Im trying to calculate the solutions to the following preflop game using the indifference principle.
Stacksize = 120BB

A) SB-opens 3BB : X frequency
B) BB-3bets 8BB : Defends 0.38
C) SB-4bets 20BB : Defends 0.36
D) BB-5Bets 50BB : Defends 0.39
E) SB-Shoves 120BB : Defends 0.4
F) BB- Call 70BB : Defends 0.41

BB needs can maximally exploit SB by calling with all hands > 29% equity ((170BB/70BB) = 2.4:1) and also needs to get to point E with (0.38*0.39*0.41) = 6.1% hands

I used the hand range calculator in equalab to find SB range satisfies both of the BB's conditions. SB needs a range of JJ+(3.0%) to give BB about 6.5% hands with > 29% Equity.

So if SB gets to point E with 3.0% they need to open with 21% hands.

Did I do this correctly? It seems like open folding 79% of hands at equilibrium seems like a lot.

Does anyone know of any utilities I could use so I wouldn't have to do it by hand all the time?

Thanks

First, practically, solns to this game probably don't have much to do with nlhe, since the option to call changes things a lot.

Anyhow, could you explain how you got those continuing frequencies? You seem to have made some assumptions that will be clearly not true if you write them down explicitly.

Ok. Could this type of game theory still be useful in identifying exploitive play?How does having a call option change the properties of the solution? Does position have an effect as well. For example Blind vs Blind in a ring game or heads up.

Here was my process:
I assumed that we don't know what the SB opening frequency is but we could calculate the BB frequencies based on how often they need to defend to keep the SB indifferent with their whole range at any point.

A) EV sb fold = EV sb raise/fold
S-0.5 = f(S+1.5)+(1-f)(S-3) where f = BB folding frequency, S=Stacksize
f=0.625, so BB needs to defend 38% of the time here.

I did this type of calculation for all of BB's and SB's points untill SB shoves and BB's best response is to Call with all hands that have appropriate all in equity. Greater than 29% by my calculation (risk 70BB to win 170BB), and to also call with enough hands to prevent SB from jamming as a bluff with their whole range.

So I took all of BB's defending frequencies 0.38*0.39*0.41 and got 0.06. I then tried to find a range that SB needs to get to point F so that the bottom of BB's 6% calling range has 29% equity (I assumed a linear range for BB). It was 3% for SB, so I worked backwards using SB's appropriate defence frequencies vs the BB. SB opens X hands*0.36*0.40 = 0.03. I got X = 21%.

It was also more of an exercise to see if I could figure it out using my own example. Obviously not LOL!

Thanks

What indifference equations did you write corresponding to points E and F, and to what hands do they apply?

E) EV SB shove bottom of range = EV 4bet/fold
(S+50)f1 + (1-f1)(2S)(Eq) = (S+9.5)f2 + (S-20.5)(1-f2)

Where f1 = BB's folding % at point F, f2 = BB's folding % at point D, Eq = all in equity.
Solving for Eq = [(S+9.5)(0.61)+(S-20.5)(0.39)-(S+50)(0.59)/(0.41*2S)
Eq = 18%
I think I know where I made one error though. I calculated BB's folding % at point F by setting EV SB 4bet/fold = EV SB shove/fold which is Obv wrong because you dont fold when you shove lol. So in this indifference equation at Point E do we have 2 unknowns? Eq, and f1?


F) EV BB call w bottom of range = EV BB 5bet/fold
(2S)(Eq) = (S+21.5)f + (S-50)(1-f)

Where f = SB folding % at point E

Solve for Eq = [(S+21.5)(0.6) + (S-50)(0.4)]/(2S)
Eq = 47%

I think those equations are both incorrect for essentially the same reason. Let's focus on one.

You've essentially set equal the SB's
- EV of jamming some hand starting at point E
- EV of 4bet/folding starting at point C
We've no reason to expect those to be equal. Generally, we'll look to take advantage of situations where the SB is indifferent between his two options in a particular spot.

