Concepts: Semi-Bluffs - Theory & Practice

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A semi-bluff, just like its pure bluff counterpart, has the aim of getting the opponent to fold better hands. In contrast to the pure bluff however, you have a hand with moderate to good chance that it is still the best, or may become the best.

With a semi-bluff, we are trying to maximise our expected profit by combining fold equity and hand strength.

In this article we will discuss the mathematical foundations of semi-bluffs using examples, and explain the conditions where they are suitable at a poker table.

Example:

Pre-flop: Hero is MP2 with A , Q
4 folds, Hero raises, 4 folds, BB calls

Flop: (4,5 SB) T , 7 , 2 (2 players)
BB bets, Hero calls

Turn: (3,25 BB) J (2 players)
BB bets, Hero raises …

You have a monster draw: 9 outs to the flush, 3 more for the kings (we exclude the king of spades) plus two overcards, which can count as another ~3 outs - makes a total of 15 outs. You have a good chance of having the best hand by the river, but are unlikely to have the best hand on the turn.

You decide to raise here on the turn because you know that there are better hands that the opponent can fold here. How often must they fold a stronger hand here in order to make our semi-bluff profitable? How do we know whether a raise is better than a call? How do we assess whether the opponent folds better hands sufficiently often? These and many other questions concerning semi-bluffs will be discussed in the next section.

The Mathematics of Semi-Bluffs

The mathematics behind semi-bluffs is very complicated. Classical EV calculations are difficult to apply. We therefore concern ourselves with answering the following questions:

Is raising the turn better than a call? How much fold equity is required to make a raise rather than a call?

First we look at the expected values of raising and calling individually and compare them to get a general formula. As with all formulas we first clarify the terminology:

P = Potsize at the start of the betting round, without counting the opponents bets - for example, 5BB is P = 5

V = Betsize of Villain (Opponent) - if he bets 1 BB, ten V = 1

H = Betsize of Hero (You) - a raise in fixed limit corresponds to H = 2, EQ = your equity; Usually we count this in outs, but this can be easily converted: 1 Out represents an equity of about 2.2% - with a simple flushdraw on the turn our equity is roughly 20% i.e. EQ = 20% = 0.2

P(F) = The probability of the opponent folding a stronger hand - if he always calls then P(F)=0; if he always folds, P(F)=1

Poker is a very complex game. An exact formula taking into account every factor would be too long and cumbersome. There are so many uncertain factors that we could not obtain any useful result. We must therefore be a little bit abstract. Thus we consider the possibility that villain 3-bets, or that we have implied odds on the river, as negligible.

We hope that these inaccuracies balance each other out, or are so small they have no influence on our final decision. For the sake of completeness, we are assuming that we have a worse hand than our opponents.

Enough with the small details - on to the EV! First consider the call. The article on odds and outs provides the necessary background.

EV(Call) = (P + V + V) * EQ - V

The term in brackets represents the size of the entire pot at the end of the betting round, namely „pot at the beginning" plus „bet of the opponent" plus „our call", naturally, each of these bet sizes is the same as the opponents raise.

We win this amount with probability EQ. Logically speaking, equity is nothing more than our probability of winning. Now we have our profit and we need to deduct our costs. The value we are left with is the expected value of a call.

According to this principle - amount to be won times the probability of winning minus costs- we can calculate the value of a raise:

EV(Raise) = P(F) * ( P + V ) + ( 1 - P(F) ) * ( ( P + 2 * H ) * EQ - H )

In this case we have to include the chance that the opponent calls or folds. There are two possibilities, so we work them out separately and weight them according to their likelihood:

If villain folds [P(F)], then you win the whole pot [(P+V)]. You also „win" your raise, so this cancels out it being a cost. We are done with the first part.

If the opponents call however, then P, V, H (your raise) and H-V (the opponent's call, whereby he has already put some of the money in) are in the final pot. Simplified, this comes to ( P + 2 * H ).

