Formula of Poker 2

In the last post Constructing the Formula of Poker we established the formula of poker
Ca=C+(p*B)+((1-p)*E)

Where

Ca=Chipstack AFTER result of hand

C=Current chip stack
p=probability steal attempt wins

x=probability it doesn't win blinds

B=chips steal attempts gain if successful.
We will more clearly define E.

E=equity gained/lost if opponent responds=(Pr*Er)+((1-Pr)*Ec)

Where

Pr=Probability opponent raises, given he responds

Er=equity gained/lost if he reraises (3bets)

(1-Pr)=probability opponent calls, if he doesn't fold

Ec=Equity gained/lost if he calls.

We can thus notate it from
Ca=C+(p*B)+((1-p)*E)
and substitude E for  (Pr*Er)+((1-Pr)*Ec) to get
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}

The problem is, we aren't done yet. Afterall the equity gained/lost if he doesn't fold is still contingent upon a lot of various options before and after the flop.

The equity gained/lost if opponent raises has it's own formula. The equity gained/lost if opponent just calls has it's own formula.


Equity gained lost if he raises equals to (-amount risked*percentage of the time you fold)+(equity you gain when you call*probability you call)+(Equity you gain when you raise*probability you 4bet raise).
Percentage of the time you fold equals (1-(probability you call+probability you raise))
Therefore
Er=(-Ar*Pf)+(Ec3*Pc3)+(E4*p4)
OR
Er=(-AR*(1-(Pc3+p4)))+(Ec3*Pc3)+(E4*p4)

Where
Er=Equity gained/lost if he 3bets
Ar=Amount risked in raise
Pf=Percentage you fold GIVEN this line of play
Ec3=Equity you gain when you call a 3bet GIVEN this line of play
Pc3=percentage of the time you call a 3bet GIVEN this line of play
E4=Equity you gain when you 4bet
p4=percentage of the time you 4bet.

Make a mental note that we still have to calculate the Ec in the formula
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}
Right now we will finish "Er" and the various formulas that need to be derived from other variables to truly complete the formula.

At this point the "raise" will be assumed to be an "all in" so we can put an end to what otherwise would be an indefinite formula since we have not limited by chipstack or number of additional raises over the top of the previous raise. We have finally just about finished one possible line of play that goes raise, 3bet followed by you pushing all in. We only need to determine the equity gained/lost if you push which will be contingent upon whether opponent calls or fold and finally if he calls what percentage of the hands that you win. Also, the amount of chips in the pot is won if he folds. At some point you may be able to condense the formula and relate it to terms of a particular hand range.

However, in doing so we created several more possible outcomes for the way the hand could be played out. In some cases, our opponents 3bet will be all in. In that case, the probability of a "raise" will be zero and the equity if you call will have no future value other than the equity between hands.

Equity gained from all in 4bet= (probability opponent folds*chips in the pot minus the chips you have put in)+(Probability opponent calls*Probability your hand wins*Chips in pot minus your own)
E4=(Pf4*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))
probability opponent folds can be expressed as
(1-Pc4)
Therefore
E4=((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))
Finally we have a basic possible "end" (of many) to the equation aside from having to plug in the amount of chips we have and somehow having a way of determining or estimating the percentage of hands our opponent our opponent will call when facing an all in.

Now that we have come to an "end" we can go back and plug this equation in by substituting E4 for ((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu)) to previous parts and then go and begin to complete the remaining unfinished parts of the problem.
So we were previous left with
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}
Where
Er=(-AR*(1-(Pc3+p4)))+(Ec3*Pc3)+(E4*p4)
AND
E4=((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))
OR
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}
Where

Er=(-AR*(1-(Pc3+p4)))+(Ec3*Pc3)+([((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))]*p4) 
and we can eliminate E4 from the equation.

Now before we integrate this into the original formula, we still must solve for Ec3 or equity you gain when you call a 3bet. This will start to get really tedious and complicated beyond this point as it will introduce FLOP play, several more variables and then from those variables, turn and river play.

This work will basically be the same as the "Ec" just with a slightly different starting pot size and likely much different hand ranges. As such the analysis and expectations of hand ranges will thus be different and the results will be different, but the equation will look very similar.

Ec and Ec3 BOTH are basically equal to equity post flop. We can express them as the same equation PLUS preflop pot size and express bets as a size in relationship to the pot size if we want.

Actually, we could go back to the initial equation and come up with bet size assumptions and express it in number of big blinds. But I think we are getting ahead of ourselves and still have a long road of plenty of math in front of us.

Also remember that we started this formula with the assumption that you always open up for a raise. This may actually not be ideal with certain stack sizes as the opportunity to limp raise all in as well as gain more implied odds to maximize your post flop edge may provide a larger outcome. At some point we still may want to introduce another formula and it's potential outcomes if we just limp in. I don't necessarily know if it's needed. Afterall you can just express the amount raised as the call amount with zero percentage chance that the big blind opponent folds and the equity gained/lost will just change.

Additionally this formula was is will leave open the possibility to play with different raise sizes if we can estimate how our opponent may differ in strategy given certain bet sizes.

For now I must save additional calculations for another post as I am beginning to grow tired and need to refresh my thoughts with a new blank sheet.