As I mentioned in my blog, I've been playing around with a model for the 3bet/4bet/5bet/shove game.

Here are my assumptions

i) UTG and button have 100bb stacks.

ii) We ignore everyone else, but take into account the 1.5bb of dead money in the pot (Not sure this is v important though)

iii) UTG open raises to 3.5bb with 12% of his hands.

iv) Button 3bets to 10bb with KK+ balanced by some mid suited connectors (long tirade in my blog from various people telling me that flatting QQ and AK IP is more +EV than 3betting (I play NL10 rush)).

v) UTG 4bets to 25bb with KK+ and also with some hands that will be bluffcatchers (e.g. JJ,QQ).

vi) Button moves AI with KK+ and some fraction of his bluff range.

vii) UTG calls with KK+ and some fraction of his bluffcatchers.

viii) No card removal effects.

At the river, there are four possibilities

1) UTG value range v button value range = KK+ v KK+ = 0.5/0.5

2) UTG value range v button bluff range = KK+ v e.g. 76s = 0.775/0.225

3) UTG bluffcatchers v button bluff range = e.g. JJ v 76s = 0.775/0.225

4) UTG bluffcatchers v button value range = e.g. JJ v KK+ = 0.1825/0.8175

Some notation

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v1: proportion of hands that UTG opens with and has in his value range = 0.14 (12 combos out of 164)

v2: proportion of value hands in button's 3bet range (1-v2 = bluffs)

b1: UTG 4bets with v1+b1 of hands he raises, and folds 1-v1-b1

b2: proportion of bluff hands that button 5bets (he raises v2+b2(1-v2) and folds (1-b2)(1-v2))

b3: proportion of bluffcatchers that UTG calls AI with (calls (v1+b1b3)(v1+b1), folds (1-b3)b1/(v1+b1)

After some algebra (ugh!), which I'd be very happy for someone to check, I get that the EV for each player is

UTG = -3.5(1-v1-b1)+11.5(v1+b1)

+b2(1-v2)(-11.5(v1+b1)-25(1-b3)b1+56.16(v1+b1b3))

+v2(-11.5(v1+b1)-25-(1-b3)b1+0.75v1-63.23b1b3)

Since UTG chooses b1 and b3, the best he can do is to choose b1=b3 = 0, i.e. value only.

Button EV = 5(1-v1)-10v1(1-b2)(1-v2)+0.75v1v2-54.66v1b2(1-v2)

+b1(-5-10(1-b2)(1-v2)+26.5(v2+b2(1-v2)))

+b1b3(-26.5(v2+b2(1-v2))-54.66b2(1-v2)+64.73v2)

Now we can make the button indifferent to UTG's choice of bluffcatching frequency by making the factors that multiply b1 and b1b3 zero. This gives v2 = 0.28, b2 = 0.18. This is the Nash equilibrium strategy, with which the button gets +2.5bb and UTG loses 1bb (the power of position, and don't forget the 1.5bb from the blinds).

This means that the unexploitable way for the button to balance his 3betting range is to 3bet with KK+, which is 12 combos, and since v2 = 0.28, about 31 combos of midsuited connectors (e.g. 45s, 56s, 67s, 78s, 89s, oops, ran out, so maybe need some one gappers too!). UTG should only 4bet for value, and, since b2 = 0.18 the button should shove with KK+ and 18% of his bluffs (about 5 combos, e.g. 76s and 87h).

Some questions?

---------------------

1) Can anyone check this? I've been known to bugger this sort of thing up before!

2) How can we change the assumptions to make things more realistic? Remember, this is mathematical modelling, so the name of the game is

Model -> results -> compare with reality -> change model -> repeat until bored/satisfied.

3) Can I be arsed to redo this with variable bet and stack sizes??

Here are my assumptions

i) UTG and button have 100bb stacks.

ii) We ignore everyone else, but take into account the 1.5bb of dead money in the pot (Not sure this is v important though)

iii) UTG open raises to 3.5bb with 12% of his hands.

iv) Button 3bets to 10bb with KK+ balanced by some mid suited connectors (long tirade in my blog from various people telling me that flatting QQ and AK IP is more +EV than 3betting (I play NL10 rush)).

v) UTG 4bets to 25bb with KK+ and also with some hands that will be bluffcatchers (e.g. JJ,QQ).

vi) Button moves AI with KK+ and some fraction of his bluff range.

vii) UTG calls with KK+ and some fraction of his bluffcatchers.

viii) No card removal effects.

At the river, there are four possibilities

1) UTG value range v button value range = KK+ v KK+ = 0.5/0.5

2) UTG value range v button bluff range = KK+ v e.g. 76s = 0.775/0.225

3) UTG bluffcatchers v button bluff range = e.g. JJ v 76s = 0.775/0.225

4) UTG bluffcatchers v button value range = e.g. JJ v KK+ = 0.1825/0.8175

Some notation

------------------

v1: proportion of hands that UTG opens with and has in his value range = 0.14 (12 combos out of 164)

v2: proportion of value hands in button's 3bet range (1-v2 = bluffs)

b1: UTG 4bets with v1+b1 of hands he raises, and folds 1-v1-b1

b2: proportion of bluff hands that button 5bets (he raises v2+b2(1-v2) and folds (1-b2)(1-v2))

b3: proportion of bluffcatchers that UTG calls AI with (calls (v1+b1b3)(v1+b1), folds (1-b3)b1/(v1+b1)

After some algebra (ugh!), which I'd be very happy for someone to check, I get that the EV for each player is

UTG = -3.5(1-v1-b1)+11.5(v1+b1)

+b2(1-v2)(-11.5(v1+b1)-25(1-b3)b1+56.16(v1+b1b3))

+v2(-11.5(v1+b1)-25-(1-b3)b1+0.75v1-63.23b1b3)

Since UTG chooses b1 and b3, the best he can do is to choose b1=b3 = 0, i.e. value only.

Button EV = 5(1-v1)-10v1(1-b2)(1-v2)+0.75v1v2-54.66v1b2(1-v2)

+b1(-5-10(1-b2)(1-v2)+26.5(v2+b2(1-v2)))

+b1b3(-26.5(v2+b2(1-v2))-54.66b2(1-v2)+64.73v2)

Now we can make the button indifferent to UTG's choice of bluffcatching frequency by making the factors that multiply b1 and b1b3 zero. This gives v2 = 0.28, b2 = 0.18. This is the Nash equilibrium strategy, with which the button gets +2.5bb and UTG loses 1bb (the power of position, and don't forget the 1.5bb from the blinds).

This means that the unexploitable way for the button to balance his 3betting range is to 3bet with KK+, which is 12 combos, and since v2 = 0.28, about 31 combos of midsuited connectors (e.g. 45s, 56s, 67s, 78s, 89s, oops, ran out, so maybe need some one gappers too!). UTG should only 4bet for value, and, since b2 = 0.18 the button should shove with KK+ and 18% of his bluffs (about 5 combos, e.g. 76s and 87h).

Some questions?

---------------------

1) Can anyone check this? I've been known to bugger this sort of thing up before!

2) How can we change the assumptions to make things more realistic? Remember, this is mathematical modelling, so the name of the game is

Model -> results -> compare with reality -> change model -> repeat until bored/satisfied.

3) Can I be arsed to redo this with variable bet and stack sizes??