I posted this hand in my blog and in the micros forums last week and tried to do the math behind it. Bobbo pointed out some errors I made, and so here is my new attempt to understand the math behind all of this. I will just post the results of my work without drawing any conclusions in the end because I want to think about it a bit, and I have to go to work very soon.



SB ($50.43)
BB ($144)
UTG ($50.75)
UTG+1 ($82.20)
Hero ($53.50)
BTN ($97.75)

Dealt to Hero 8 8

fold, UTG+1 raises to $1.75, Hero calls $1.75, BTN raises to $7, fold, fold, fold, Hero raises to $53.50 (AI), fold

Hero shows 8 8

Hero wins $16.50


The pot is $11.25 when I am up, and I am risking $51.75 to win it. If I am called, the pot will be $108.75.


Now, let's talk about my equity vs. his calling range. This is really the key. When I read the post, it had to so with 400nl, and I looked at the calling range they had listed, it seemed a little loose for 50nl, tbh. Range 1 is what I think is "standard" for an all in range considering the dead money in the pot already. Range 2 is some sort of amalgam range that I don't think many people really have. Range 3 is obviously a bit wide for 50nl (it was the calling range they gave villain in the thread). Range 4 is the conservative all in pre range, and perhaps somewhat counterintuitively, the range I have the most equity against because of how the card combinations of AK work out in relation to the card combinations of overpairs to my hand (16 combos of AK and 6 each of AA and KK). Once you start adding in more combos of QQ and JJ, my equity sinks, as in Range 2. What are people's actual calling range here and how much of their squeezing range are they folding? Obviously it depends on villain. At 50nl, I think some villains really do fold 99-QQ here, thinking that anyone who 4bet shoves must have AA or KK and sometimes AK. Other people will stack off fairly light in this scenario. I would like to evaluate this play for 3 of the ranges listed below and see what difference they make. I also would like to do the math for the hand in the event that there was another caller or maybe two callers ahead of me to see how the extra dead money affects the math.


Range 1

equity win tie pots won pots tied
Hand 0: 35.842% 35.66% 00.19% 20757902 108631.00 { 8c8d }
Hand 1: 64.158% 63.97% 00.19% 37243172 108631.00 { QQ+, AKs, AKo }
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Range 2

equity win tie pots won pots tied
Hand 0: 33.281% 33.09% 00.19% 22665122 129688.00 { 8c8d }
Hand 1: 66.719% 66.53% 00.19% 45567662 129688.00 { JJ+, AKs, AKo }
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Range 3


equity win tie pots won pots tied
Hand 0: 39.374% 39.19% 00.19% 37574366 181057.00 { 8c8d }
Hand 1: 60.626% 60.44% 00.19% 57952544 181057.00 { JJ+, AQs+, AQo+ }

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Range 4


equity win tie pots won pots tied
Hand 0: 39.423% 39.24% 00.19% 18812180 88981.00 { 8c8d }
Hand 1: 60.577% 60.39% 00.19% 28954370 88981.00 { KK+, AKs, AKo }



Let's start with the "tight" all-in calling range, Range 4 above, as KK+,AKo,AKs, and I have 39.24% equity in the hand.

So, the pot is $108.75 when I get called and I have 39.24% equity.

My equity% multiplied by the pot (108.75 x .3924) is $42.67. Since I am risking $51.75, I lose $9.08 on average assuming I am called 100% of the time. I also win $11.25 when he folds. Which begs the question, how often does villain have to fold for this play to be break even considering my equity in the hand against this all in calling range?

Assuming x is the fraction of times he needs to fold,

(Amount I win when villain folds)x-(Amount I lose when villain calls)(1-x)=0

or

11.25x - 9.08(1-x)=0

is the equation for breaking even. 1-x is his call fraction if x is the fraction of times he folds.

11.25x = 9.08-9.08x
20.33x=9.08
x = 0.44663 or 44.6% FE needed to break even assuming villain has a very tight AI calling range preflop.
-------------------------------------

Now, Range 2 above, which many people suggested was the more realistic calling range of most 50nl players, is JJ+, AKs, AKo, and I have 33.09% equity in the hand.

My equity % multiplied by the pot (108.75 x .3309) is $35.98. Since I am risking $51.75, I lose $15.77 on average assuming I am called 100% of the time. I also win $11.25 when he folds. How often does villain have to fold for this play to be break even considering my equity in the hand against this all in calling range?

Assuming x is the fraction of times he needs to fold,

(Amount I win when villain folds)x-(Amount I lose when villain calls)(1-x)=0

or

11.25x - 15.77(1-x)=0

is the equation for breaking even. 1-x is his calling fraction if x is the fraction of time he folds.