You now win the proportionally to your EQ. The cost of our action is our raise size, H. The probability of a call is the compliment of the probability of a fold, since we excluded the possibility of a 3-bet. The opponent folds if he does not call and calls if he does not fold. The chance of a call is therefore 1 - P(F).

The original question was whether a raise was better than a call and how much fold equity we need. We get this answer by noting whether EV(Raise) > EV(Call) for a particular value of P(F). This is mainly paperwork and some simple arithmetic:

This formula has been very general so far. Since we are only interested in fixed limit, we can be more concrete and use fixed bet sizes[ V = 1, H = 2] to get:

Application example: Let's apply our findings to the first A Q - example. Recapping:

Pre-flop: Hero is MP2 with A , Q
4 folds, Hero raises, 4 folds, BB calls

Flop: (4,5 SB) T , 7 , 2 (2 players)
BB bets, Hero calls

Turn: (3,25 BB) J (2 players)
BB bets, Hero raises …

P is 3,25 and our equity is around 34%. With our formula we obtain a minimum P(F) value of ( 1 - 2 * 0.34 ) / ( 3,25 + 3 - 0.34 * ( 3,25 + 4 ) = 0,32 / 3.785 = 0.0845.

The opponent must fold a better hand only ~8% of the time, such as Q2 to make the raise better than a call. Who would have thought?!

A formula alone does not give us a feeling for all situations - experience often helps more than a thousand numbers. Here we have used our equation, assuming P = 3.25 to derive a graph showing the fold equity required for various pot sizes. Points falling in the grey area advocate a raise, whereas a raise is inadvisable for those situations in the light blue area.

We look for the equity required in our AsQs-example, and following the black arrows from 0.34 EQ we end up at 0.0845 for P(fold).

At this point we would be almost lost, if it were not for the fold button. This option may be the best choice so we must compare against the two alternatives. The expected value for fold is 0 since we can't win anything and we don't invest anything. This makes comparison very easy, since we already have all three EV values.

A raise is better than a fold if:

• EV(Raise) > EV(Fold)
• P(F) * ( P + V ) + ( 1 – P(F) ) * ( ( P + 2 * H ) * EQ – H ) > 0

And a call is better than a fold when the odds and outs suggest it:

• EV(Call) > EV(Fold)
• (P + V + V) * EQ - V >0

We can apply this formula to Fixed Limit once again, assuming P = 3.25, and insert it into our old chart. The new light brown area represents when a fold is the best option.

The decision between calling and folding is of marginal interest to us in an article on semi-bluffing since you will nearly always have the odds to make the call. What is important here is the line between the grey area and the light blue area, and not just for P = 3.25. Figure 3 combines the profitable areas for semi-bluffing for different values of P.

With a gutshot on the turn you have an estimated equity of 10% and if the pot size at the start of the betting round is 9 BB then it is profitable to raise if villain folds 7.5% of the time. If the pot size is 7 BB, however, then we need 10% fold equity, and with just a 3 BB pot, we need the opponents to fold 25% of the time.

The mathematics behind semi-bluffs is quite complicated because the various outcomes of our actions must be taken into account, resulting in some quite complex formulae. Only with the help of graphs, which provide a clear result, can we apply our analysis.

Even if the fold equity against our opponent is never accurately determined, we can get a good feeling of how much fold equity we need to make a profitable semi-bluff with the aid of the graphs and our analysis.

Now we shall explore semi-bluffs using some specific game situations. Here we will also show you how can determine how much fold equity you actually have.

Practical Applications of Semi-Bluffs

Naturally, there are many opportunities for using semi-bluffs. In contrast to the pure bluff, there are some situations where a semi-bluff is a standard move. Semi-bluffs are basically more effective, since we also have some outs. One might even say that with strong draws such as OESD's and flushdraws, a semi-bluff is almost always +EV.

This does not, however, mean that a semi-bluff is always the best way to play such a draw. In position, and heads up it is generally best to make a turn-raise, since the big bets will grant you higher fold equity. In multiway pots with position, you should raise the flop, to keep you options open, such as the following:

• Value-raise-battle
• Trying to win the pot without a showdown by applying pressure
• Without resistance, there is the free card option on the turn

Out of position, you should try to retain the initiative on the flop, so that there is the chance of the opponent folding his (usually better) hand on the flop or the turn.

