Introduction
What is fold equity? Very basically, fold equity is the probability that a villain will fold his hand in the face of a bet or shove by Hero. Note that it is completely different from pot equity, which is the probability at the current time that you win a pot if it were to go to showdown with no further betting. Various other definitions of fold equity are out there, but I find this to be the most direct and logical meaning. When you hear someone say “You have no fold equity”, they basically mean there is zero chance that a villain will ever fold if you bet or raise. For this article, I’m specifically addressing fold equity as it relates to no limit hold ‘em cash games. The concepts here are basic and meant to serve as an introductory lesson to fold equity.
Theory: Fold Equity Equation
So how can the concept of fold equity be used to help our game? To eventually achieve this understanding, some theory is in order. First of all, to define fold equity mathematically, we can rely on an equation that helps us find the breakeven fold equity of a given situation. For example, say you are heads up in a no limit hand on the flop against a villain who decides to check and the action is on you. We want to find out how often we need the villain to fold to make a jam breakeven expected value (EV). The following equation represents this scenario:
PF + (1-F)(-VSl+ HSw) = 0
Here we are saying the expected value of a villain fold plus the expected value of a villain call must equal zero. The probability that the villain folds (F) times the size of the pot (P) is the expected value of our play when the villain does indeed fold. The expected value of our play when the villain calls is the amount that we win (Sw, the sum of P and the amount the villain calls) times our pot equity (H), plus the amount we lose (Sl, our bet) times Villain’s pot equity (V). Note that for this equation we are assuming that we are covered.
So, since we know all variables except for F, using some seventh grade algebra we can solve for F and come up with our breakeven fold equity for various scenarios. For a given pot equity, if a villain folds more often than this, we gain, if Villain folds less often than this, we lose. You can find various online calculators that compute this for you given certain scenarios, for example:
http://dailyvariance.com/fe-calculator.php
This is great, but it is more helpful to look at a graph to see how this equation behaves so that we can get an intuitive feel for how fold equity actually works in relation to our pot equity. The following shows how breakeven fold equity changes versus real pot equity for three different scenarios.
On the graph there are three curves, each representing a different ratio of the amount bet to the pot size. These are roughly representative of three different heads-up situations in a 200NL game. The main objective here is to see how three very different bet sizings in no limit appear to behave regarding fold equity. In the “overbet” case (the blue curve), we are shoving $155 into a pot of $90. In the “pot bet” case (the red curve), we are betting $130 into a pot of $140. Finally in the “1/2 pot bet” case (the green curve), we are betting $100 into a pot of $200.
You will probably say, “But wait, I don’t really know my pot equity, I can’t see the villain’s cards.” This is of course entirely true. But in many scenarios it is possible to fairly accurately estimate your pot equity given the villain tendencies and the line he has taken. So for this analysis let’s assume you have a rough idea of your pot equity in a hand. At the very minimum, you usually know when you have none versus some.
The points on the Y-axis where the curves intersect represent how much FE you need to break even in the scenario where you are drawing dead ( i.e., you have zero pot equity). This may be the case on the river, for example, where there are no more cards to come and you are sure the villain has at least a pair and you have nothing. Another case may be on the turn where you have undercards to the board with no straight or flush draws and you think Villain has at least a pair.
The points on the X-axis where the curves intersect represent the point where you no longer need any fold equity because your pot equity is now enough to justify your bet. An example of this would be if you have a huge draw on the flop with 45% pot equity and you shove with two cards to come. Once you get to a certain percentage of pot equity, the villain can never fold and it will still be +EV.
Everywhere in between assumes you have some fold equity and some pot equity. A couple points should be highlighted. The graph really shows how having even a little pot equity helps our bluff cases immensely. For example, in the overbet case, simply having 25% equity reduces the fold equity you need from almost 65% to about 40%. Similar effects exist in the other cases as well. This is why semi-bluffing is so much better than stone bluffing. You have some insurance against your bluffs failing. And when that insurance pays off, you win a massive pot.
When some players first start playing no limit, it is not clear to them why shoving big draws with meager equity is actually profitable. They think, “I’m winning less than 50% of the time, how can this possibly be profitable. What they are not considering is the large amount of money won when a shove gets called but sucks out anyway.
