Math, EV and Middle Pair Options (long)

[fimbulwinter]

Since I've been doing so much destructive posting, i figured i might as well do some constructive posting and put this out here for neon who requested it and any others who may want to see it.

I will go over the EV analysis, as i see it (hearty discussion always encouraged) of making preflop raises with hands which do not generally hold significant preflop pot equity edges over those hands which will call them. a few examples include things like suited connectors and mid PP's. hands like AA have even more analysis necessary as the preflop EV earn is real (getting in 10% of your stack with AA against AK and AJ is fantastic) and will not be discussed (will be ignored) here.

First, let me give a more easily applicable definition of EV WRT a particular bet in poker, but specifically NL since that's what we care about. EV means what it says: it defines the expected return on a bet given all the possible outcomes and outcomes of those outcomes ad infinitum. One of the reasons a poker bot is unlikely to be made successfully for NL is the fact that estimating these almost infinitely complex odds (and we all do this every time we sit at the table) requires human intuition to take it down to a manageable level.

An example of what might make the above more tangible is the standard move with A8o UTG in a 6max game. most all of us fold here, but our hand is likely to hold a pot equity edge. we know that raising there gets us into all kinds of crap, but that would not be instantaneously apparent to a computer looking only at preflop EV.

Now let's get down and dirty. Like I said before, I will calculate the EV of two different lines with a small PP (because calculating the times it "flops big", as opposed to a SC is easier for me). I will simply start at the beginning, look at the possible outcomes, assign them values (some of which must come from intutition and some of which are simple/given), assign them probabilities of occuring (same analysis as values) and then sum them to get the total average expected return every time that bet is made.

After that I will examine the different linearities within the constructed equation and draw out the relationships inherent (see below). I will leave it up to you all out there in MH land to plug in actual expected numbers to draw your own conclusions about when/if you should be making these kinds of bets. I do this to avoid noisy side chatter on "no, that's way less likely" etc. That's not the point of this post. take the equation and apply it to your game and make your own conclusions.

Onto the math:

We won't consider folding as we all (hopefully) agree that PP's can be played profitably in a NL game under most circumstances.



Limping

EV = total value (all options)
EV = (expected earn from hand) - (cost to pay hand)

we'll examine the second part of that equation first:

Cost =
[(1BB)*(% pot unraised)] +
[(SUM (PFRn)*(% PFRn made))] +
[(1BB)*(% pot overraised)] +
[(SUM (above))*(% pot then overraised)]

the above looks complex, but it's really not. it costs us 1BB to play every time we limp in and get to look, it costs us xBB's every time we play against/call a raise where x is the average PFR (which is described by the summation above) and it costs us 1BB to limp/fold when we don't have implied odds and xBB to call/fold to 3bet when that happens.

Note that we will have to take into account the times we pay but don't get to see the flop when we deal with the expected earn. We'll do this by discounting overall expected earn (don't you hate folding to a 20BB RR and seeing your set flop?) by the % time (when i say percent, i mean probability, so all the %'s sum to 1, not 100) we dont get to see a flop. this makes sense as our costs don't change the more we call then fold but our earn does.

now onto the earn portion:

Earn =
[(% time raised flop seen)*(average earn in raised pot)] +
[(% time unraised flop seen)*(average earn i unraised pot)] +
[(% time noflop seen)*(0 <- (earn noflop seen))] +
[metagame considerations]

I won't discuss metagame considerations here, but suffice to say that if raising all manner of hands to get invited back to a super soft, super big game is what you have to do, then do it. there are far too many other things to discuss under this heading, so I wont, i'll simply say that it can (in most cases) be something completely overborne by the other considerations or be the main motivation behind the move.

Both the average earn's above (unraised and raised) can be broken down further into (where xpot is unraised or raised

average earn =
(average earn xpot set) + (average earn xpot noset)

and into infinitely more considerations like

average earn =
(average earn xpot set vs nothing) +
(average earn xpot set vs 1 pair) +
...

This is what i referenced above. a little human intuition, and some PT analysis can give you how much you're expecting to make every time you make your set and every time you don't. when the math comes to "calculate the expectation on these million events and estimate their probabilities of happening" you have to put on your cowboy hat and resort to wisdom, rather than numbers.