11.25x = 15.77- 15.77x
27.02x = 15.77
x = 0.58364 or 58.3% FE needed to break even assuming villain has an AI calling range preflop of JJ+,AKs, AKo

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Now let's deal with Range 1 above, QQ+,AKs,AKo. My equity against this range is 35.66%. I want to deal with this range in particular because it gives me an amount of equity between the equity I had in the first two Ranges, and I think it will give us a sense of how the numbers trend (realistically, to make a graph or something, we would have to put in many more ranges and equities, but I never intended this to be a definitive study on this subject, I just wanted practice doing this type of work).

My equity % multiplied by the pot (108.75 x .3566) is $38.78. Since I am risking $ 51.75, I lose $12.97 on average assuming I am called 100% of the time. I also win $11.25 when he folds. How often does villain have to fold for this play to be break even considering my equity in the hand against this all in calling range? Let me explain some of the steps in solving the equation now that I have done it a few times, too. I will probably flub some of the terminology since it has been quite a while since I took algebra.

Assuming x is the fraction of times he needs to fold,

(Amount I win when villain folds)x-(Amount I lose when villain calls)(1-x)=0

is the equation for breaking even. 1-x is his calling fraction if x is the fraction of time he folds.
or

11.25x-12.97(1-x)=0

since the first figure minus the second figure equals zero, then the two must be equal to each other, so we can express this mathmatically as:

11.25x = 12.97(1-x)

Now, for the right side of the equation, we multiply 12.97 by both figures in the parentheses [think of it like a(b-c) = a(b) - a(c)and we get:

11.25x = 12.97 - 12.97x

Now, we add 12.97x to BOTH sides of the equation which will keep the equation balanced and allow us to isolate the variable to one side after we simplify:

11.25x + 12.97x= 12.97 -12.97x + 12.97x

Now we simplify:

24.22x = 12.97

Now we divide both sides of the equation by 24.22 to reduce the left side to x:

x = 0.5355

which we can express in percentages rounded to the nearest tenth of one percent as:

x = 53.6%
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Let's lay out all of the Equity percentages we have vs. Villain's possible preflop all in calling ranges and show them in relation to the amount of fold equity we need to be breaking even.

When villain's AI calling range is/ Our Equity with 88 is/ And we break even with a Fold Equity of

{KK+,AKo,AKs} / 39.24% Equity / 44.6%

{QQ+,AKo,AKs} / 35.66% Equity / 53.6%

{JJ+,AKo,AKs} / 33.09% Equity / 58.3%

I know this seems really obvious. The less equity you have in the hand, the more fold equity you need to break even. But take a close look at the ranges. I just added one paired hand to each range, or 6 combinations, and look at the dramatic rise in FE that you need. If we were to add TT and then 99 to villain's all in calling range, then our equity would drop by 2 points with each hand added, and I am guessing that our FE needed would be 63% and 68% respectively. Obviously that is because we are introducing more overpairs to our 88 to villain's range, and no unpaired overs to his range. If we were to add AQs and AQo to villain's widest all in calling range listed above, then our Equity vs. that range shoots back up to near 40%. If we were to add only AQs, we are at only 35% due to there being only 4 suited combinations and 12 off suit combinations.

I also think that it is worth noting that the lighter villain stacks off, the more FE we need, the more hands he has to be squeezing with that he is willing to fold to hit that break even FE threshold. For instance, if we take the bottom of the three ranges where we need the most FE (almost 60%), villain stacks off with 3.0% of all possible card combinations. That means he needs to be squeezing with 7.5% of his hands. (One example of that would be {AJo+,AJs+,KQs+,KQo+,99+} but I don't think anyone would really construct their squeezing range like that.) Whereas if he stacks off with only KK+,AKs,AKo, that represents only 2.1% of all starting hands, and he only needs to be squeezing with a little less than 4% of his hands.

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Finally, I would want to deal with the incidences where there is a little more dead money in the pot, ie, another preflop caller or two in front of us. To make things tidy, we will assume that we are on the Button, and it is always folded to us after the sB squeezes. If we open UTG, get two callers and the BTN squeezes, we have to consider whether or not the MP and CO have all in calling ranges and that just gets messy. Also, if we are in the CO, get squeezed by the BTN and the MP calls the threebet, we have to consider if he calls our jam or not, and that gets complicated, too. We will just say that there is one additional caller then 2 additional callers, and the squeezer(this time from the SB) adjusts his 3bet amount to compensate. I am very aware that by doing this, we are departing further and further into theoretical territory and moving away from reality because people's ranges are (or should be) affected by these new set of circumstances. Everyone's preflop open calling range will change, the preflop opener's range will change, and the SB squeezer's 3bet range will change, too.

