fold equity and bet sizing

[LawDude]

I was inspired to write this by the discussion over in the tilt forum about what to do about a flush draw on the board in no limit when Hero has the best hand.

There two aspects of fold equity that are tremendously significant. First, it plots as a bell curve. Second, it is partly probabilistic and partly psychological. When you think about these two aspects of it, it may not give you the answer as to when and how to bet to induce folds or force calls at incorrect odds, but it may at least get you asking the right questions about it.

First, the bell curve. Fold equity is obviously at zero on a check (unless you count the dumbbells who occasionally muck their hands even when it is checked to them). Less, obviously, though, fold equity on a shove will only be at its maximum in those situations where the Villain was already convinced that you had the best hand.

Why is this? Because a bet to induce a fold, like any other bet, is a bet with ane expected payoff, in this case, the pot size. When you shove, you are betting the largest amount possible to collect the same payoff and thus you are getting the lowest possible odds on that payoff even when you induce the fold.

Thus, to collect the best price on your money, theoretically, when you want to induce a fold, all other things being equal, you would want to bet the minimum amount necessary to induce that fold. If you could induce that fold betting half the pot, you are getting a better payoff on your money than if you induce that fold betting all the pot. If you bet below that amount, you don't induce the fold. If you bet above that amount, you are reducing your payoff odds on your bet. You are at the crest of the bell curve.

(In addition, betting too far past the bell curve may start looking like a sign of weakness rather than strength-- a bet that is "too large to be believed" may in fact invite callers.)

And the reason those odds are important is because in some instances, your reads will be wrong and the player will stay in the hand and beat you. Whatever percentage of time that happens, it will be cutting into your profits on your successful induced folds.

There is, however, one exception to this. And that is when you put Villain on a draw. The optimal folding price might be less than is necessary to price Villain out of his draw. If that is the case, then you have to bet enough to price Villain out of his draw, even if that is more than the optimal amount to induce a fold.

So now, we have a more refined rule: when seeking to induce a fold, bet the minimum to induce that fold unless you believe Villain has a draw to a better hand, in which case bet enough to price out the draw.

Finally, though, this is only the math. The psychology is key. Who folds to aggression? Who calls? Which players will tend to read a large bet as a sign of weakness rather than strength? One of the things that killed me for awhile after I first discovered the mathematical concept of fold equity was placing these bets and watching people call me (or even raise me), knowing they had the best hand.

Probabalistically, x percent of players with better hands will fold to bet y in situation z. But what about THIS player? Your fold equity against this player with the optimal bet may actually be zero (because he knows he has you beat or because he is a calling station) and it may actually be much greater than you think it is (thereby necessitating a smaller bet).

This is where observing the table and doing reads is once again key. When someone shoves or overbets the pot in a no limit game, I observe everything that happens. I want to see how the other players in the hand react, and I want to know what the hands are. I also construct my own reads of what the players have or their ranges, both to see if I have identified their tendencies and also to see what works against them in what situations.

Eventually, if I have sat at a table long enough, I want to have a bell curve in my own mind for each player. What will I need to bet to get them to fold a better hand? How likely will it be that they will do it? If they call or raise me, what does that tell me?

[Otter_Brothface]

I am confused by all of this. If we bet too much to get someone to fold all the time it means that the times we are wrong and he has a strong hand is very costly to us. So maybe if someone doesnt often fold we should not bluff them instead of betting more.

Otter

[LawDude]

I am confused by all of this. If we bet too much to get someone to fold all the time it means that the times we are wrong and he has a strong hand is very costly to us. So maybe if someone doesnt often fold we should not bluff them instead of betting more.

Otter

If someone never folds, then you should never bluff them. If someone has great reads on you, you should also rarely or never bluff them. You need to read those types of players and assume no bell curve for them.

But most players will fold in situations they perceive to be unfavorable. They may not have great reads, and they may not make mathematically correct folds, but they will fold if you make the hand too costly. The trick to taking advantage of fold equity is to make a reasonable estimate of what their "price" is.

[loonychune]

I think people are going to be confused by the concept of bell-curves without seeing what they are.

I have just MatLabbed a quick idea as to how we might conceptualise fold equity using the Bell Curve.

The dotted lines represent the optimal bet sizes necessary to get certain players to fold.

The curve representing a FISH, where optimal bet size is assumed to be 1.2x the pot, is going to be shorter and wider, since:

i) he folds less
ii) there is a wider range of bet sizes that will get him to fold since his decisions are usually based on his hole cards with respect to the board.

The curve representing a TAG, where optimal bet size is assumed to be 0.8x the pot, is going to be taller and narrower, since:

i) a TAG will fold more often when he misses the board
ii) a TAG will peel if you bet too small and find something suspicious if you bet too much. In this example, 0.8x is neither too big nor too small.

This is a vague general overview. The heights and widths of the curves representing each individual player are situation-dependent, so the curve will look different in different spots (they will also be shifted left or right depending on the amount to be bet in that spot).

[LawDude]

Thanks Loony-- that's a great illustration!