Sunday, December 11, 2011

Fold equity and Float equity



I once heard Antonio say,"Poker game theory dictates that no one ever makes a hand. So just bet and win!"  While I understand the subtle truth behind his sarcasm I wanted to quantify it. I took a reasonable range of hands for a Tag  and a Lag in holdem' and actually calculated the percent of times that they hit these flops with hands that they would want to continue with (straight draws included). This is a rough guideline and it ignores flush draws but I believe it is a good reference.  The percentages represent the "steal ability" of the hand. For instance The Lag will miss and probably fold to a 3 bet on a 552 board 78% of the time.

         
FLOP                    TAG                   LAG
A72                        47  %                  66%
K72                        59 %                   63%
Q72                        51 %                   67%
J72                          61 %                   53%
T72                         73  %                  61%
552                          73  %                 78%
JT3                          48  %                 29%
QQ3                        57  %                 71%
This chart does however take into account card combo odds. For example while pocket jacks are in the Tag's raising range and there are 6 different combinations of jj ♠♣, ♠♥, ♠♦,  ♣♥, ♣♦, ♥♦. However if there is a jack on the board there are only three combinations possible. This is reflected in the percentages.

So what to do with this information? Its obvious that people do not hit low boards or paired boards often. So does this mean that you can steal often? I believe with good taste you can carve out money in some situations.
NL HOLDEM 1-2$
Situation 1:  classic
A couple of limpers and you limp in with jt♦, the button a standard LAG
raises to 11$ the two limpers call and you call pot is now 47$
flop is 552 rainbow
Action checks to the button who bets a standard 30$
The two limpers fold.
pot is 77$
(Of course this scenario requires that you have a decent image and enough effective stacks to make a bluff plausible, in this scenario well say that the villain has 250$ and you have him covered.)
So there is 77$ in the pot and we know that he is likely to fold to a 3 bet 78% of the time. or (3 times out of 4) or 3-1 so if you raised you would have to raise at least 30$ more meaning that you would put in 60$ I believe to raise here I would probably make it about 80$. so I'm risking 80 to make 77or about 1-1.
Obviously profitable.  3 times he folds for +77$ or 3*77=231
                                  1 time he calls for - 80$   or 1*-80=-80
                                                                231-80= 151$(in 4 hands)so 151 divided by 4=+37.75 per hand (or per situation)
This situation is very lucrative according to the math. I understand that this doesn't account for any other players  or re-bluffs or other intangibles. I would like to think that these pitfalls are cancelled by the value created on future hands. Such 3 betting "light" is easily observed by the table and if you are looked up your image has become that of an "action" player. With this in mind lets say that there is one more player behind you and he limp-called pre-flop so his range is something like 22-99, qj-qt, jt-j9,kj-k9 a9-a2suited (288 combinations) and 22 55 and A5 are the hands that he will call your 3 bet with or (16 combinations) that's about 6% of his range. So the equity of this hand has gone down from 78% to about 72%. With out laying out the math its easy to recognize that this does not make the play negative expectation. It should be noted that a 3-bet into 2 players is more credible than a "heads up" play.
Also if you are in position you can float one bet and bet 40 on the turn. Or maybe you feel strongly enough about the situation that you can float out of position. The math generally stays similar except for the chance that  he will hit one of his overs (13%). But in reality you don't like any card higher than  a jack ( you really do not like a jack because the type of hand we are saying he is most likely to have is one that frequently has you dominated) so the  actual odds of an unfavorable card is 15/47 or 32-15 or 32% or about 2-1. So the float play (if you could say for sure that if you flat-call the flop then 90% of the time he folds the turn if no scare card comes is 90% of 68% of 78% or about 48% equity. so about 1-1 which means that flat calling the flop should be about even money. Which once again I believe is still positive ev because of the implications on future hands.
     I would like to make clear that this is a simplistic idea. It was only meant to give an idea of how comfortable you should feel in these situations. If you had a hand like AK in this spot you might (correctly)  feel like you are value betting instead of bluffing.

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