You Couldn't Sell Two Copies if You Pressed a Double Album

My last post on bet/call theory was a little duct taped toegether and left out some important concepts which i will address here. First we will aim to find the overall EV needed to do have both stacks go in the middle given the pot size and stack depth. then we will see the impact of bet folding and bet sizing has on our needed equity to call. lastly we will Lastly will will show the impact of P2 folding and any equity edge we have on later streets to paint a picture of the actual "game" being played.

The first step is draw a decision tree to show what all possible paths. here is the one i have drawn:

 

Note the payoff [y] is a nebulous concept i havent done much work on but it is player ones approximate EV on later streets given P2 calls. it is bounded [-1,1], one could look at it having a few components: skill advantage and chances of making mistakes on later streets. For instance huge draws will probably make way less mistakes (if any) on a drawy board while a strong made hand like top set or bare nut straight can make quite a few wrong plays later. The skill advantage i wouldnt put to any number until you're decidedly more skilled than you're opponents and you're dropping down.

From here we can make a few substitution to make game more analytical. 

1. B1+B2=S

2. B1=Sx

3. B2=S(1-x)

4. Sz=P

5. F2+R2+C2=1

6. R2=1-C2-F2, later we will see this is not needed but i am following the math i used.

in the above all S,x,F2,R2,C2 are bounded from [0,1]

the next step will be to subsititute in then divide through by S thus putting all payoffs in terms of percentages of inistial stack size.

the resulting tree:

 

Step 1: equity needed to bet/call

The first step in evaluating any game theory tree is to start at the last nodes and work backwards here its #2 on the raise branch, here well solve for the EV of a call. EV(call)=E[z+1}-[1-E]=E[z+2]-1. solving this yields the required equity needed to get stacks in the middle: E> 1/[z+2]. For example if we are in a 3bet pot where each player still have 100bb left and 25bb are in the middle z=25/100=0.25, thus the required equity to bet/call is E>1/2.25=4/9=0.4444. Therefore any equity over 44.44% will lets bet/call become a viable option.

Step 2: correcting for folding

Now we will analyze the #2 branch of Player 2's possible actions. What we do here is put the EV we just determined as the payoff for calling and then go through the same process as step 1.

Calculating EV (facing raise)=(1-F1)(E[z+2]-1)-F1*x

from here we can solve for F1 or E to determine when our expected value is greater than zero. Solving for F1 yields  confusing results as wed graph something showing us the more we fold the higher our EV. while this isnt untrue it doesnt yield anything illuminating. for those interested we determine when F1> (1-E[z+2])/((1-x)-E[z+2]) yields positive expectation. if one were to graph this theyed be looking at a picture showing whenever we fold more than this surface its +EV. I say this isn't too helpful as we could start folding just to fold to increase our folding percentage, it tells us nothing about our hand range composition.

solving for required Equities yield much more interesting results. E> (1-F1(1-x))/((1-F1)(z+2]). there are two ways to look at this equation. the first is to think how often youd fold a specific hand, or how often youd fold hand x when facing a raise. This is both a bit short sighted and too advanced for many beginners and intermediate players. at these levels people are mainly either they are or arent calling a raise, while those at higher limits will talk about folding some hands x% or y% of the time against the same villain. The equations main power comes from the second way to look at it. if we look at the overall picture for player one we see the more

here is a 3d graph of a 4spr situation where we bet and now have been raised. so the equation used is E> (1-F1(1-x))/((1-F1)(2.25)), whats a bit confusing about this graph is i used F1 on the x-axis and x on the y axis

the graph is a bit simplified of a situation because some raise size choices by villian will let him raise fold (especially if we bet < 1/3 pot or < 1/12 of our stack, but a simplifying assumption to get rid of these pesky situations is to say this graph assumes were getting called if there's room left to raise him and we can deal with the outlying bet sizes separately later. 
In this graph any space above the surface yields positive expectation for P1. we need to remember this is given weve already decided to bet, we get to choose the size. We can see the interesting behavior as we choose to bet larger and fold more often we actually need more equity when we decide to call in order to make up for lost equity when we fold. 
we look at the end behavior (when x or y = 0 ) we see the equities we found in step 1. both borders are @ E>1/(z+2). I find it very interesting that this is the minimum equity. that the more we fold the more equity we need when calling. my initial assumption was the corrections would have dipped our equity lower, not higher. to me this is close to mind blowing as i generally have good mathematical intuition. The goal is to make up huge chunks of equity by having large equity edges, not make our mistakes smaller by calling with closer odds more.
Note that this isn't equities needed to make the calls after betting, these are equities to bet/call with, so these are the required equities to begin betting with.
Step 3:
now we can condense the raising node down to just 1 payoff and evaluate the payoffs of each action of P2. Here's P1's tree:

Now well use the same steps as before using the numbered variables as percentages and bolded as payoffs.. From there i solved set greater than zero and solved for E. After A LOT of algebra i arrived at a neat little solution of the game.

The required equity to bet/call with is:

E> (R2[1-F1*(1-x)]-F2*z-C2*y)/((z+2)*(R2(1-F1)+F1))

 While this is a complicated solution we can analyze its meaning without graphs.  The higher our advantage on later streets and the more often villain calls, the lower our needed equity, the more often and larger the initial pot, the lower required equity we need. but the more p2 raises and more we fold the more equity we need while this is already known conceptually we can look at relative sizes to determine which have a larger impact. We can also see the impact of our hand type. if we have a made hand that plays worse (relatively) on drawy boards, taht will raise our needed equity to bet/call, while if we have the draw it actually lowers the equity.

From this equation we can actually see the strategy profiles emerge. for Player 2 his profile becomes (F2, R2, C2) and Player 1's strategy profile becomes (F1, x, W). W is actually a concept talked about but not quantified and that is his range of hands he does the actions with (in this case betting). for instance if a player is always bluffing, he cant be calling much if at all. its actually a concept that builds itstelf into the game at each stage, but we were able to factor out in the algebra because its the same everywhere. now it is these variables which we can categorize villains by and see very observant opponents change themselves and everyone adjusts.
there are also inputs into this game (z,y). Z is the inverse of the Spr and Y is the future street gain by player 1 or on the river, P1's equity v P2's calling range.
more to follow later as others ask questions and i find more detail to add.