As mentioned, an important part of semi-bluffs is developing your ability to estimate the fold equity you have against an opponent. Even if this may appear too abstract to quantify, there are ways and resources of making an approximation.

When considering a semi-bluff, you should ask the following question: Specifically which better hands can fold to a raise?

To answer this, you must estimate your opponent's hand range. To help you with this, you can use the opponent's stats, take note of the board and your history with this opponent. Then you must consider which of these hands he could potentially fold. You then estimate how likely it is that he has a hand in his range that he will fold in order to get an approximation of your fold equity.

It can also be good for you if the opponent folds a worse hand, because he'll often have outs. An argument against this is the possibility of bluff induction. An extra bluff-bet obtained might compensate for any potential losses through the possibility of your opponent hitting an out. In general, you'll want worse hands to fold if they have the correct pot odds for a call.

Pre-flop: Hero is Button with Q , 9
3 folds, Hero raises, SB 3-bets, 1 fold, Hero calls

Flop: (7 SB) 3 , 6 , J (2 players)
SB bets, Hero calls

Turn: (4,5 BB) T (2 players)
BB bets, Hero raises ...

Now we want to do a precise analysis. We must first ask the following questions:

• 1) How many outs do we have?
• 2) What is the opponent's range?
• 3) How much fold equity do we need from a better hand to make EV(Raise) > EV(Call)?
• 4) Specifically what better hands can villain fold? Is the potential fold equity more than that needed for a profitable semi-bluff?

1) Clearly we have 8 outs fro the OESD, but how many outs do we assign ourselves for the Q or the 9? The exact calculation for these outs is beyond the scope of this article, however. This issue is covered in the platinum article:

Heads-up on the flop Flop OOP: C/C Flop without Initiative - without Showdownvalue

We estimate a little pessimistically and give you 2 outs for the Q and 9, making a total of 10.

2) The opponent's range. Since we do not have accurate information, we assume the 3-betting standards of a TAG. His range should be like this:

33+, A2s+, KTs+, QTs+, JTs, T9s, A5o+, KTo+

As the opponent made a preflop 3-bet from the SB, we can assume that he'll bet on both the flop and the turn with his entire range.

3) Your required fold equity. We could, of course, read from the corresponding graph to get the required fold equity. For practice, however, we shall go through the exact calculations required. You'll see that with a bit of practice it is not so difficult. As shown above, we apply the formulas for a semi-bluff raise in fixed limit:

EV(Raise) > EV(Call), when P(F) > ( 1 - 2 * EQ ) / ( P + 3 - EQ * ( P + 4 ) )

P = potsize at the start of the betting round, without the opponents bets - e.g. with 5BB, P = 5

EQ = your equity; usually we count our outs, but this can be easily converted: 1 out represents about 2.2% equity - with a simple flush draw on the turn this gives roughly EQ = 20%, or EQ = 0.2

P(F) = probability that the opponent folds a stronger hand - if they always call, then P(F)=0; if they always fold, P(F)=1

Now we replace the variables by the known parameters to obtain P(F):

• P = 4,5 BB
• EQ = 10 Outs x 2,2% = 22% = 0,22
• P(F) > (1 - 2 x 0,22) / (4,5 + 3 - 0,22 * (4,5 + 4)

P(F) > 0,56 / 5,63 = 0,099 ~ 10%

In this example, we have shown that we need 10% of better hands to fold to a raise to make EV(Raise) > EV(Call).

4) The calculation of what your opponent folds, of course, is never precise because we do not know how willing they are to fold, but we can imagine what we would do if we were them.

We assume at this point that the opponent bets again on the turn. We take a look at the board and villain's range:

Pre-flop: Hero is Button with Q , 9
3 folds, Hero raises, SB 3-bets, 1 fold, Hero calls

Flop: (7 SB) 3 , 6 , J (2 players)
SB bets, Hero calls

Turn: (4,5 BB) T (2 players)
SB bets, Hero raises ...

