Fold equity

Fold equity refers to the value that a hand has simply by betting or raising such that all opposing hands, including better hands, may fold. The term is typically used in the context of call-versus-raise or bet-versus-check decisions. Checking and calling inherently offer no fold equity, because your opponent will not fold to those actions. However, if you bet or raise, there is a chance your opponent will fold and that chance is your fold equity.

For instance, say your opponent has shown himself to be a weak player and tends to fold under pressure. You think you're behind, but you have draws. Normally you'd check, but you also figure this guy will fold about 30% of the time to a strong bet. In addition to the chance you have of making your draw, there's the estimated 30% chance he'll fold. 30% is your estimated fold equity in this situation.

Another example might be when a scare card hits and you're first to act. You have a bit of fold equity if that card didn't help anyone.

Theoretical Analysis
The fold equity of a bet is essentially the gain in raiser’s expected post-raise fractional pot take, resulting from the likelihood that one or more of her opponents fold to the bet.

The concept is easiest explained for the situation in which the raise represents an all-in move of the shortest stack. Let’s assume I am short-stacked in a betting round with several active opponents. All players before me check, and I decide to move in my whole stack. It could happen that all opponents call my all-in, thereby creating a total pot of size S. This is the reference case which has fold equity zero. In this reference case, the expectation value of my hand can be written as a constant E, the show-down equity of my hand against all opponent’s hands, times the size S of the pot: EV = E S.

However, following my all-in, odds are that one or more of my opponents might decide to fold rather than call. In that case my expectation value EV’ will differ from the reference value EV as A) the total pot available to me will not have increased to S, but to some smaller size S’, and B) because due to some opponents folding, the show-down equity of my hand has increased to E’. We define the fold equity FE of my all-in move as the difference in expectation value between all opponents calling and not all opponents calling, measured in units of the reference pot size S resulting from all opponents calling: FE = (E’S’ – E S)/S.

The mathematics is easy to work out in case of heads-up situations. If at some moment in the game I go all-in with a stack of size B that is called by my opponent thereby creating a total pot S, my expected take will be

EV = E S

However, if there is a chance Pfold of my opponent folding to my bet, my expected take changes into:

EV’ = Pfold (S – B) + (1 – Pfold) E S

The two terms on the right-hand side represent the two distinct contributions to my expected take. The first term is the expectation that comes from my opponent folding, and the second term is my expected take resulting from my opponent calling and me winning the show-down.

It follows that the fold equity (EV’ – EV)/S of my all-in is given by:

FE = Pfold(1 – B/S – E)

As 0 < 2B < S, the ratio B/S is positive number not exceeding 1/2. Hence, if I am the underdog with a show-down equity E < 1/2, the fold equity of my raise is always positive (independent of the size of my bet).

However, if I am the favourite (i.e. my show-down equity E > 1/2), there is a critical stack size, above which the fold equity of my all-in turns negative. This critical stack size follows from equating B/S to 1-E (i.e. from demanding that the pot odds B/S for calling your all-in equals the pot equity 1-E of your opponent). This critical size of the all-in move can be written in terms of the pre-raise pot So = S – 2B as: B = So (1 - E)/(2E – 1). All-in moves larger than the critical move have a negative fold equity. Such negative fold equities are associated to hands that are favourite in the show-down and do not improve substantially by one or more opponents folding.

The conclusion of all this is that the fold equity of my all-in move is largest when I am the underdog (E < 1/2), and my all-in bet B is large enough to yield a decent Pfold, but not so large to cause B/S to raise too close to 1/2. Hence, all-in bluffs with stacks just deep enough to be feared by the opponent, yield the highest fold equity.

Discussion threads

 * Fold Equity & Bluffing (2+2, 26 Sept 2005)