Scenario 1:


BB ($50.43)
UTG ($144)MP ($50.75)
CO ($82.20)
Hero ($53.50)
SB ($97.75)

Dealt to Hero 8 8

fold, MP raises $1.75, CO calls $1.75, Hero calls $1.75, SB raises to $9, fold, fold, fold, Hero raises to $53.50 (AI), fold

Hero shows 8 8

Hero wins $14.75.

Scenario 2:

BB ($50.43)
UTG ($144)MP ($50.75)
CO ($82.20)
Hero ($53.50)
SB ($97.75)

Dealt to Hero 8 8

UTG raises $1.75, MP calls $1.75, CO calls $1.75, Hero calls $1.75, SB raises to $11, fold, fold, fold, Hero raises to $53.50 (AI), fold

Hero shows 8 8

Hero wins $18.50.


Villain's AI calling range is {KK+,AKo,AKs}

Scenario 1

My equity is 39.24% vs. this range. Total money in the pot when called is $110.50. $110.50 x .3924 = $43.36. Since I am risking $51.75, I lose $8.39 on average assuming I am called 100% of the time. I also win $14.75 when he folds. How often does villain have to fold for this play to be break even considering my equity in the hand against this all in calling range?

Assuming x is the fraction of times he needs to fold,

(Amount I win when villain folds)x-(Amount I lose when villain calls)(1-x)=0

or

14.75x - 8.39(1-x)= 0
14.75x = 8.39(1-x)
14.75x = 8.39 - 8.39x
23.14x = 8.39
x = .03625
x = 36.2%

Scenario 2, same all in calling range.

$112.25 x .3924 = $44.05. I am risking $51.75 to win this, so I lose $7.65 on average assuming I am called 100% of the time. I win $18.50 when villain folds.

Assuming x is the fraction of times he needs to fold,

(Amount I win when villain folds)x-(Amount I lose when villain calls)(1-x)=0

18.50x - 7.65(1-x) = 0
18.50x = 7.65(1-x)
18.50x = 7.65-7.65x
26.15x = 7.65
x= 0.29254
x= 29.3%


Villain's all in calling range is {QQ+,AKo,AKs}

Scenario 1

My equity is 35.66% vs. this range. Total money in the pot when called is $110.50. $110.50 x .3566 = $39.40. Since I am risking $51.75, I lose $12.35 on average assuming I am called 100% of the time. I also win $14.75 when he folds. How often does villain have to fold for this play to be break even considering my equity in the hand against this all in calling range?

Assuming x is the fraction of times he needs to fold,

(Amount I win when villain folds)x-(Amount I lose when villain calls)(1-x)=0

or

14.75x - 12.35(1-x) = 0
14.75x = 12.35-12.35x
27.10x = 12.35
x = 0.4557
x = 45.6%

Scenario 2, same all in calling range.

$112.25 x .3566 = $40.02. I lose $11.73 assuming I am called 100% of the time. I win $18.50 when villain folds.

18.50x = 11.73 - 11.73x
30.23x = 11.73
x = 38.8%

Villain's all in calling range is {JJ+,AKo,AKs}

Scenario 1

My equity is 33.09% against this range, or $36.56 of the total pot when called. I risk 51.75 to win this money, so I lose $15.19 on average when called 100% of the time. I win $14.75 when villain folds.

14.75x = 15.19 - 15.19x
29.94x = 15.19
x = 50.7%

Scenario 2

My equity is 33.09% of a $112.25 pot when called, or $37.14. I lose $14.61 on average assuming I am called 100% of the time. I win $18.50 when villain folds.

18.50x = 14.61 - 14.61x
33.11x = 14.61
x = 44.1%

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Now, let's put this all together and look at it.

Range / Equity vs. that Range with 88 / FE Original Scenario / FE Scenario 1 / FE Scenario 2


{KK+,AKo,AKs} / 39.24% Equity / 44.6% / 36.2% / 29.3%

{QQ+,AKo,AKs} / 35.66% Equity / 53.6% / 45.6% / 38.8%

{JJ+,AKo,AKs} / 33.09% Equity / 58.3% / 50.7% / 44.1%


So, as I noted before, the less equity you have in the pot the more FE you need to break even. However, this is counteracted by the addition of dead money in the pot (in my example an extra caller or two extra callers and a bigger sized 3bet), which decreases the amount of FE you need just as dramatically. I would like to ruminate a little bit about what this all means for everyone involved before drawing any conclusions. I also have to head off to work, and so I won't realistically be able to comment until tomorrow afternoon. But I will post this in my blog and feel much better about having done all the math correctly (I hope) this time.