Mathematical Concepts for No-Limit Holdem (4) - Bluff Equity & G-Bucks

Introduction

In this article

• The equity of a 5-bet
  
• Bluff equity
  
• The G-Bucks-Analysis
  

Having looked at bluffs, in the fourth and last part of this series, special sample applications as well as an alternative form of analysis will be introduced: G-Bucks.

5-Bet-Analysis

In the last part, pre-flop bluff 4-bets was one of the topics. You have learned which hands you should 4-bet-call and which you should 4-bet-fold. Pocket pairs had a special place in this analysis due to their relatively high equity.

In this article, bluff 5-bets will be discussed as a possible counter against 4-bets. It is immediately obvious that we are talking about a semi-bluff, not a pure bluff.

You will be unable to 5-bet and then still find a fold in situations with roughly 100BB effective stack sizes. Consequently your equity will play an important role again. A rough approximation will illustrate the approach: A frequently occurring scenario heads-up includes a 3-bet of 12BB as well as a 4-bet of 28BB. How much fold-equity does the all-in of the 3-better have to create, to have a neutral EV given 35% equity?

EV = 0 = pFold*(12+28)BB + (1-pFold)*(0.35*112-0.65*88)BB
0 = pFold*40BB - (1-pFold)*18BB
pFold = ~0.3

In this particular scenario the 4-better needs to fold in only one of three cases to make a re-bluff profitable. Here is a short repeat of pre-flop equities, so that we can get back to them later on:

Small pairs obviously have a good equity, even against tight ranges. We were aware of this. Furthermore, a strong hand like AQ offers itself against slightly looser ranges. The card removal effect will not be discussed at this point: AQs has about 35% equity against a TT+, AQ+ range, similar to a small pocket pair.

The card-removal effect makes AQs a push that is preferable to 55, because with AQs the "TT+, AQ+ range" becomes 30% less likely compared to a 55, as shown in the previous article. This applies for A2s similarly. However, the card-removal effect in the A2s case is less significant. Consequently the much lower equity of A2s compared to pocket pairs makes them more suitable candidates.

Different kinds of all-in ranges were also not taken into account:

If a player goes all-in pre-flop with many pocket pairs but not a lot of broadways, it is a good idea to resort to stronger pairs yourself. The reason is simple: As long as you don't want to call the pair against a 4-bet, but want to 5-bet it, small and big pockets have roughly the same equity against a range that does not include pockets smaller than your own (22 vs. TT+, AQ+: 36% equity, 99 against the same range: 37% equity).

Even if his range includes only a few smaller pockets, your equity increases significantly (22 vs. 88+, AQ+: 36% equity, 99 against the same range 41% equity). Hence there are two cases: Either both hands have the same equity or 99 is better. Hence it should be clear which hand you should choose.

SH, 100BB Stacks

Pre-flop: Hero is BU with XY
CO raises 3.5BB, Hero raises 11BB, 2 folds, CO raises 28BB, Hero is All-In

Roughly the situation that we used in our approximation earlier. Hero's EV can be calculated depending on his equity as well as given fold-equity of an all-in:

EV = pFold*40.5BB + (1-pFold)*(Equity*112.5-(1–Equity)*89)BB

If you set EV = 0, as commonly done in this form of analysis, the equation can be solved for pFold. You get pFold as a function of the stack- (fixed) and raise-sizes, as well as the equity, which can be illustrated using a graph.

pFold = (112.5*Equity-89*(1–Equity)) / (112.5*Equity-40.5–89*(1-Equity))

For the relevant range of equity from 25% to 45%, the function looks like this:

Evaluation for equity exceeding 45% is pointless, because negative fold-equity is an unrealistic result. The area under 25% equity is also not relevant, because you should have more than that in most cases. Some exact values are:

• pFold ( 25% Equity ) = 0.48
• pFold ( 30% Equity ) = 0.41
• pFold ( 35% Equity ) = 0.31
• pFold ( 40% Equity ) = 0.17

As expected, the required fold equity varies depending on your equity. In this situation you need a fold probability of 32% to 25% with a semi-bluff hand.

It is also possible to determine the influence of stack size of the relevant players on the EV. All we need to do is modify this term in the formula:

Equity * 112.5BB - ( 1 – Equity ) * 89BB

Once again we can create functions that can be illustrated. Here are some examples:

From bottom to top the effective stack sizes in BB are 90, 100, 110, 120, 130. Below the representative value of required fold equity is given an equity of 35% for each individual stack size:

• pFold(35% Equity, 90BB Stacks) = 27%
• pFold(35% Equity, 100BB Stacks) = 31%
• pFold(35% Equity, 110BB Stacks) = 34%
• pFold(35% Equity, 120BB Stacks) = 37%
• pFold(35% Equity, 130BB Stacks) = 40%

It was obvious from the start that the required fold equity increases with stack size. However, the increase in my opinion is lower than most people think it should be in order to go all-in with 130BB. Often players will go all-in more than 1.48 times more for 130BB than for 90BB. This ratio (27/40) however, exactly determines the profitability of the bluff.

