Mathematical Concepts for No-Limit Holdem (3) - Fold Equity & EV Advanced

Introduction

In this article

• How a possible fold influences EV
  
• A close look at standard post flop moves
  
• Profitable 4-bet bluffing
  

In the second part of this series implied as well as reverse implied odds were introduced and calculated. You saw the first really complex EV calculation for a pre-flop call for set value. In this article, the focus will be on your opponents' moves, their associated ranges and possible counter moves.

Expected value taking possible folds into account

As bluffs and semi-bluffs play an important role in this article, it is important to introduce the possibility of an opponent's fold to the EV calculation. This formula:

EV = Equity * (Win + Investment) - Investment

will not be sufficient. There is no variable for an opponent's fold in this formula. We therefore need a more general formula to calculate the EV:

EV = Pfold * Pot + (1 - Pfold) * (Equity * (Win + Investment) - Investment)

The probability that you win the pot is Pfold; if you don't win the pot immediately we use the EV. Pfold is the probability that your opponent will fold. Under the assumption Equity = 0 the formula can be simplified to this:

EV=Pfold * Pot - (1 - Pfold) * Investment

Loss in this case is your continuation bet. This is the case for all pure bluffs. Given the definition of a pure bluff, you can only win if your opponent gives up.

Standard moves post-flop: Continuation Bet

After pre-flop raises, continuation bets are probably the most frequently used standard plays. When examining their EV, we particularly think about how frequently a player hits a hand of a given category. At the same time, we need to make assumptions about pre-flop ranges and post-flop behaviour given different hand categories. Let's look at a specific example:

SH, 100BB Stack size

Pre-flop: You are MP with XY
You raise 4BB, 1 fold, BU calls 4BB, 2 folds

Flop: (9.5BB) K, 5, 3 (2 Players)
You bet 7BB

The flop is fairly uncoordinated, and therefore a classic board for a continuation bet. Not taking into account that you should frequently make a hand here, we now analyse the following situation.

You always lose if BU stays in the hand. Is the bet still profitable and if so, how profitable? What assumptions need to be made?

The determining of ranges is not the designated topic of this article, but you can't find a way around giving BU a certain hand range. Using this example another concept is introduced. The frequency with which an opponent does something. It could be possible that BU would cold call with any pocket pair and only with 30% of suited connectors. Certain players obviously won't cold call with any suited connector whilst others will cold call all of them.

Without further information, there can still be more realistic results if instead of 0% or 100% of cold calls you use a value somewhere in between for a certain hand category. You can modify this parameter later on to see how the total EV varies.

In order not complicate matters, we assume there are three different possible hand categories for BU, with the relative probability of whether he will play them (p(Category)):

• Pocket pairs: 22-99: p(Pocket)
• Suited connectors 45s-KQs: p(SC)
• Stronger broadways AQ, AJ, KQo: p(BW)

How many combinations are there for each of these ranges? As calculated in the previous article, every pocket pair offers six combinations so there are 42 combinations for the pocket pairs.

Every suited hand can be built in four different ways so there are 33 combinations for suited connectors. There are 16 combinations for non-paired hands, so 41 combinations for the broadways.

The Button holds (hand combinations) with (probability):

The cold-calling probabilities lie between 0 and 1. This construction can be illustrated by looking at a player who only cold-calls pocket pairs. For him p(SC)=p(BW)=0 and p(Pocket) = 1. So the probability of him holding a pocket pair is:

42 * p(Pocket) / [33 * p(SC) + 41  * p(BW) + 42 * p(Pocket)] = 42 * 1 / (33 * 0 + 41  * 0 + 42 * 1) = 1

What will BU do when faced with a bet? This depends on his own hand which raises the question: How frequently will BU hit one type of hand with his range?

The given ranges open the possibility for BU to either hold nothing, i.e. less than ace high, or hold a gutshot with 67s or ace high itself.

As another option is he might hold a pair/top pair with 45s, 56s and 22, 44, 66-99. Additionally, there is top pair with KQs and both sets with 33 and 55. Because we are only interested in the EV of the bet regardless of our own equity, we only need to determine the probability of a fold for his different hands.

All misses can be summarised as pf(Miss), gutshots as pf(Gutshot), ace highs as pf(AceHigh), pairs as pf(Pair), top pair as pf(TopPair) and sets as pf(Set). The last two will almost always be 0 and are only listed to keep the equation complete.