Fold Equity and Bet Size
So how do most players decide whether or not to fold? There are multiple hands that villains can hold. Generally, at some particular bet size as bet size increases, villains will start to rapidly fold more and more hands. For a simple case, take the case of a bet on the river after there has been action on previous streets. Usually this starts to happen around a half pot bet. Once the bet size gets past a pot size bet and into overbet territory, almost all hands are folding except extremely strong almost nut hands or occasional bluff catchers. This concept is represented by a logistic curve, with the X-axis representing bet size and the Y-axis representing probability of a fold. For more information, I suggest No Limit Holdem Theory and Practice by Sklansky.
This phenomenon can be exploited in no limit in certain cases. For example, if you have the absolute nuts, and it is highly probable that Villain has a near-nut hand, you exploit the fact that against such hands you have almost no fold equity. You do this by shoving, even if your shove is a massive overbet. Due to the lack of fold equity, this is profitable when the profit from the overbet gained overcomes the extra money to be made by getting weaker hands to call with a smaller bet more often. The idea is, sometimes those weaker hands are never calling anyway or are not in Villain’s range, so the only time you get a call is with a very strong hand. Those times, you are free to set the bet size, so bet as much as possible. A common situation here is having quads versus a likely full house, or straight flush versus a flush. This could probably be analyzed with further mathematical rigor, but is left as an exercise for the reader. I suggest reading up on the Zeebo theorem for more information.
There are certain exceptions to the above graph. For example, you may find that a certain fish always bluffs with the bet pot button. In this case, the fish may actually have less fold equity against you then he would with a ¾ pot bet. Another exception that is relevant here is the “post oak” bluff, which is a concept turning the above idea on its head. The idea is that in some circumstances, a smaller bet may actually have more fold equity than a larger bet, because Villain may think, “This bet has to be for value, otherwise he would not bet so small. Therefore, I fold. I might have called if the bet were larger because bluffs are in his range in that case.” These exceptions are interesting, but for the most part you should retain the above as your fundamental model.
Using Fold Equity to Advantage
Against bad players who are bad because they are very passive calling stations, you have no fold equity. Use this fact by relentlessly value betting big, and simply don’t bluff.
Against good players who can hand read and are known to have the ability to fold, use fold equity by bluffing in spots where you can hold a strong hand and they most probably have mediocre holdings. Additionally, spots where you can represent a very strong hand resulting from the turn or river card combined with your line combined with a Villain line that is weak or indicative of a mediocre holding are great spots to bet big and let fold equity do the heavy lifting. Yes, you will occasionally get trapped, but for most villains the probability of a trap is so small given a certain line that it’s still fine to abuse them. I will again stress that to have fold equity, you usually want a villain that can actually think above level 1 and hand read. Overbet bluffing a river card that is terrible for a villain’s range when you have air is all well and good, but the old analogy of playing Mozart to a cow applies. If the villain has no ability to even recognize that his range is probably crushed based on board texture and lines, it’s probably a bad idea to bluff.
In no limit once you bet larger than pot, most solid players will start to think that the bet is polarized, meaning it’s either the nuts or close to complete air. In many cases this is true, but occasionally you will find players who will call slight overbets just as often as pot size bets. Abuse these players by getting more value with value hands. You also may find players who refuse to call overbets without the nuts. Amazingly many of these players do not adjust and will continue to allow themselves to be abused because they continue folding to overbets. Simply continue to abuse them until they fight back. It’s hard to have the nuts.
In certain situations you may use image to manipulate fold equity to your advantage. This is probably most applicable in a live setting where everyone is paying attention to every hand, but may also apply online if your villains are paying attention and not mass tabling. For example, if you getting caught running a huge bluff at some point, there will probably be period of time where your fold equity against villains may diminish significantly. This can be exploited if you do indeed pickup a monster because your monster will get paid off lighter. As well, the inverse case can be exploited. If you have shown down very few hands and the ones that were shown down were indeed solid, it will be much harder for villains to call down bluffs, so feel free to bluff until your FE gets reduced due to people seeing you get caught. There can be some leveling involved here, but due to psychological phenomenon of recency effects, most people tend to remember the most recent events. Remember that what people expect will usually be based on what most recently happened.
On Stakes and FE
Some may disagree with this, but I’ve found as you move up in stakes you slowly transition from having betting for value be your bread and butter to having more and more bluffs make money. As players become stronger they will be reluctant to put in a ton of money with weak/mediocre hands. Value becomes more scarce, hence bluffing becomes more profitable. The reason for this is that there is more fold equity to be found as players become better thinkers and hand readers and know that they should fold in certain situations.
