Equity from playing a small pair against a big pair (very long post)

[Adrian20XX]

I've been thinking a lot about this, it's pretty obvious that a huge number of the situations in which one player gets stacked come from the typical set against huge overpair, so I'd like to express my ideas here and most importantly would like to know what other people think about this.

So, I'll analyze the case in which the hero has a small pocket pair, and the villan has a huge pocket pair. The normal plan is for the hero to flop a set and stack the villan, if the flop does not bring a set, hero is out of the hand.

So I'll analyze all the possible flops for no sets, sets, and quads, but not for straight draws or flush draws.

We'll call C(n, m) to the possible combinations of m elements taking from n (in Excell this is the funcion COMBINAT).

So, here are all the odds for the different cases:<ul type="square">[*]Villan flops nothing, Hero flops nothing: C(44;3)/C(48;3) = 76,57%[*]Villan flops Quads, Hero flops nothing: C(2;2)*C(44;1)/C(48;3) = 0,25%[*]Villan flops Quads, Hero flops full house: C(2;2)*C(2;1)/C(48;3) = 0,01%[*]Villan flops Set, Hero flops nothing: C(2;1)*C(44;2)/C(48;3) = 10,94%[*]Villan flops Set, Hero flops Set: C(2;1)*C(2;1)*C(44;1)/C(48;3) = 1,02%[*]Villan flops Set, Hero flops Quads: C(2;2)*C(2;1)/C(48;3) = 0,01%[*]Villan flops Nothing, Hero flops Quads: C(2;2)*C(44;1)/C(48;3) = 0,25%[*]Villan flops Nothing, Hero flops Set: =C(2;1)*C(44;2)/C(48;3) = 10,94%[/list]These are all the possibilities. Now the Hero is facing a bet pre-flop and has to decide if it is profitable to call and later get all the stack if he flops a set (no matter what else comes on the flop), or if it is not profitable and he has to fold.

In all the cases that hero does not flop at least a set, as I said he'll be out of the hand, so the equity for all this cases is the value of the call pre-flop (negative, of course).<ul type="square">[*]Villan flops nothing, Hero flops nothing: C(44;3)/C(48;3) = -76,57%[*]Villan flops Quads, Hero flops nothing: C(2;2)*C(44;1)/C(48;3) = -0,25%[*]Villan flops Set, Hero flops nothing: C(2;1)*C(44;2)/C(48;3) = -10,94%[/list]These cases total 87,77%.

Now we do have to consider all the other cases, the ones in which the Hero is ahead an has a slim chance of being outdraw, and the ones in which hero is behind with his set, and he either is drawing very thing to one out or he is drawing dead (we do not count here the BET that hero has already invested). <ul type="square">[*]Villan flops Quads, Hero flops full house: Hero is drawing dead. The probabity for this case is 0,01%, and he’s equity in this case is 0%, but he still is willing to put all his stack in, so he also loses -STACK, and hence the equity is -0,01% * STACK.[*]Villan flops Set, Hero flops Set: Hero is drawing to one card, and begging for the other Quads not to fall. The probability of this case was 1,02%, and the probability to outdraw is C (1;1)*C (43;1)/C (45;2) = 4,34%, and as he is also is willing to put his stack here his equity is 1,02% * (4,34%*(STACK+2*BET+INITIAL POT) - 95,66%*STACK)[*]Villan flops Set, Hero flops Quads: Villan is drawing to one out. The probability of this case was 0,01%, and the probability that Hero is still ahead after the turn and the river is =C(44;2)/C (45;2) = 95,56%. So the equity in this case 0,01% * (95,56%*(STACK+2*BET+INITIAL POT) - 4,44%*STACK)[*]Villan flops Nothing, Hero flops Quads: Life is sweet Villan is drawing to runner runner quads Draw boy, draw !!!. The probability of this case was 0,25%, and the probability that hero is still ahead after turn and river is =1-C (2;2)/C (45;2) = 99,90%. So the equity for this case is 0,25% * (99,90%*(STACK+2*BET+INITIAL POT) - 0,10%*STACK)[*]Villan flops Nothing, Hero flops Set: The probability of this case was 10,94%. Hero here has 3 different ways of wining: either with two blank cards, with one card for his Quads and one blank, and with one card for Villan’s Set but also one card for Hero’s Quads. So the probability of this is =(C (42;2)+C (1;1)*C (42;1)+C (1;1)*C (1;1))/C (45;2) = 91,31% and hence his equity here is 10,94% * (91,31%*(STACK+2*BET+INITIAL POT)-8,69%*STACK)[/list]So if there is an initial pot that we will call INITIAL POT (composed by blinds and limpers), the total equity for Hero to play the hand with our plan is:
EV =
-BET- 0,01% * STACK
+ 1,02% * (4,34%*(STACK+2*BET+INITIAL POT) - 95,66%*STACK)
+ 0,01% * (95,56%*(STACK+2*BET+INITIAL POT) - 4,44%*STACK)
+ 0,25% * (99,90%*(STACK+2*BET+INITIAL POT) - 0,10%*STACK)
+ 10,94% * (91,31%*(STACK+2*BET+INITIAL POT)-8,69%*STACK)
=
10,30% * INITIAL POT - 79,40% * BET + 8,36% * STACK

This gives us a formula for the EV of the small pair:
EV = 10,30% * INITIAL POT - 79,40% * BET + 8,36% * STACK

If we want the hand to be EV+, we need that
10,30% * INITIAL POT – 79,40% * BET + 8,36% * STACK &gt;= 0
79,40% * BET &lt;= 10,30% * INITIAL POT + 8,36% * STACK
BET &lt;= (10,30% * INITIAL POT + 8,36% * STACK) / 79,40%

And the final formula for EV+ we get is:
BET &lt;= 12,97% * INITIAL POT + 10,53% * STACK

Note that the initial pot has a bigger coeficient than the stack. This is reasonable, because when the small pocket pair wins the initial pot adds to the equity because it’s dead money, but when the small pocket pair loses it does not affect the equity. The stack has the same positive impact when it wins, but the negative impact of when it loses results in the lower coeficient for the stack.