Raising

Now onto the analysis of raising. you'll notice that in some areas it's much more interesting (earn) and some less (cost) but the math is the same.

EV = total value (all options)
EV = (expected earn from hand) - (cost to pay hand)

we'll examine the second part of that equation first:

Cost =
[(SUM (PFRn)*(% PFRn made))] +
[(SUM (PFRRn)*(% PFRRn made))] +
[(SUM (both of above))*(% pot then overraised)]

Notice cost simplifies significantly as we're no longer dealing with limps. we either see a flop raised or we dont.

Let me throw out one thing that may be confusing: we are analyzing a single bet here. the real answer, which the math gives, is obviously not to play all PP's for set value alone and not to raise all PP's. this analysis is for one particular action with one particular hand at one particular table at one particular time.

onto the more interesting earn portion:



Earn =
[(% time raised flop seen)*(average earn in raised pot)] +
[(% time noflop seen due to folds)*(average limps/blinds)] +
[(% time noflop seen)*(0 <- (earn noflop seen))]

the fun part here is breaking down "earn in raised pot"

earn in raised pot = (EV times original raiser) + (EV times called raise) etc.

again the EV calculation for EV of, say a single, isolated continuation bet is another one of these inifinte rabbitholes of something like:

EV =
(earn noset + 1 calls) +
(earn set + 1 calls) +
(earn he folds)

earn he folds can be looked at as a pure bluff on the river, namely if i bet the pot, he must fold >50% of the time for that facet of the EV equation to be +EV. pragmatics of this are discussed below.

Obviously this continues to branch like the roots of a tree into trillions of different possibilities if we keep looking at all the bets and probabilities until the river hits. we can't do this. computer's cant. but what we have now is adequate to take a look back.

Conclusion:

Well, what can we draw out of all of this mess? Let's look at different game factors and see how they affect the earns:


Stack depth:

This one is really interesting in that it is a kind of parabola where really short games, like say if you were playing 25BB poker, dictate that limping 22 UTG is bad as you will likely face a raise and must fold, making the cost side of the equation too large to conpensate for the pots you win. This is seen in tournament style play where implied odds are greatly diminished.

Then med-short games where one does not worry about folding TPGK (i dunno where this is, maybe 40-60bb's?) make limping OK but rule out raising in some cases, especially if opponents won't lay down 2nd pair etc.

Finally deeper games have a much bigger return on the earn from raised pots portion (in BB's) and so the additional cost of raising mid PP's can be accounted for in their increased earn, both in flopping sets and allowing for a continuation bet against which most opponents will play very tightly.


Position:

For reasons discussed above, as well as increased chance of continuation bet, increased earns across the board, decreased chance of overraise/3bet, increased earn from preflop takedowns, etc. makes this raise much more attractive on the button than UTG.

Here's something i really like: say you always raise 4bb+1/limp and you're afraid of raising your 77 on the button because you'll have to raise 11BB which doesn't gel with your 5/10 ciaffone rule or your 3/8 FSU/ML4L (others too, sorry if i forgot) rule. this is still an ok option as the marginal loss of EV from "overpaying" for your set draw is by far overcompensated by the times your bet takes it down preflop and the EV of your continuation bet. you've probably all seen players, myself included, "steal" the limps preflop. if there are 7.5BB's in the pot (6 limps plus blinds) and you raise to 10, winning 50% of the time, your actual "cost" of your set draw is (10-7.5)/2 = -1.25 BB plus the value of flopping a set headup in a heavily raised potand the value of continuation betting (obviously removing the times you get LRR'd out). Obviously this is a nicely profitable play with the right image.


Table Tightness and Skill:

Not that the two above go hand in hand, but they can kind of be discussed together.

At a table where there is little PFR'ing, lots of limping and tight play against raises, then raising is clearly best as you will not get paid when you do flop a set, and when you are getting called in an unraised pot they are likely drawing to hands that beat yours. This is not what you want. When they're playing tightly the value of the continuation bet especially goes up, so raising pairs, especially with position, seems like the way to go in weakloose and weaktight games.

At a loosepassive table say of new players, limping is probably best as continuation bets get called correctly and players will pay off, so a raise gains really no value in deception.

