#fold equity

It is a general rule that you should not try to bluff more than 1 or 2 opponents. Therefore, you are more likely to find bluffing opportunities in a Heads-up game than in a 6max or full Ring game. But, for a Ring game, what if all opponents but one fold? Then you are essentially heads-up so should that be equivalent to a two-man game? The answer is no and here’s why.

In a Heads-up game, neither player is likely to have a strong hand so players are forced to play weak hands more often. Therefore, with a weak hand, they will often have to win by inducing a fold through bluffing. In a game with more players, it is likely that at least one has a strong hand so a bluff is less likely to succeed. Thus, in a Heads-up game, your one opponent probably has a weak hand whereas in a Ring game, your one opponent probably has a strong hand. The former calls for bluffs while the latter suggests otherwise unless special conditions exist.

For a final thought, assuming a bluff has zero equity, the general rule for bluffing is the following:

If villain fold frequency is greater than Risk/(Risk + Reward), then a bluff is profitable, where Risk is the amount you bet and  Reward is the pot before you bet; thus –

Bluff if Fold Equity > Bet/(Pot+Bet).

The following table/graph displays some results.

Example: If you bet 0.75 Pot, with an air-bluff hand, you need fold equity of at least 43% to be profitable.

Introduction

In the process of developing a fold equity software program, I looked at extreme values for card equity and fold equity to see how they affect EV. The following formula is applicable for a hero bet assuming villain’s call amount equals the bet amount:

EV = FE * Pot + (1-FE)*(eq*(Pot + 2*Bet) – Bet)

where

FE = fold equity

eq = card equity

Pot is the pot amount before hero bet

Bet is the bet amount which villain must call

FE for Equity Extremes

The following shows EV outcomes for the extreme values of FE and equity and for each at 50%, assuming Call = Bet.

  We note the following:

Hero EV is always negative if both FE and eq equal 0%, since villain never folds and hero never wins

Hero EV always equals the pot if villain always folds

If hero equity = 0 he can still have positive EV if villain folds enough. For FE = 50%, hero EV is positive if the bet is less than the pot

EV is always positive for equity >= 50%. At equity = 50%, EV cannot exceed the pot and at equity = 100% EV will always be at least equal to Pot.

If villain never folds, FE=0, hero’s EV range is -Bet to Pot+Bet as his card equity ranges from 0 to 100%

Minumum FE for +EV (EV>=0)  When Card Equity < 50%

It can be shown that the critical minimum fold equity value to assure +EV when your card equity is less than 50% is the following:

FEcrit = (Bet-eq*K)/(Pot+Bet-eq*K),

where K = Pot + Bet + Call

Example: If hero bets 30 into a pot of 40 and he estimates his equity to be 25%, then the minimum fold equity he needs assuming villain must call 30 is as follows:

K = 40 + 30 + 30 = 100

FEcrit >= (30 – 0.25*100)/(40 +30- 0.25*100) = 5/45 = 11%

So, even though hero has only a 25% chance to win at showdown, if villain folds at least 11% of the time to a 3/4 Pot bet, hero will show a profit.

The table below present the critical minimum fold equity for various card equities and bet sizes. Blanks mean EV >=0 for any villain fold probability. Note that for card equity > 50%, hero always has positive EV. Even if hero has complete air, he can still have positive EV if villain folds enough, the underlying case for bluffing along with the need to have balance.

It is seen that for the low card equity situation, the minimum fold equity increases with the bet size. That is somewhat tempered by the fact that a player is more likely to fold as the bet size increases. Here is where the art of bet sizing comes into play.

Summary

Folding is an important factor is doing an EV analysis. We showed in this article how it interacts with card equity and bet size to define EV. Fold equity is included in a number of other blog posts. Four in particular are the following:

When Is A Villain Fold Better Than A Call?     

Bet Sizing. Two Case Re Fold Equity.

 EV Analysis When Fold Equity Is Greater Than 50%

The Cost of Folding

You are short stacked with a drawing hand. Should you shove?

Introduction

To answer this question for preflop, there are so-called Push-Fold charts, which assume heads-up players can only push or fold. The recommended decision is derived using the equilibrium criterion, which depends only on the stack size and your hand. Here we consider an exploitative strategy where you can estimate either your card equity or fold equity for the current street to arrive at a +EV decision based on shove size and an equity estimate.

The EV Model

The applicable EV equation is as follows:

EV = fe*P + (1-fe)*(eq*(P + 2*B) – B))

where

fe = fold equity

eq = card equity

P = pot before hero bets

B = shove amount

We consider the following two cases and use as an example: P=100, B= 300:

(1) estimating required card equity given fold equity  

(2) estimating required fold equity given card equity

Case 1: Assuming a given fold equity, the minimum card equity that provides +EV is the following:

eq_min = 1/(P+2B) * (B –fe*P/(1-fe))

Example: Assume you estimate the fold equity to be 20%

eq_min = 1(/100+2*300)*(300-0.20*100/(1-0.20) = (1/700) *(300- 25)=39.3%

(As a check, EV = 0.20*100 + 0.80*(0.393*(100+2*300)-300)= 0)

Case 2: Assuming a given card equity, the minimum fold equity that provides +EV is the following:

fe_min = (B- eq*(P+2B))/ (B+P-eq*(P+2B))

Example: Assume your equity estimate is 40%.

fe_min =(300 - 0.40*700)/(400-0.40*700) =20/120 = 0.167

If you are bluffing with air so that eq = 0, the following holds:

fe_min = B/(B+P), a term known as Alpha, which is the fold equity a bluffer needs for break-even. For the example, with card equity = 0, villain has to fold 300/400 = 75% of the time.