Villain: 33+, A2s+, A5o+, KTs+, QTs+, JTs, T9s, KTo+

The following hands should definitely fold to a raise on the turn: A2s, A4s, A5, A7, A8, A9

Because the board offers no flushdraws, it may well be the case that A high can be folded. Not every opponent will play A-high for showdown value. Paying 2BB for the call-down each time is expensive on a board which is not A-high friendly.

Furthermore, hands such as 44, 55, A6 or even 77-88 bring folding into question, but first we consider those which will definitely fold. We must now answer the question: What % of the range do these hands comprise? - A2s, A4s, A5, A7, A8, A9.

In order to determine this, we consider the number of combinations which could form each hand. A pocket pair has 6 combinations, an off-suit hand has 12 and a suited hand has 4 combinations.

We must take into account the cards on the board and those we possess (dead cards). A hand such as A9 theoretically has 16 combinations, but in this example there are only 12, since we posses a 9, removing 4 possibilities. We must consider this scenario for all possible hands.

Here we tabulate our analysis:

Our analysis of the possible hand combinations shows that villain could potentially fold 68 of the 234 card combinations = 29% of possible hands to a raise on the turn. And that is without even considering that he might fold small pairs.

We have previously shown that in this spot we need only 10% fold-equity in order for a raise here to be a profitable play. Our expected fold equity is miles above the required amount, advocating that a raise is the right move. Some players, of course, will not easily fold A high here on the turn, but certainly not everyone will make such a call-down.

You can see that a rigorous analysis is time consuming, but is certainly possible, since we can estimate even the abstract quantity of fold equity. Remember that it is what happens in the average case that counts, and that there is no shame in getting called down by A7 and losing, for instance.

If you happen to know that the opponent will never fold A high in this spot, and thus your fold equity against better hands on the turn is 0, then a semi bluff is, of course, not the best option.

Pre-flop: Hero is BB with J , 9
2 folds, MP2 calls, Button calls, SB completes, Hero checks

Flop: (4 SB) 3 , 4 , 8 (4 players)
SB (Semi-TAG) bets, Hero raises and bets every turn when MP2 and the Button fold

It seems very likely that SB has a better hand than you. It is questionable, however, whether he will want a showdown with a hand such as J 4 if he encounters resistance.

You can try to get your opponent to fold better hands on the flop - your chances are quite good. The pot is not very large, and unraised pots are generally not as fiercely contested.

It is important when doing this that you continue the bluff on the turn. On the flop, villain can call a pair with the 7 to 1 odds available. On the turn, he will get only 4,5 to 1 and is more likely to fold a small pair. On the river, we should not make any further bets, since it is unlikely that villain holds a busted draw on such a board. Only a Q-A high flush draw are the likely draws and he would probably 3-bet these on the flop.

Unless the opponent has played the hand passively, he will no longer fold to a river bet if he calls on the turn. In addition to scaring away SB with your raise, you can potentially force out better hands held by MP2 and the button behind you. With 2 SB to pay and pot odds of 3,5 to 1, hands such as 66 and A3 can fold. In the event that SB holds a draw such as 56, you could well buy the best hand by making a raise on the flop.

In addition, you could buy outs, if hands which have you dominated fold, for instance K9 or QT. You should not simply call on the flop because you expect better implied odds in later streets. In this example, you have a realistic chance of getting better hands to fold, so a semi-bluff is always the better alternative!

Pre-flop: Hero is MP3 with T , 9
1 fold, Hero raises, 3 folds, BB calls

Flop: (4,5 SB) 6 , 7 , J (2 players)
BB bets, Hero calls

Turn: (3,25 BB) Q (2 players)
BB bets, Hero raises ...

On the flop we see the beloved donkbet from the opponent. Your gutshot, backdoor flushdraw, T and 9 are worth 7-8 outs. You can loosely call the donkbet on the flop. On the turn you have not only run into an OESD, but the arrival of the overcard also contributes to the scare-factor.