At a very ramped up table where raises and reraises are the norm, then implied odds go way down and a hand like AJo is better than 88, but both should be played as the best hand if they are to be played as there are no implied odds to justify playing them for set value alone. in some cases the small pairs don't justify playing (I recently played at LC where $100 PFR's with KT were the norm over a 2/3/5 blind structure, no way limping 33 UTG makes a profit) at all


you can use the equations above either literally and plug in some numbers you make up to see which is best, or more importantly just take the above trends into consideration when deciding what to do the next time you're dealt 55.

In reality doing this kind of in depth rebuild is totally unnecessary. a simple bit of mental math using guesstimates is sufficient in all but the most complex of cases. I just got into it deep here because neon asked for it nicely. hope this as what you wanted.



gl and thanks for reading.


fim

[AZK]

Nice post.

[muckmaster]

Here's my take on you fim...your some college professor who teaches at some so-so school (most likely a state school) and has a chip on your shoulder because you didn't have the intellectual acumen to make it to the big time. And your right ..most of your posts are destructive. Don't worry though I'll probably end up the same way...I pick my dissertation committee in the fall...

[Go_Blue88]

hahaha you created an account just to say that?? wow...be a lil more pathetic plz...

[TheWorstPlayer]

Here's my take on him: he's in med school.

[muckmaster]

what difference does it make to you what my first post is? I noticed you didn't comment on his post...could understand it?

[Go_Blue88]

[ QUOTE ]
what difference does it make to you what my first post is? I noticed you didn't comment on his post...could understand it?

[/ QUOTE ]

yeeep that must be it. actually i'm going to read it when i get home from work, and it makes a difference because you created a name just to insult someone. it's pathetic, and you are pathetic. anyways you can respond to this and get the last word in, have fun.

[neon]

Wow.

Thanks, fim. Awesome post.

I just woke up from a nap, and so my brain isn't functioning well enough to absorb all of this right now, but I'm going to give this post a good pore over (or four) later today, and I'll post any questions/comments that may come to mind.

Thanks again.

-neon.

[Rotating Rabbit]

Jeeeesus thats what I call a post and a half !

Everyone would do well to read the conclusion several times, some gold there.

I have a thought or two to make on this discussion:

1) --- With 100bb stacks, it seems to me that the critical preflop pot size is 10bb. That is to say, if there is 10bb in the pot just as the flop comes down, there is a material chance of stacking your opponent if you hit a set. Three pot sized bets postflop is a good result. So if the pot is 10bb on the flop, its not likely to be more than 30bb on the turn, and on a good day 90bb before river bets, leaving both players with 60bb, and a good chance of a stacking.

Less than this in the the pot before the flop, and it will be difficult to get your opponent committed enough the call on the all important final bet, the one thats accounts for the majority of the profit.

I put this really as a supplement to the traditional 1/10th esque rules regarding calling for set value. If you are the raiser with a medium pp you WANT the flop to be 10%+ of relevant stacks in the pot when the flop hits.


2)--- The other point, perhaps obvious but nobody's mentioned it, is that 77-type hands are relatively VERY EASY to play. In nearly all cases with a set there's little discussion, and when we dont hit a set its pretty easy too, shutdown or maybe a continuation bet if we raised preflop. It doesnt require any finesse. And thats great because it means we wont make mistakes. TPTK or overpairs, as we all know, can be difficult to play, and easy to get money in as a big dog. In my opinion this pushes the value of medium pocket pairs up considerably (the same argument, to a lesser extent to connectors).

[AZK]

[ QUOTE ]

Here's my take on you fim...your some college professor who teaches at some so-so school (most likely a state school) and has a chip on your shoulder because you didn't have the intellectual acumen to make it to the big time. And your right ..most of your posts are destructive. Don't worry though I'll probably end up the same way...I pick my dissertation committee in the fall...

[/ QUOTE ]

A lot of you need to chill out. He acknowledged the fact that he was a dick in another post. It seems like he is trying to move on and be more constructive now. He writes a very informative, educational post and you guys still give him [censored]. Give it up already. What more do you want from the guy? If you have beef with a specific post or disagree with a certain poster, just ignore them, either by not reading their stuff or by clicking the ignore button. Don't flame them in the forum (this isn't OOT), it's a waste of everyones time and contributes nothing.

/rant over