If we discard the extra equity from the initial INITITAL POT and we take for it only 10,53%, we get to:
STACK + INITIAL POT &gt;= 9,50 * BET

If Villan is not willing to get stacked (let’s say players are playing deep), then STACK becomes the money that Hero can get Villan to put after the flop.

Three things are not considered in these analysis:[list][*]The possibility of flushes[*]The possibility of straights[*]The possibility that the small pair flops an underfull house, and the big pair outdraws when a third card converts the underpair to a bigger full house.[list]

All these things can not be determined without knowing the exact cards that both opponent have. There are weird cases in which the small pair does not flop a set but is ahead after the flop, for example if the small pair is TdTh and the big pair is AcAs, on a flop of 9h8h7h the small pair is an exact 60% favorite. There are also weird cases in which the small pair flops a set and has only a coin flip, like when the small pair is TdTh, the big pair is JcJs and the flop is Tc9c8c where the set has only a 50,9% equity.

Well, this is the start of the analysis, but I think it gives us a very clear result. If I get more possitive feedback than flames, I will post a few additional ideas about this.

[hasugopher]

This actually looks like a very good and helpful post and it's obvious it took a lot of time to write it. It's also going to take a long time to read it and understand it, but I definitely will either tonight or tomorrow. [img]/images/graemlins/smile.gif[/img]

good organization btw

[Parlay Slow]

You need what's called an "executive summary". This is a short description of what you've found for busy CEO's like myself to be able to make an informed decision on whether the rest of it is worth reading.

[SuperPokerJedi]

[ QUOTE ]

All these things can not be determined without knowing the exact cards that both opponent have. There are weird cases in which the small pair does not flop a set but is ahead after the flop, for example if the small pair is TdTh and the big pair is AcAs, on a flop of 9h8h7h the small pair is an exact 60% favorite. There are also weird cases in which the small pair flops a set and has only a coin flip, like when the small pair is TdTh, the big pair is JcJs and the flop is Tc9c8c where the set has only a 50,9% equity.


[/ QUOTE ]

Thank you for your efforts - I found the above the most intersting! Any more of this will be kindly welcome!

[img]/images/graemlins/cool.gif[/img]

[smartalecc5]

[ QUOTE ]
You need what's called an "executive summary". This is a short description of what you've found for busy CEO's like myself to be able to make an informed decision on whether the rest of it is worth reading.

[/ QUOTE ]

aka: cliffnotes

[ArturiusX]

I've always been interested in how flopping things other than sets effects your equity. Whether its intensely practical or not is another thing :P

[Enzyme]

I haven't checked the numbers, but it seems like one significant factor is missing from this analysis, which is way more important than flopping quads, etc. In case where small pair flops a set, but there is an overpair on the board to a big pair, big pair is not stacking off, which would reduce our set equity significantly and would happen pretty often. Would be cool to see the numbers when accounted for that(even simplistically), but i'm too lazy to do it =)

[Adrian20XX]

Cool, a lot of possitive feedback and no flames.

Altough I have double checked the formulas it would be great if someone develops them independelty in Excell to see if we get to the same value.

Now I'll address some comments:

Parlay Slow: Executive Summary
The idea is to know when it is EV+ for a small pair to call a bet pre flop in order to flop a set and stack a big pocket pair, and to be able to quantify that EV. At least this formula should (and does) consider that sometimes when the small pair flops his set also the big pair flops his set and gets stacked, and also that the small set can get outdrawn when is ahead after the flop.
The results are:
It is EV+ to play when STACK + INITIAL POT &gt;= 9,50 * BET (this is the simplified formula that should be used in a game).
And the EV is EV = 10,30% * INITIAL POT - 79,40% * BET + 8,36% * STACK

Enzyme: Paired board avoids opponent from being stacked
One of the hypotesis was the the big pair can not avoid to be stacked, but I also mentioned that when the big pair is not willing to put all his stack then STACK should become the ammount that the small pair gets him to put into the pot. There is also a more simple case that is when the opponent has KK and an ace flops, the big pair is out on the flop cheaply. There is also the case in which the flop comes King high when the big pair has AA and the small pair starts to doubt if the big pair has AA or KK, and hence loses value but from his own reticence (from the small pair) to put all the stack on the pot.

ArturiusX: How other flops affect the EV
I've put two examples of this as you have seen. Do I get a free coaching lesson for this ? :-)

I do have some ideas for followup that will go into one or two more posts (also very long). One is how to get the money into the pot after flopping a set, here I've got to a very good idea of what is good for the small pair and what is good for the big pair (not regarding cards but only how to play), and I also get to a formula. And the other one was for me a shocking revelation, that the value of the EV for the typical hand that according to the formula is only 5BBs on the typical case, and here I'm double guessing and I'd like to hear more opinions either if you agree or if you disagree with this very small EV+.

Thank you all for the possitive feedback, it really made me happy.

Any taker for the idea of develpoing the formula independently to confirm the results?

Regards ...

[FiSheYe]

I would love to see such an analyse for suited connectors [img]/images/graemlins/smile.gif[/img]
I will still use the 10% rule for setmining!