The BB is unknown here and naturally it is difficult to assess whether he is likely to fold a pair. You should not, in fact, have great hopes that he will fold pairs, but this board and development are well suited to such a scenario.

The opponent could well fold hands such as 22-55, A6, 65 or other combinations consisting of low pairs. There also exist strange fish who could bet twice with a hand such as A3o and then fold to a raise. In the average case, you definitely have enough fold-equity against better hands, so a semi-bluff is the right course of action.

Now we will look at a slightly modified situation.

Pre-flop: Hero is MP3 with A , K
1 fold, Hero raises, 3 folds, BB calls

Flop: (4,5 SB) 6 , T , J (2 players)
BB bets, Hero calls

Turn: (3,25 BB) 3 (2 players)
BB bets, Hero???

You are again confronted with a double donkbet. Even now you have a strong draw with a gutshot and 2 overcards, which will give you top pair, top kicker. The parameters are different, however:

• The turn is not a scare card, but rather a total blank.
• Your hand has showdown value - you beat silly bluffs, such as A4 and semi-bluffs such as Q9 and 98.

Here, the range of better hands that will fold on the turn is relatively thin, so a semi-bluff has a smaller chance of success, and we will only really get worse hands than ours to fold.

On the turn you should invest only 1 BB. It could be that the opponent checks a stronger hand on the river but would not fold to a turn raise, so we save a BB. In addition, we could induce another bluff from a hand which would otherwise fold to a turnraise (e.g. A4).

The main argument which speaks against a semi-bluff in this hand is that you hold the strongest non-pair hand and that it is unlikely that villain will fold a stronger (pair) hand.

Turnplay: Hero calls

Pre-flop: Hero is SB with A , 9
6 folds, Hero raises, BB (26/17/2,3/34) calls

Flop: (4, SB) 5 , 7 , J (2 players)
Hero bets, BB raises, Hero???

You are in a blind war against a TAG with a small WTS and have the initiative on the flop, but get raised. You have the nutflush draw and the question is whether to call or to 3-bet.

Once again we have to ask which better hands the opponent will fold. You have A high with a passable kicker, and in the absence of a preflop 3-bet, it is unlikely that your opponent has a better A high. Thus the better hands we want to force out consist of small pairs.

When the opponent has a tendency to bluff, we could continue this hand in a passive manner. Since we have the flush draw, the opponents often only have 4 outs in this case, provided we are ahead on the flop.

It is a question of whether your opponent is able to fold a pair. There are TAGs who will play the so-called information-raise on the flop, and fold to a 3-bet now, or on the turn, since this means a relatively low cost of 1-1.5 BB instead of the 2.5 it would normally cost for a calldown. The low WTS value suggests such a player.

On the flop you should do the following:

• 3-bet flop, bet turn against the opponent described here and all those opponents you would describe as weak.
• 3-bet flop, bet turn against unknown. In case of doubt take the aggressive approach. Your draw here is strong with over 12 outs.
• Call flop, check/call turn and check/fold river against passive, showdown happy players. Here the fold-equity against better hands is virtually 0.
• Call flop with the intention of a showdown against aggressive LAGs, when the board is not particularly nasty. You don't want to raise the turn and will rather build your showdown value via bluff induction. Because of your flushdraws, a freecard is not so bad because 1-2 of your opponent's potential outs are contaminated.

When to fire again, if the bluff does not work

This is a difficult issue, because it is hard to come up with a standard recipe. We take any example whereby we have made a bluff or semi-bluff, but the opponent hasn't folded and we are on the river with a poor hand. Another bluff? Maybe he folds the river, but who knows?

Nevertheless, we shall try to shine some light into the darkness. You should consider the following things in such a situation.

• The board is structured so that villain himself may hold a busted draw with little showdown value, but that a bet from you would not suggest you were drawing yourself.
• On the river there is a scare card which completes many draws. Even if your draw didn't arrive, you can hope that the opponent will get out of the hand believing that you just made yours.
• The opponent makes bad calls because he is simply curious as to what the rivercard might be, for instance, drawing to two pair with a low pair which he will fold unimproved on the river.
• The opponent has a high PT fold to riverbet value (e,g. > 40).

