[Lefort]

So I was in my Financial Risk and Stock Portfolio Analysis class and boredom ensued, so obviously seeing the references to equity and expected values led my thoughts to poker theory..

5/10, $1000 eff stacks

Folded to you in the hijack with A[img]/images/graemlins/heart.gif[/img]K[img]/images/graemlins/heart.gif[/img] and you make it $40. BB (22/18 regular) calls.

Flop: A[img]/images/graemlins/club.gif[/img] 7[img]/images/graemlins/club.gif[/img] 5[img]/images/graemlins/heart.gif[/img] (pot = $85)
BB checks, HERO bets $70, BB raises to $250.

Ruh roh!

Alright, this is obviously a "player dependent situation". But lets not cop out completely, and do a bit of analysis.

Analyzing the line of calling and "re-evaluating turn" is much more complicated than analyzing shove/fold so for now lets assume that our two options are to shove or to fold. We can work in the calling option later.

Second of all, lets define a range of hands where villain is doing this for value with intentions of getting all-in (calling a shove). Keep in mind that these must all be within villain's preflop calling range from the BB in this spot.

Hands he would want to felt here are presumably: 77, 55, 75s, A7s, A5s, 86cc, 98cc, 64cc, 65cc. I think we can safely assume that if he did check-raise top pair for value, he's folding to a 3bet shove without metagame... but if you don't buy that, then we'll just assume he doesn't CR TP for value. Due to the fact that he might fold/3bet some of the suited junk in this range, we'll say he can show-up with them 75% of the time he's dealt that part of the range by leaving out a hand.

PokerStove tells us we have 24% equity against his felting range on this flop. By shoving we are putting up $890 to win $1115 and thus need 44%+ equity so assuming he never CR-bluffs here, shoving is bad. How bad?

How Bad = 0.24*($2005) - $890 = -$406.60

Now lets look at the other scenario after you shove... he folds! In this situation you will win exactly the pot of $405.

He folds = +$405

The numbers (coincidentally) indicate that you need your opponent to fold atleast half of the time you push for this play to be profitable.

So when does he fold? Well.. we are assuming that if he decided to check-raise a weaker draw (flush/straight draw) he will fold to a push. (He'd be getting roughly 38% equity so we'll assume he plays *correctly* and folds out the weaker draws). He might also choose to CR weaker pairs to "see where he's at". And most notably, he's going to sometimes CR complete air. This is where the "player dependency" comes into play. Everyone and their grandmother knows that the pre-flop aggressor is going to fire a continuation bet most of the time. So some people (and some of their grandmothers) counter this by CR-bluffing players with high (possibly, too high) CB%s. This CR-bluff% is what we want to know. My personal intuition tells me that this stat ranges very much from player to player, yet our typical HUD stats don't give us much to base a decision on.

Nevertheless, we can still fiddle with ranges and numbers to get an idea of what %'s look like in our given situation.

PokerStove tells us that the villain's flop felting range (that we determined) is 2.1% of the total range (ie. any two cards) that he can dealt.

So now lets give the villain a pre-flop *calling* range in this situation where we raised from the Hijack with no other callers. This is again a player dependent spot but I think we can hammer out a reasonable range, and the fact that some people will opt to fold/3bet some of the hands in the range yet possibly add other hands means that the overall range% will still be roughly the same across the board. If we give a range of: 22-99, ATs-, AJo-A7o, 65s+, KTo+, K6s+, broadways.. (again obv I'm missing some but thats to counter the fact that some of this range might get folded/3bet).. and fiddle with it we see that a reasonable range is 15-19% of the possible dealt hands, so we'll say 17%.

So villains range on the flop before he CRs is 17% of the combinations. After he CRs, if he's never bluffing (always felting), he is at 2.1% of the combinations. That is to say, 2.1/17 = 12.4% of his flop range is CR-felting for value.

So for villain to fold (be CR-bluffing) more than 50% of the time, he needs to be CRing you here more than 12.4 x 2 = 25% of the time.. (half of the time he's felting, half he's bluffing and folding..)

This post doesn't have any clearcut conclusions about "what to do" or "when to do it" but I think it atleast gives some tidbits into what type of thinking you should be doing about situations like this. It might also get you thinking about optimal CR frequencies from the villain's perspective.