Talk:Fold equity

Opening comment

I don't like the first line: "Fold equity is a concept in poker strategy that is especially important when a player becomes short-stacked in a no limit (or possibly pot limit) tournament.[1]" This seems to imply that the concept of fold equity is more important in tournaments than in cash games. I don't have a problem with mentioning the importance of fold equity when a player is short stacked in a tournament but I don't think it should be mentioned in the first line. Later on in the article would be more appropriate. With the line in the first sentence, it may prejudice the reader to assume that fold equity is not "especially important" in cash games when, in fact, it's a critical concept. A cash game player may read the first line and think that the concept doesn't apply to them and quit reading. —Preceding unsigned comment added by 64.33.169.228 (talk) 18:30, 28 May 2008 (UTC)

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This isn't really accurate. What's listed is an example when fold equity comes into play, but not the only one.

Fold equity is essentially the amount of expectation you gain by the possibility that they'll fold to a bet. The chance that they'll fold to a bet or raise adds value (fold equity) that wouldn't be there for a call. —Preceding unsigned comment added by 68.108.174.102 (talk • contribs)

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I agree. The current article is rather vague on the concept of fold equity. I think we can't avoid some math to reach a more precise definition. Don't want to overhaul the whole article without reaching consensus that we indeed need to do so. So I present on this talk pagew hat I think is the correct concept. (Not written in Wikipedia style yet, but can work on that later.) Here goes:

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.

JocK 00:30, 17 December 2006 (UTC)

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I think JocK's explanation is too long and confusing.

My problem with the article is just that the example doesn't take into account the fact that the pot size increases when you get called. In the example given, your "hand equity" may increase from 31.5% to 80% when you bet, but that doesn't necessarily mean that your expected value has increased at all, in fact it could have actually decreased! The point of poker isn't to win hands, the point is to win money. If you based your play on the assumptions of this article, you would think that you should just go all-in on every hand, because of the added fold equity. The problem is, if you do this you will win many small pots (when your opponent has nothing and just folds), but you will lose all the big pots (your opponent will only call you when he knows he can win). Deepfryer99 18:08, 15 November 2007 (UTC)

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Deepfryer99, I think we are saying exactly the same. I have updated the article to correct the issues highlighted above. (In a - hopefully - simpler way than expressed in my text above.) See what you think. Cheers -- JocK (talk) 18:03, 26 February 2008 (UTC)

It looks a lot better now. However, I think there may have been a problem with your new example. Instead of this:

70% times (1 - 0.315 X 2), shouldn't the fold equity be equal to this?

70% times (1 - 0.315 X 3)

Because, when you make a pot-sized bet and get called, the new pot is 3 (not 2) times the size of the original pot. I adjusted the percentages accordingly. Let me know if this looks right to you. Deepfryer99 (talk) 16:51, 29 February 2008 (UTC)

Let's see. I think the following describes the two situation (EV when betting compared to EV when checking) correctly:

EVbet = -Bet + Pf (Pot + Bet) + (1 - Pf) E (Pot + 2 Bet)

EVcheck = E Pot

It follows that:

EVbet - EVcheck = (2 E - 1) Bet + Pf ((1-E) Pot - (2 E - 1) Bet)

The first term is the 'value betting' contribution to your equity, the second term represents the fold equity. For a pot-size bet (Bet = Pot) this results into a fold equity of Pf (2 - 3 E) Pot.

I think we need to expand on this in the article. -- JocK (talk) 19:56, 29 February 2008 (UTC)