So for example, after facing the 5-bet, there's some borderline hand of his that's indiff between folding and jamming. At that point, folding gets him S-20, jamming gets him f1(S+50)+(1-f1)(2S)(Eq).

If this is hunl, there's no room for a 5bet here, it's just a shove. It's not like you can raise 50/fold any hand anyway.

oh great that cleared things up for me, So now our indifference equations are

E) for SB - f1(S+50)+(1-f1)(2S)(Eq) = S-20

F) for BB - 2S(Eq) = S-50 ; Eq = 29%

Now we try to solve for f1 by assuming that BB gets to point E with (0.38*0.39) 15% hands and assume BB calls with hands at point F with > 29% equity.

Im not sure where to go from here

Im starting to see this

Right, so, so far, if SB's air is indifferent between each of its lines:
- open-fold
- minr-fold
- minr-4bet-fold
then those constraints give us BB's 3bet freq and BB's 5bet freq. (The only part of his strategy that is left is his shove-calling range.)

And if BB's air is indifferent between each of its lines:
- fold
- 3bet-fold
- 3bet-5bet-fold
then those constraints give us SB's 4bet freq and his shoving freq. (The only part of his strategy that is left is his opening range.)

All we need now is SB's initial opening range, since that fixes his jamming range, and from that we can get BB's shove-calling range, since it's just all hands that have pot odds to call all in.

So:
- we used all the indifference relationships, and the SB's opening range is still undetermined. How do you think we can do that?
- I agree w/ shou that the indifference relationships don't actually hold here, since we end up being too shallow for BB to 5bet-fold, but it's not too hard to see what does happen in that case, so I was going to ignore it for now...

Ok here we go.

In order to specify what the SB's opening range is we need to determine what equity the SB would have at point E with the part of their shoving range that is going to be indifferent between shoving and folding.

We could find how many hands the BB get's to point E with by looking at their defending frequencies given by our previous indifference equations. This is going to be (0.38*0.39) = 15%.


Their final calling frequency depends on what hands have > than 29% equity, and also enough hands to make the SB indifferent from Jamming with all of their air at point F.

Is this sort of the right track? Im going to give it another go tomorrow I am just struggling to visualize what to do at the moment.

Now that I think about it some more. SB open's X, 4bet/folds 0.36, shoves 0.40 for a total of 0.14X of all hands that SB could have when they go all in. If BB also gets to point E with 15% of hands then it is reasonable that X = 1.0 for the SB to maximally exploit the BB at point E, and of course SB never folds to the shove because every hand in their range has > 29% equity.
So in the end SB can open 100%. Now to just show this using math.

Yea, you have the right idea here -- SB's opening range isn't constrained by the indifference relationships, so he gets to choose it, and he'll choose the one that's most profitable for him. Doesn't have to be X=1, though.

Here's a related thread, btw:
http://forumserver.twoplustwo.com/15...-game-1305801/

Just sifting through your psuedocode there, Do we want find an equation for SB's EV = as a function of X opening hands, take the derivative and find what opening frequency yields the highest EV for our entire line?

SBEV={ (1-openperc/100)*(S-0.5) + openperc/100*((1-BB3BPERC/100)*(S+1) + (BB3BPERC/100)*( (1-SB4BPERC/100) * (S-2) + (SB4BPERC/100) *((1-BBCALLFREQ)*(S+5) + (BBCALLFREQ)*2*S*sb_allin_equity ) )) }

How come this isn't OPENPERC - SB4BPERC

Also what type of language do you use for these types of programs?

Thank you so much for helping me understand!

Well, I guess we could find the EV as a function of X and use calculus to maximize it. But in that thread, I just found EV(x) for a bunch of x, plotted them, and looked at the graph to locate the maximum.

SB4BPERC is the frequency SB 4bets given that he has the option to, i.e., when he is facing a 3-bet. (1-SB4BPERC) is the chance he folds when facing a 3-bet. I think that's what we want there...

I believe I did that calculation in bash, with a separate utility to do the equity calculations. Bash isn't really a good option for this.. it's just quick and easy for me.

np