July 05, 2010

Evaluating Bet/Call-theory

often in any card game we are faced with situations or decisions where we know if were raised in a certain spot well be playing for stacks. For instance, imagine youre playing PLO and you've 3bet OOP and now flop a mediocre hand, what do to? if we think check/folding is out the the question, we have to evaluate the line bet/call line. I cant post a decision tree here as i dont know how to post charts. IF YOU KNOW HOW TO POST CHARTS TELL ME AND ILL MAKE IT EASY TO VISUALIZE.

1. Preflop pot [P] 2. P1 bets {B1] 3a. P2 folds F%, P1 wins [P] 3b. P2 raises [B2], P1 calls 3b1. P1 wins [P+B1+B2] E% of the time 3b2. P1 loses (1-E)% and wins -(B1+B2)

To make things easy we will now put each bet in terms of the Preflop pot size:
B1=Px, x domain [0,1]
B2=[P+2B1]y=[P+2Px]y, y [0,1] (again here it could in theory go up to the spr of the 2nd bet

a common term to come up in analysis will be [B1+B2]=Px+y[P+2Px]=x+y[1+2x]=z

now we will divide thru by P thus putting all bet sizes will be in terms of percentage of original pot
from here we will evaluate the win and loss outcomes:
Win: [1+x+y[1+2x]]=1+z
Loss: -[z]

evaluating EV of a call of raise
EV(call)=E[1+z]-[1-E][z]=E[1+2z]-z

evaluating original bet: this is the final EV of the overall bet/call line
EV[b1]=F+(1-F)[E[1+2z]-z]=F+E+2zE-EF-2zEF-z+zF

from here well solve for min Fold equity and Equity needed to bet/call

E>(z-F[1+2z])/([1+2z][1-f]) visible here: (y is the spr, X is the fold equity) (note: the graph is of the lower boundary surface, any point above this surface is +EV for us

here we can see the interaction of adding folding equity decreases the needed equity. We can quickly see how littel fold equity we need around the break-even point, for instance a hand with 37.5% equity against a range  and a 1.6 SPR only needs to fold out 4% of hands.


F>(z-E[1+2z})/([1+z]-E[1+2z]) visible here (y is the spr, X is the raw Equity

a thorough analysis demonstrates Z is the SPR before any action on the flop. We could probably stretch z to 6 seeing that then well be giving him 2/11 or ~18% on a call. so this holds when either villain is a nit who's never folding once he raises, or a smaller spr against another player.(higher spr would provide situations where our Equity becomes a function of SPR as the villain can raise/fold. but in in fact this analysis can play out for us if we were in his shoes.

From here we can see we've made a +EV line, showing when bet-calling gains chips, but can we improve on it? but how? a quick look will show if we're pure bluffing with 0 equity then its much better for us to bet/fold than to bet call any amount, can we incorporate that into what weve developed so far? To do that we must determine if we can bet/fold at any point to improve our EV.

Now we will determine the boundary between bet/folding and Bet/calling:

EV (B/c)= E(1+2z)(1-F) + F(1+2z) - z
EV (B/f)= F + (1-F)x, x is the same x from B1=Px, b1=x

now well had another option to our tree: 1. Preflop pot [P] 2. P1 bets {B1] 3a. P2 folds F%, P1 wins [P] 3b. P2 raises [B2], P1 calls 3b1. P1 wins [P+B1+B2] E% of the time 3b2. P1 loses (1-E)% and wins -(B1+B2) 3b3. P1 Folds F1% and wins -[b1]

our first step is to determine when EV (b/c) > EV (b/f): from above: E(1+2z)(1-F) + F(1+2z) - z > F + (1-F)x

now we can do some algebra and solve for E and F. these will be from the same equation so the equation for one will contain the other term
E> [(x+z)-F(x+2z)]/[(1+2z)(1-F) and
F>[(x+z)-E(1+2z)]/[(x+2z)-E(1+2z)]

now with both x and z in the equation we have to substitute in for z=x+y(1+2x), since we cant use a 1/2 psb on the turn to get a 4spr stack in by being raised once.

E>[(2x+y(1+2x)-F(3x+2y(1+2x))]/[(1+2x)(1+2y)(1-F)]
F>[((2x+y(1+2x))-E(1+2x)(1+2y)]/[(3x+2y(1+2x))-E(1+2x)(1+2y)], in these equations x,y,F ~ [0,1], i am using ~ as the domain symbol

thus we have solved for the border between bet/folding and bet/calling. therefore the b/f range wrt E is [0,x(2(1+y)-F(3-4y)]+y(1-F)]/[(1+2x)(1+2y)(1-F)]] and the b/c range is [x(2(1+y)-F(3-4y)]+y(1-F)]/[(1+2x)(1+2y)(1-F)], 1] and

the b/f range wrt F is [0, [(2x(1+2y)+y)-E(1+2x)(1+2y)]/[x(3+4y)+2y-E(1+2x)(1+2y)]] and the b/c range is [[(2x(1+2y)+y)-E(1+2x)(1+2y)]/[x(3+4y)+2y-E(1+2x)(1+2y)], 1].