We have collected all the required variables. As mentioned, we look at the continuation bet as a pure bluff, i.e. the EV of the bet itself without a chance of winning the pot any other way. The EV can be calculated through case differentiation based on pre-flop hand categories, so we can continue to vary the cold-call probabilities.

Explanation: The first term of the three subdivisions represents the probability for one of the three hand categories of BU's pre-flop range. The part that gets multiplied by the first term gives us the EV for a pure bluff after the flop, against the opponent's hand range. The respective probabilities i.e. p(Set) still have to be calculated. As mentioned above, pf(Set) is the fold probability for this hand category.

BU gets a set if he holds 33 or 55. The probability of holding 33 or 55 out of the 22-99 range is initially 2/8. However, we need to take into account that one of the 3s and 5s is actually on the board, so that there are effectively only three, rather than six, possible combinations for holding 33 or 55.

Overall there are (6*6 + 2*3 =) 42 possible combinations, six of which are a set: 6/42 = 0.14. In all other cases BU has a pair < top pair. p(Pair1) is therefore 36/42 = 0.86.

p(Pair2) is the probability for a pair/top pair with one of the suited connectors (hits with 45s and 56s). Out of the initial four for 45s, 56s and KQs there are three left over, so a total of 3*3+6*4 = 33 combinations are possible. As a result p(Pair2) = 6/33 = 0.18.

For the gutshot, four out of 33 combinations are possible so p(Gutshot) = 0.12. For top pair there are 3/33 = 0.09. The other 20/33 cases BU has hit nothing so p(Miss) = 20/33 = 0.61.

For the broadways there are nine possible combinations for KQ (KQs is calculated with SCs) given the blocked king. Add to this 2 * 16 combinations of AQ and AJ and you get a total of 41 combinations. Of these, the probability of ace high is given by p(AceHigh) = 32/41 = 0.78 and that for p(TopPairBW) = 9/41 = 0.22. Now all probabilities are known. Before we start discussing the results, for the sake of clarity, all probabilities are again illustrated in this table.

Now we need to make assumptions for the behaviour of BU with regards to folding. His range on the flop has been separated into categories: set, top pair, pair, gutshot and miss. Obviously the bet EV will be very much influenced by the probability of a fold.

Because such a high number of variables cannot be illustrated in a graph, we will look at five different sets of values together with their respective EV. You can by all means assume different values and use them for your calculations.

We expected this tendency: the looser our opponent the lower the EV. However, the relatively high EV for the continuation bet against the medium player is quite remarkable. He does not fold a lot and a bluff against him, not taking into account our equity, is profitable. The border line lies between the medium and the loose player. Against him you should seriously think about the profitability of a continuation bet or even leave out pure bluffs altogether.

Standard moves post-flop: Bluff versus continuation bets on the flop

Now let's look at the reverse scenario. Think of where in a similar situation you cold-called pre-flop, of whether you can bluff, and how profitable it would be.

The method of calculating the result is similar to the previous example. To start with, we define probabilities for certain post-flop hands given the hand range and board. Given the different possible outcomes we can approximate the EV. Contrary to the previous example, we won't look at every step with the same level of detail.

6max, 100BB Stacks

Pre-flop: You are BU
2 folds, CO raises 4BB, You call 4BB, 2 folds

Pot: 9.5BB

Flop A: K, 9, 7

Flop B: A, T, 9

Flop C: 2, 7, 9

CO bets 7BB, You raise 23BB

There shouldn't be any exceptional history between CO and you. The range of CO pre-flop is assumed to be 22+, A2s+, K9s+, Q9s+, J9s+, T8s+, 98s, 87s, 76s, 65s, 54s, A8o+, K9o+, Q9o+, J9o+, T9o, 98o, 87o, 76o, therefore  roughly 31% of hands. We can assume that CO doesn't really know you, so this would be 100% continuation betting. This situation is much easier to describe and requires fewer assumptions than a possible check-call, check-fold, and check-raise scenario. The following hand categories come into play:

• Top Pair+
• Middle Pair
• Pair < MidPair (P < MP)
• OESD, FD (sDraw)
• Gutshot (wDraw)
• Everything else (Air)

In this scenario, even ace high against the CO's unknown opponent shouldn't be worth more than any other hand with top-pair outs. For simplicity's sake, the formula to be used is:

pf * Pot – (1 – pf ) * Raise = pf * (Pot + Raise) – Raise

Raise in this case is 23BB, Pot is 9.5 + 7BB.

