be missing your point entirely.

[rzk]

no, i'm saying if you are making this mistake then it costs .11bb/hand, so given that you are making a mistake with a 50% probability (such as playing KJo when in fact it should be folded or vice versa), it costs 50%*.11bb/hand = 0.055bb/hand.

i downgraded it a little to .05bb/hand because of the shape of the distribution, but that's a minor point.

[Oink]

[ QUOTE ]
i downgraded it a little to .05bb/hand because of the shape of the distribution, but that's a minor point.

[/ QUOTE ]

I was gonna comment on this.

I am no stats wiz. But we need to know the distribution to estimate the average mistake, no?

Do we know the distribution or do we just assume its normally distributed or whatever?

If I am wrong just say I am wrong. I wont understand a technical answer anyway [img]/images/graemlins/smile.gif[/img]

BTW. The subject of your post is highly important and your conclusion should be stickied somewhere. People just play by their starting hand charts and are like: "Sweet a good player came up with this, now I dont have to worry about preflop decisions"

I'd guess that preflop is where your average player makes the most mistakes and looses (or fails to win) the most.

[rzk]

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Do we know the distribution or do we just assume its normally distributed or whatever?

[/ QUOTE ]

EDIT:
i'll revise:

our win for a given hand has a discrete distribution, but after a sufficient number of hands the difference between observed and actual winrate is distributed normally.

so the cost of the error is the average of a half of a normal distribution. in my post i guesstimated that to be 0.9sigma. but i just did a slightly better estimate and it looks like it's closer to 0.7 sigma.

[mute]

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no, i'm saying if you are making this mistake then it costs .11bb/hand,

[/ QUOTE ]

And when we're not making a mistake, then we will win .11BB/hand, yes?

[rzk]

[ QUOTE ]
[ QUOTE ]
no, i'm saying if you are making this mistake then it costs .11bb/hand,

[/ QUOTE ]

And when we're not making a mistake, then we will win .11BB/hand, yes?

[/ QUOTE ]

on average yes, that's correct (with the proviso in my previous post).

but note that if we are not making a mistake we are not winning anything relative to the perfect player.

[mute]

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but note that if we are not making a mistake we are not winning anything relative to the perfect player.

[/ QUOTE ]

OK, I get your point your now.