Also note how this graphs don't care about board texture, all that it takes to make this graph is a decision about bet sizing. we can take the same graph to represent a much tough board texture say 9c8c4d, if we choose again an 80% psb on the flop we have the same graph. now if we only have say a 10% FE, the Required equity to b/c is now 48%. Thus nearly everytime were betting to bet call, we almost have to be a favorite!

the "shame" of these equations are they are 4d, thus i cant represent anything visually except for specific representations, thus the best way to go about this is to either pick a bet size on the flop or turn you want and look at the how bet sizing on the other street affects the E,F variables.

the equations for F and E should be used in concert as once one determines the equity for certain situations and the boarder btwn b/f and b/c we then play around with different ranges on the flop or turn to see how much we can move F.

EXAMPLE: lets say we 3b preflop and had and A flop so we know our FE is relatively high, lets approximate 40%, and were betting 80% of pot, leaving a psb behind. putting x=0.8 and y=1 into E>[(2x+y(1+2x)-F(3x+2y(1+2x))]/[(1+2x)(1+2y)(1-F)]  we reduce the equation to E>(4.2-F*(7.6))/(7.8*(1-F)). from here we can graph this equation in 2D we can see the break even Equity w/ 40% FE is 26% equity. Above 26% equity we should be Bet/calling, below 26% equity we should be bet/folding.

User Uploaded Image

Two math things first, the domains i have given in the later formulas for x,y are [0,1] and that goes up to an spr of 4, we could probably stretch this out to 6 by letting y go slightly over 1. i cut the domain at 1 becauase i was working with this for PLO, but in theory x,y can go from [0,oo) to incorporate themselves of NL games where overbetting can be used. we must be careful with letting y stray too far as then we allow for other lines available such as min-raises or calling.

The two easiest places to visualize these equations being applicable are on the flop in a 3b pot or on the turn in a single raised pot. ill be making some of my own charts once i get my office suite up and running, have fun with finding your own borders.

 Excluding the bottom right square thats yellow under the curve, the entire yellow region is the bet/call region. thus anytime we estimate our Equity over ~26% w/ 40% fold equity its an automatic bet/call. the Peach portion is the bet/fold region, if we end up having only a gutshot and a backdoor flushdraw (~15%) its a bet/fold. but notice how quickly we pick up equity as we pick up pieces. add another gut shot (double gutshot) and your at ~25%, even discounting if your against sets youre still up to 18-20%. add another bdfd and youre at ~24%, we can see how quickly this adds up. Remember, this graph is not an end all/be all. this is for our specific bet sizing. if we change our bet sizing to half pot, thus x=0.5 and y=1.75 (we allow y to venture up around 1.8 to 2 as long as z stays under 6). we see the needed equity jumps up to 31-32%


Example 2: lets say again we decide on a certain flop texture, say KT6r we decide to bet 80% of pot on the flop, setting up a the last bet of 1psb.

User Uploaded Imagehere in example two (can someone tell me how to enlarge font? i really dont know how to do this)


in example two we see the folding equity vs raw equity graph of the same situation where we posit to make a n 80% pot bet on the flop, leaving p2 to decide if he wants to shove it on the flop or call it on the turn. it in fact is the first graph rotated around the line y=x since our x, y didnt change. I like this version of the graph and equations better personally because the equity contain F, a quite subjective term, where we were guessing, hoping to be close to the right answer. here we can paint a clearer picture with the equity in the equation as we can pick pieces to add to improve our hand or subtract them as we wish to see the resulting effect

Its interesting to me to see our equity ever have to be over 50% as to me it should cap out at 50%, IM GOING TO INVESTIGATE THIS FURHTER AND DETERMINE IF THIS IS AN ERROR ON MY PART OR AN UNEXPECTED FINDING.

note: the reasons behind this become clearer when we post the fold equity vs equity graph. although we may be a slight favorite when we have say 52% equity, we actually make more by him folding, but theres a region from 50% to y% (specific to each set of (x,y, E)) that although we dont mind a call, wed like a fold even better, even though were a favorite, our opponent simply makes a larger mistake by folding and giving up his equity than by calling.

part of this may be a simplification from a 4d problem to a 2d problem, but i am accounting for our decisions, x, y vs  hand properties, E, F

heres a graph assuming z=(spr)=4 showing the effect of various bet sizing and its effect on the bet/call-bet/fold border.

User Uploaded Image

 

after putting more thought into it i realized x and y need not be ratios of pot bets strictly between [0,1]. if you take x as the % of pot of the initial bet and y as the % of pot left behind to be put in later, we get a clearer picture of how the money left behind impacts equity and FE needed as most bets in PLO are not pot, but we simply use the assumption that up to a certain spr, a repot of our cbet means theyre not folding and we get the above graphs.

ill come back later and post graphs for z=2, 2.5, 3, 3,5 and 4 and we can get an even more detailed view

 

Posted By lifes3ps at 03:41 AM

184 Comments

Tags: theory 3bet raised bet/call

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