We note that 1-Alpha is known as the minimum defense frequency (MDF), the minimum frequency with which one should defend to prevent his opponent from profiting with a pure bluff (eq=0). Alpha and MDF are most applicable for river bet situations.

Results

The following tables provide results for various combinations of bets, in pot size units, and card and fold equities

.Example: If you estimate that villain will fold to a shove 1/3 of the time with a bet of 2 times the pot, you need card equity of at least 30.1% for +EV.

Example: If you estimate your card equity to be 33% and your stack is about twice the pot, you need villain to fold to a shove about 26% of the time

Summary

While the focus of this article is on shoving, one can use the results for a non-shove bet, recognizing that future betting is not considered and therefore the result is exact only if the hand is checked down. Nevertheless, it does provide a first-cut look at profit potential, which should then be modified for factors not directly included in the model. One of these factors is that the card equity estimate should be realized equity if future action will occur. While a low pair and high suited connector may have approximately the same showdown equity against a certain villain range, the low pair is much more likely to be folded if a set is not hit on the flop, so the suited connector will almost surely have a higher realized equity. Another factor to consider is that fold equity generally increases with the size of the bet so a small bet, say ½ pot or less will typically have a fold equity lower than if the bet was twice the pot or greater.

You’re in a heads-up situation and out of position on a later street. Suppose you decide to bet for you don’t want to possibly give villain a free look at the next board card if he is on a draw. You know that villain may call or fold and that the larger the bet the more likely he will fold. On the other hand, if villain has the better hand, you will lose more with a larger bet. You also know that if your bet is a bluff, a fold is obviously the desired outcome.

Let’s do some math.

The EV equation in this situation is as follows:

EV = Pr(V folds)* Pot + Pr(V calls)*EVcall

EV = fe(B)*Pot +(1-fe(B))*(eq*(Pot + 2B) – B),

where

fe(B) is the probability villain folds when you bet B, which is also the amount villain must call

Pot is the pot before your bet

eq is the chance you win if villain calls

We seek the bet size, B, that maximizes our EV. But to do that we have to quantify fe(B). As noted earlier, players will normally fold more frequently with a larger bet size. However, if they have a very good hand, that is not true and it might not be true if they suspect you are bluffing or betting with a marginal hand hoping to cause a fold.

We’ll consider four cases where your hand is basically a bluff,  a so-so hand, a 50-50 race hand and a monster hand represented by equities of 10%, 33%, 50% and 90%, respectively. We’ll also assume the following fold equity function where the bet, B, is in pot size units:

fe(B)= B/(1+exp(B/2.7)

The fe function graph with a table of values follows:

We see that fe increases with bet size until about a 3.5Pot bet and then fe starts to decrease as villain suspects a bluff. At a ½ Pot bet, villain folds about 23% of the time and his maximum folding probability is about 75%.  Solving for the break-even bet size (one that sets EV to 0) analytically is a challenge I’d rather not tackle. Instead I solved for EV for various bet sizes for each of the assumed equities and the results are shown below:

The following is noted. A bet of 0 means hero checking; it is assumed the hand is then checked down. For each equity, the bet size that maximizes EV is bolded. It is 2.75Pot for both the 10% and 33% card equities and increases to 3.5Pot for the 50% case. Even with a bluff hand of only 10% equity, EV can be positive, in this case for a number of middle bets.  For the two high equity hands, bet the effective stack. It is seen that EV increases with the card equity as would be expected.  It should also be noted that showdown equities were used so unless the bet was all-in, these results are just a first cut at the EV evaluation.

What about the question posed in the title – do you want a fold or call? Remember, a fold earns you the pot, defaulted to 1 in this analysis. So, a call is more desirable if EVcall > 1. Well, for no equity less than or equal to 50% is a call better than a fold. Even for equities over 50%, a fold may be more desirable. For example, if the card equity was 60%, only for a bet of 2Pot or greater is EVcall greater than 1.  Naturally, these results are based on the fold equity function we used and adjustments to that function should be made to reflect your opponent’s tendencies.

“I found out that Resource xxx[1] suggests folding 55% against 3x open in a heads-up match. Isn't this exploitable? I mean, if we are defending less than half the time, then villain should raise with any two cards simply because he is making money immediately. What do you think?”

We’ll consider this question but make our hero be the bettor.