• You have a busted draw which has SD value against the busted draw of an opponent.
• The board completes no draw, the villain has showdown value if he calls and is no longer likely to fold to a bet.
• We have a read that the opponent always wants to see a showdown after calling a raise on the turn.

Take the analysis from example 1:

Pre-flop: Hero is Button with Q , 9
3 folds, Hero raises, SB 3-bets, 1 fold, Hero calls

Flop: (7 SB) 3 , 6 , J (2 players)
SB bets, Hero calls

Turn: (4,5 BB) T (2 players)
SB bets, Hero raises, SB calls

River: (8,5 BB) 7 (2 players)
SB checks, Hero???

Here we have to ask: How much fold-equity do we need? This case is easy. It is the formula from the Pure-Bluff-Article.

Bluff is +EV when: P(fold) > Bets/Pot

Bets = the cost of the bluff = 1BB Pot = the pot size after the bluff = 9,5 BB

Therefore: Bluff is +EV, when P(fold) > 1 / 9,5 = 10,5%

Does villain fold a better hand in >10,5% of cases? What better hands could he call on the turn but fold to a bet on the river? As a reminder, his range is as follows:

Villain: 33+, A2s+, A5o+, KTs+, QTs+, JTs, T9s

There are ultimately only 3 hands in question that villain could fold at this stage - AK, AQ and KQ. Could he fold these? Actually yes! What hands could he beat on the river? There is no flushdraw on the turn which could now be busted, the 7 has completed a 98 straightdraw. If villain folds AK, AQ and KQ on the river, then we now need to see whether these three hands comprise more than 10,5% of his hand range.

On the turn, villain had 234 different card combinations in his hand range. We assumed, however, that he would fold A2s, A4s, A5, A7, A8, A9 on the turn. This removes 68 of the combinations. In addition, the 7 on the river means that the hand 77 has 3 fewer possible combinations.

Furthermore, we must remove those hands which he would fold to a 3-bet on the turn, that is JJ, TT, 66, QQ-AA, JTs and AJ. Taking into account the board and your hand, this is another 38 possible combinations.

Ultimately there remain 234 - 68 - 3 - 38 = 125 combinations on the river. Now we need to know how many combinations AK, AQ and KQ comprise. We must bear in mind that we ourselves have a Q. There are 12 combinations each for AQ and KQ and 16 for AK, giving a total of 40.

And thus: 40 foldable combinations / 125 possible combinations = 32%

If you are in a position where villain will fold AK, AQ and KQ on the river in this spot, then you should fire a final bluff here. You need >10,5% fold-equity, and have a massive 32%. Even if KQ is the only hand that villain will fold here, you still have 12/125 which is almost the 10% fold-equity you need for a profitable bluff.

Conclusion

For most semi-bluffs, you require very little fold-equity against stronger hands in order to make a profitable move. For good draws it's almost always <20%, and for very good draws you often require less than 10%. Therefore it should not bother you if you find that opponents are not folding too often.

It is inevitable that, for example, that 10 of your semi-bluffs in a row get called, or you keep getting 3-bet. This does not necessarily mean that the move is incorrect. You must adapt psychologically and remember that the percentage of times the semi-bluffs and bluffs fail is outweighed by the long term benefits when they succeed, so that you are making moves which are mathematically profitable in the long term.

On the other hand, you should always make sure that it is better hands which you are trying to get to fold. If you blindly bluff with any draw, without thinking about what concrete better hands the opponent might throw away, you can quickly end up with a bloody nose.

You have now seen how with a small amount of effort you can estimate the so-called variable fold-equity. With the knowledge of the presented concepts you now have the ideal tools for analysing semi-bluff situations. You should do this regularly, since it is only after analysing many situations that you start to get a feel for them at the table. With the appropriate amount of experience, you start to recognise situations at the table and be able to make good decisions intuitively.

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