Just as in the last example:

EV =
P(TopPair+) * ( pf(TopPair) * ( Pot + Raise ) - Raise)+
P(MidPair) * ( pf(MidPair) * ( Pot + Raise ) - Raise)+
P(P < MP) * ( pf(P < MP) * ( Pot + Raise ) - Raise)+
P(sDraw) * ( pf(sDraw) * ( Pot + Raise ) - Raise)+
P(wDraw) * ( pf(wDraw) * ( Pot + Raise ) - Raise)+
P(Air) * ( pf(Air) * ( Pot + Raise ) - Raise)

The probabilities required for different flops:

On Flop B small straight draws, as well as small flush draws, get treated as weak draws. Top pair + flush draw is counted as top pair. Even here enormous differences between flops stand out. On Flop A all categories of made hands exist, but not many draws and a small bit of air. On Flop B the aggressor will usually have hit something, while his hand range on the low card flop is highly polarised.

For better understanding, the table also lists the combined probabilities of an air-like hand (air, weak draw and pair/midpair) for each flop.

In order to calculate the EV you need to approximate the fold probabilities of the possible hands. The share of pf(TopPair) ≠ 0 come from situations where tight players could fold weak top pairs (especially on Flop B and C with weak aces or a 9 with a weak kicker).

This result, or a similar one, was to be expected. Some players might have considered Flop C to be the best bluff flop. The fact that this isn't the case here is largely caused by the fact that it's 9 high. On a 7 high flop, a bluff would be much more profitable because a top pair is much less likely.

All in all, it becomes clear that the continuation bet alone is not a sufficient post-flop strategy. A player needs to play back occasionally (like the medium player) and rarely fold his draws or made hands, in order not to be directly bluffable.

Against a 'Medium' opponent, despite the positive EV in two out of three cases, it isn't always advisable to invest 23BB to fight for an uncertain profit of 0.7BB (when looking at the estimated variables).

As a further possibility against the tight player, the float should be taken into consideration on Flop C. Because of the increased number of approximations of your opponent's behaviour, the calculation of the float only offers itself against a player choosing very simple lines. We still assume that our equity is 0%. If the player bets again, you should give up. If the player checks, you should bet 15BB into a 23.5BB pot and if he gets called, he will lose.

Hence the EV is:

EV=p(Check/Fold)*(Pot+BetFlop) - p(Bet)*(YourCallFlop) - p(Check/Call(Raise))*(YourCallFlop+YourBetTurn)

Now we need to determine the probabilities with which CO will play the following lines on the turn:

• Check-fold
• Check-call(raise)
• Bet

These are our approximations. Now for the slightly more complicated part: the determining of probabilities depending on the different hand categories for different turn cards.

The EV of the float works out to +5.28BB. It's less than the bluff-raise in the same situation, however, it is on average also less risky. The EV is lower, particularly because the assumption gives him less of a chance of giving up hands ~top pair on the flop. If you frequently expect a 9 to Check/Fold and not Check/Call, the EV of a float will improve.

The 4-Bet Bluff

Short-handed 4-bet bluffing is an especially important topic. Because of the aggressiveness before the flop, it is important to look at 4-bets as, among other things, a bluff . Your opponents' ranges will depend a lot on your own image and your history combined. That's why if you want to incorporate the results given here into your game, you need to be particularly careful about your opponent's ranges for a 3-bet or an all-in against you.

Obviously the results will only match yours if the ranges are relatively realistic. The situation is the following:

SH, 100BB Stacks

Pre-flop: You are BU with ??
3 folds, You raise 4BB, fold, BB raises 14BB, You raise 31BB

Let's look at your EV given the assumption that you will always lose if your opponent does not give up (which in this case is a realistic assumption):

You would call 69BB into a 131.5BB pot so you need a minimum of 34.41% equity. For comparison, see this table with equities of different hand ranges. The hand's equity is given in the hand row.