Fold Equity in Practice: Examples
Here are a few examples where the concept of fold equity is important in actual play.
Example 1: Fold Equity Preflop
A very standard play is isolating a weak player with a substandard hand because of the fold equity involved. With no fold equity the play would not be profitable because your hand strength is just not that great. For example, you may find fish who limp/fold constantly and/or fish who may sometimes limp/call but then fold to 85% of cbets. Find out who these fish are and then isolate them with marginal hands letting fold equity make the play profitable. In this hand, I know that the villain limp/folds and folds to cbets way too often, so I decide to isolate him with a marginal hand, knowing that most times I will win the hand without even worrying about my actual hand strength.
Full Tilt Poker $1/$2 No Limit Hold'em - 9 players
2+2 Hand Converter
Powered By DeucesCracked
Hero (BTN): $307.15
SB: $260.50
BB: $152.00
UTG: $183.50
UTG+1: $914.30
UTG+2: $140.65
MP1: $199.10
MP2: $294.10
CO: $74.90
Pre Flop: ($3.00) Hero is BTN with Q
7
3 folds, MP1 calls $2,
2 folds,
Hero raises to $10,
2 folds, MP1 calls $8
Flop: ($23.00) J
A
8
(2 players)
MP1 checks,
Hero bets $14, MP1 folds
Final Pot: $23.00
Hero wins $21.85
(Rake: $1.15)
Example 2: Semi-bluffing
In this example, the hero uses the tailwind of pot equity combined with fold equity to win the hand. This example represents the middle part of the graph where we have both some fold equity and some pot equity. This line is profitable here because sometimes the hero makes a hand, and sometimes the villain folds. Only the combination of the two makes the line profitable. It should be noted that we must be sure the villain is good enough to fold at some point, i.e. not a complete fish.
Poker Stars $50.00 No Limit Hold'em - 9 players
2+2 Hand Converter
Powered By DeucesCracked
UTG+2: $66.55
Hero (MP1): $50.75
MP2: $97.00
CO: $62.45
BTN: $43.50
SB: $60.75
BB: $50.00
UTG: $65.60
UTG+1: $18.50
Pre Flop: ($0.75) Hero is MP1 with A
5
2 folds, UTG+2 calls $0.50,
Hero raises to $2,
5 folds, UTG+2 calls $1.50
Flop: ($4.75) K
4
9
(2 players)
UTG+2 checks,
Hero bets $3.00, UTG+2 calls $3
Turn: ($10.75) 3
(2 players)
UTG+2 checks,
Hero bets $6.00, UTG+2 folds
Example 3: Shoving with Draws
Here we flop a hand with decent equity, given our nut flush draw outs and gutshot. When the villain raises we can put him on a range that includes enough bluff raises that we should 3-bet and get it in given our equity and the chance that we have some fold equity here. I will leave the Pokerstove as an exercise for the reader, but we can comfortably estimate our real equity at around low to mid 40s.
Full Tilt Poker $1/$2 No Limit Hold'em $0.30 Ante - 8 players
2+2 Hand Converter
Powered By DeucesCracked
SB: $310.90
BB: $200.00
UTG: $114.95
Hero (UTG+1): $230.10
MP1: $138.70
MP2: $199.65
CO: $100.90
BTN: $200.00
Pre Flop: ($5.40) Hero is UTG+1 with 5
A
1 fold,
Hero raises to $8,
1 fold, MP2 calls $8, CO calls $8,
2 folds, BB calls $6
Flop: ($35.40) J
2
4
(4 players)
BB checks,
Hero bets $22,
MP2 raises to $100, CO folds, BB folds,
Hero raises to $221.80 all in
Summary
Hopefully this article has given you some better understanding of what fold equity is and how it can be used to profit. So try using the concepts in this article and don’t be discouraged when you have a bluff that gets called and you lose a big pot. If you have valid reasoning for the play (as opposed to spaz shoving) it’s usually +EV. Having the confidence that the play was well thought out will prevent you from tilting and saying that a play was horrible after a bluff gets called and going on tilt because of short term results. Good luck.
P.S. Thanks to damntra for much help in reviewing, and joopjan for a bit of help on the samples.
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