If you are betting as hero and if villain folds 55% of the time heads up, then hero wins at least 55% of the time. Assume a pot of 1 and an open hero bet equal to Bet that villain either calls or folds:

EV = 0.55*(1) + 0.45*(eq*(1+ 2Bet) – Bet),

where eq is hero’s card equity.

The most pessimistic case is eq = 0;  i.e., hero never wins if villain calls. Then

EV = 0.55 – 0.45*Bet  

In this case, EV is positive if Bet < 55/45 = 1.22 Pot when betting with complete air.

What equity is required so hero is assured of winning, given the bet size?

EV = 0.55 + 0.45*(eq*(1+ 2Bet) – Bet) > 0

Eq > (0.45*Bet- 0.55)/(0.45 + 0.90Bet) = (9*Bet-11)/(9+18*Bet)

The following table and graph show some results:

Example. If villain is expected to fold at least 55% of the time, a bet of twice the pot requires card equity of at least 16% for +EV.

For a 3x bet, EV is positive if hero’s equity is at least 25%, which is generally true for any two cards preflop if the hand goes to showdown. So, poster is almost certainly correct in stating that a high folding probability exposes one to being exploited provided that the exploiter can realize his equity.          

Key Points:

In general, in any situation where hero has better than 50% showdown equity for a particular action, his showdown EV will always be positive. If fold equity is high, say over 50%, then in most cases a bet of 3x or less will be a +EV move. For larger bets, which may result in a higher fold equity, the table provides guidance on minimum card equity.

Future posts will generalize the relationships between card equity, fold equity and bet size relative to expected value.

Finally, whenever showdown card equity is included in an EV analysis, it is important to adjust a result or conclusion to account for equity realization;  that is, a showdown equity assumes hero does not fold in a future round, an assumption that may not be true.

The absolute nuts means you are 100% sure of winning the hand. Therefore, having such a hand at any point guarantees you win at least the current pot. We will assume you have the lead bet on the river. Your options are to check or bet. We will explore the math of sizing a bet so that you maximize expected value.

A general EV equation for a bet by you is the following:

EV = fe*Pot + (1-fe)( eq(Pot + Bet) - (1-eq)Bet),

where fe is fold equity (i.e., fold probability) for a bet of Bet into a pot of Pot with card equity of eq.

We conditioned eq to equal 100% and will assume Pot = 1, so Bet is in pot-size units.  Then,

EV = fe + (1-fe)(1+Bet) = 1+ (1-fe)Bet

This simple equation has a complication, namely the fold equity will generally be a function of the bet size. It is reasonable to assume a “lazy” S type function (    ) where fe, the fold equity, is small for small bets and approaches 1 as the bet size reaches multiples of the pot size. (If villain suspects a bluff with a very large bet, there may be a dip at the end).

We’ll use the following fold equity function:

fe = 1-1/(1+exp(4*Bet-4)).

Then

EV =1+Bet/(1+exp(4*Bet-4)).

The fold equity function and EV values for various bet sizes are shown below:

Clearly EV is positive for any bet size. EV is maximized when a 0.8Pot size bet is made and it is equal to 1.55Pot. The EV for various bet sizes is shown in the following graph.

 What about the check option? A check might be a good option if villain is likely to bet hoping to bluff you off the hand or if he believes he is ahead. Of course, the danger is that villain may also check thus resulting in a minimum win amount. Here the math has to be supplemented with good poker thinking. Is villain passive or aggressive? What has he done in the past in similar situations? Did the river card possibly help him so he might think he has a winner? Has he bluffed in the past? Is your winning hand pretty well concealed? If he does bet, you’re basically in the same situation as before in deciding now how much to raise rather than bet. Of course, you now have a bigger pot to win.

Let’s now further explore the fold equity function which plays a big part in determining your EV. The one we used says that villain will almost certainly fold as the bet size approaches 2Pot. It came about to ensure that at smaller bets the fold equity was about right, in this case a half-pot bet, for example, had fold equity of about 20%.

Clearly you cannot expect villain to always fold at 2Pot many times with you holding any two cards for he certainly will catch on and change his folding range.  Let’s now consider a more aggressive villain who folds as shown below:

  Again, fe is about 20% at a half pot bet but rises much more slowly than the previous function, never gets to 100% and actually dips down at about 4.5Pot indicating villain suspects a bluff at such a large bet. A graph of the resulting EV equation follows:

We see here that EV is maximized at 1.61Pot for a bet between 1.25 and 1.5Pot, a bet up to almost twice as large as the previous example with an increase in EV of only about 4% (1.61/1.55). The EV starts to increase again at about 4.0Pot because villain starts to call more frequently for very large bets but we remind the reader that we are assuming hero has the absolute nuts so villain calling very large bets still favors hero.

Summary

Formula della fold equity e blockers

In molti spot quando si subisce un re-raise è molto difficile capire se la mossa giusta sia il call, il fold o addirittura una 4-bet. Per capire con quali range poter effettuare una 4-bet dobbiamo assolutamente calcolare la folding equity di quello spot in quanto è essenziale che l’avversario abbia qualche possibilità di foldare una mano di partenza migliore della nostra.

Per risolvere questo…