As you can see, in this situation you cannot 4-bet/fold small pocket pairs against any range, but you always can with A2s and 67s, and with AQ only against the tight range. Note that if you 4-bet you also have to fold against the tight range. Against the loose range you can get your money in with AQ. You should know this in order not to make a mistake if your bluff fails. Obviously we still have to determine the EV of the bluff. It can simply be calculated using the formula from the beginning of this article:

EV=Pfold * Pot - (1 - Pfold) * Investment

The pot is the 18.5BB that we can win, Loss is the -27BB which will be lost if BB does not give up. The required fold probability is therefore around 59%. For possible ranges this means the following.

If your opponent goes broke pre-flop with QQ+, AK (34 combinations) he has to 3-bet at least 83 combinations to make a 4-bet profitable. This would be given for a range of 77+, T9s, AQ+ or a range like 44+, AK or TT+, AJ+, KQ+. If he goes broke pre-flop with a range of AQ+, TT+ (62 combinations) however, he must 3-bet at least 151 combinations to be profitably bluffable. That would be for 22+, A2s-A5s, A9s+, AQo+, KQ+ or 22+, AQ+, A2s+, T9s or similar ranges.

The Card Removal Effect

Apart from the equity of your hand against your opponent's all-in range, there is another factor that affects the profitability of a 4-bet: the likelihood of its success.

So far we have ignored one aspect in this context: the probability of a push is not independent of your own hand (we will just assume here that the opponent is 100% likely to fold or push). Considering this, your opponent's range basically consists of two parts, the "broke range" and the "fold range". The fact that you do or don't hold certain cards will shift the probabilities between the two ranges more or less strongly. This effect is known as the card removal effect. Specific cards are removed from the opponent's range because they are part of your own hand.

If you are in doubt in a potential 4-bet bluff situation, always play those hands that will minimise the probability of your opponent going all-in. If you want to quantify this difference instead of just saying "I have an ace, this is good because the opponent will go all-in with many AX hands", you will also have to take into account that the probabilities for various 3-betting ranges have shifted due to your own hand.

In practice, you do this by counting all existing combinations. We are going to introduce some charts representing the probabilities of various 3-betting ranges and all-in ranges. We will look at two different all-in ranges, a tight one with QQ+, AK and a loose one with TT+,AQ+.

We will also look at two 3-betting ranges (which can look extremely different in practice). One that heavily stresses pairs with 22+, AQ+ and another one with few connectors: 77+, AQ+, 67s-89s. The following chart shows the number of the different combinations for the specific ranges, with and without the card removal effect. This is done for 4 different holdings: AQ, AJ, KQ, A2.

For a better overview, we will refer to the respective ranges as 1) & 2) and a) & b). As you can already see, the number of possible combinations can shift extremely. For instance, holding AQ will decrease the probability of the tight all-in range by (34-24) / 34 = 0.294, i.e. by almost 30%.

From this chart we can also derive other charts for various probabilities. These will show how likely it is for the opponent to go all-in for each of the 4 possible combinations in the 3-betting range and all-in range. To be able to draw a comparison, we will look at each scenario with and without the card removal effect. The percentages are calculated by dividing the number of combinations of the all-in range by the number of combinations of the 3-betting range. If an opponent 3-bets with a) and goes all-in with 1), the probability for this is 34/110 = 0.309, i.e. around 31%, if you don't take into consideration the card removal effect.

Assuming you hold AJ, you would get the following chart:

If you have A2, the chart would look like this:

If you have KQ, it would look like this:

In most cases, the probability of an all-in will decrease by around 10% due to the blockers. It should be clear that in game theory, those hands that you can't call profitably will be more suitable for a 4-bet bluff. Out of position, this is often the case for all the holdings we have mentioned here. In position, you should obviously consider calling with stronger hands.

Always keep in mind the equity of your own hand during the game. As was illustrated before, it is not possible to play 4-bet/fold with hands that perform well against various ranges. Instead, you will often have to call an all-in which will not improve your EV noticeably (your call is only slightly +EV). However this would increase the variance enormously. Whether you should 4-bet/call with AQ or 4-bet/fold with A2 in a specific situation has to be decided during the game. The pros and cons of each will not be part of our analysis here.

Summary

This is it for the third part. In the next part we will look at the 4-bet bluff from a different aspect. We will also analyse several post-flop situations and look at G-Buck analysis.

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Related Articles: Mathematical Concepts for No-Limit Holdem (1) - EV & Ranges Mathematical Concepts for No-Limit Holdem (2) - Combos & Odds Advanced Discussion Forums: Discuss the content of this article in the Forum