Bluffing and Fold Equity

The romantic story about bluffing in poker is that the sly shark senses weakness in his opponent.  A sweaty brow, dilated pupils, shaky hands all clues to the expert player that his rival doesn’t have as strong a hand as he is representing.  Acting on this sixth sense our hero makes a big raise at a critical point in the hand holding perhaps only 8 high and his pray folds.

The truth about bluffing is much more mundane and is, of course, embedded in very simple math.  A successful bluff is not trying to outsmart an opponent holding top pair or getting players to make bad lay downs.  Bluffing is simply determining the range of possible hands an opponent won’t showdown with, figuring out what percentage of those hands are among all the possible hands he could have and if the pot is laying me good enough odds given that percentage to make my bluffing bet worthwhile.

Another conventional piece of thinking about bluffing is that it’s just not possible in limit hold’em given the small size of the bets relative to the pot.  As I’ve talked about before, hard and fast rules in poker lead to habits that less talented players fall into that cost them a lot of money in the long run.  Using the above analysis bluffing is absolutely possible in limit hold’em.  In fact, if players aren’t incorporating bluffing into their play the results will ultimately be devastating to their bankroll.  Here’s an example:

Hero is in the cutoff seat with A-J off suit.

UTG + 2 limps

Highjack limps

Hero raises

Button calls

Big Blind calls

UTG + 2 calls

Highjack calls

Five players 10 small bets.

Flop 9d-8d-2c

Checks to Hero who bets

Button and BB fold

UTG calls

Highjack calls

Three players 13 small bets

Turn Ks

Checks to Hero

At this point there are 6 ½ big bets in the pot.  Assuming at least one opponent will call there will be nearly eight big bets by the end of the hand.  To bluff out the hand it will cost me only two big bets at this point, so I’m getting 4:1.  I may be up against a pair an 8 or 9x up to a king or ace.  Also very likely is a straight or flush draw that threatens to brick off since a disconnected K rolled off.  So, probably around 30% or so of the one opponent who call’s range he is unlikely to show down with (assuming the flush or straight draw doesn’t come in).

Another element of this analysis is the idea of fold equity.  On top of the 30% or so of hands an opponent won’t show down with in the example above is the 10 or 15% chance that he could fold.  Keep in mind I raised preflop, C-bet and a king rolled off on the turn.  Let’s say my opponent has 8-10 and peeled the flop.  What happens if a Q comes off on the river?  What is he going to think of his pair of 8’s?  Now getting close to 40% that my opponent either won’t show down or will fold, my 4:1 to bluff out my hand has become the mathematically correct play.

It’s important to note that 40% is not 100%.  I’m still loosing more than half the time in a spot like this.  But, the math shows that I don’t have to win that often to make my betting correct.  If this situation was played out over and over 100 times with the same flop, my same cards and random cards making up my opponent’s hands, I will come out very far ahead because of the size of the pot.  I will only win 2 times in five but the amount of those wins will more than compensate for the times I lose.