At the same time, it is also true that many players will 4-bet bluff more for 130BB than for 100BB because they expect your range to be significantly tighter so they give themselves more fold-equity. This is a point where the concept that we discovered through maths could be directly used at the tables. Whether you want to 5-bet bluff pre-flop for 130BB is obviously up to you.

Finally we want to discuss the influence of different possible pre-flop raise sizes. We expect the following to be true: the smaller the 4-bet, the greater the fold-equity (with constant equity). That's exactly what we observe:

From top to bottom:

2.5BB → 8BB → 20 BB;

3.5BB → 11BB → 28BB;

4BB → 14BB → 32BB

Again the sample values for Equity=35%:

2.5BB open → pFold = 42%,
3.5BB open → pFold = 31%,
4BB open → pFold = 24%.

As you can see, raise sizes have a big impact. Another approach to varying your game is to pay close attention to the 4-bet/fold statistics of players for different raise sizes.

Some players won't necessarily be aware of this fact since the 4-bet of 32BB isn't particularly big. At the same time you should be aware of this circumstance. It means however, that you only need to change your game against opponents that are also aware of this fact. Against bad opponents, who are unaware, you don't need to make any changes.

Bluff vs. 2nd Barrel

You are probably aware of the principle of the 2nd Barrel: Often it means playing a bluff aggressively, twice. Since the term originates from a second continuation bet on the turn, often this bet is simply called a second barrel regardless of the hand of the aggressor. Often the hand is going to play bet-fold. A possibly increased frequency of this particular line is the basis for the analysis of the following situation:

SH, 100BB Stacks

Pre-flop: Hero is BU with A, T, CO has K, Q
CO raises 4BB, Hero calls 4BB , 2 folds

Flop: (9.5 BB) 6, Q, 5 (2 players)
CO bets 8BB, Hero calls 8BB

Turn: (25.5 BB) 9 (2 players)
CO bets 17BB, Hero raises 50BB

It is out of question that Hero can play this hand differently. A raise on the flop is possible just like a call on the turn or a pre-flop 3-bet. We won't look at those lines however, and focus on Hero's raise on the turn. In this particular case the following needs to be clear: this line only makes sense as a bluff or semi-bluff, if Hero could in theory, and would in practice, play a made hand the same way.

Otherwise his hand is polarised towards bluffs so much that he will get light calls from hands that he wants to get out of the hand. If this is the case however, then it makes sense to analyse this unconventional semi-bluff.

Just like for any semi-bluff the EV is given by the following:

EV = pFold * 42.5BB + ( 1 – pFold ) * ( 0.2 * 113.5 BB – 0.8 * 88BB )

Assuming about 20% equity against a range that CO will shove on the turn, Hero cannot fold against a push hence the EV is portrayed correctly. Solving the equation for pFold given EV = 0 gives us a minimum fold equity of 53%. Is this achievable in our example?

It needs to be emphasised that this is only an example: it's not the goal of this article to use perfect hand reading, but rather to explain the nature of this and similar types of analysis. We set a certain betting range and all-in range to approximate a value for pFold. If you want to choose different ranges, you are more than welcome to. You can use a different range to calculate a fold probability and get a different result.

One possible range could be the following: CO will bet the turn with top pair + i.e. QT, QJ, KQ, AQ, 55, 66, 99, AA, KK, QQ. Hence there are 63 possible combinations. CO will go all-in on the turn with sets and overpairs (hence 55, 66, 99, KK, QQ, AA). This would be 21 possible combinations. Hence the fold probability would be 1-21/63 = 67%.

If we were to add 56s and Q9 to his betting and all-in ranges, we would get 63+3+9 =75 combinations for a turn bet as well as 21 + 9 + 3 = 33 for the all-in. Even here the semi-bluff would be slightly +EV with a fold equity of 1-33/75 = 56%. To the disadvantage of Hero, not taken into account are further possible hands that CO might bet-fold like TT, JJ, 9X or weaker draws (despite beating these hands Hero will definitely profit when they fold).

On this, another example with a variation:

SH, 100BB Stacks

Pre-flop: Hero is BU with T, 9
MP raises 4BB, CO folds, Hero calls 4B, 2 folds

Flop: (9.5 BB) 8, Q, 3 (2 players)
MP bets 6BB, Hero calls 6BB

Turn: (21.5 BB) 7 (2 players)
MP bets 13BB, Hero raises 38BB

As mentioned earlier Hero could obviously play completely differently, but he won't. We will analyse the semi-bluff on the turn. The circumstances in this case however, are different: Hero's equity against an all-in range is only 18% but he needs 25% for a call against a push from MP. The EV of this move depends heavily on how frequently MP will only call on the turn and how often he will re-raise.

For simplicity's sake, assuming MP will never fold on the river when he only calls the turn (he has got about 50BB left in a 100BB pot) EV becomes a function of the fold probability and the probability of an all-in:

EV = pFold*34.5BB + (1–pFold)*[-p(Push)*38+(1-p(Push))*(0.18*111.5–0.82*38)]BB

In case of a miss, Hero will only lose 38BB but he will win the whole pot plus the remaining stack of MP in case of a hit. For EV = 0 the relationship between fold probability and all-in probability can be illustrated the following way:

Here the characteristical values are:

• pFold(p(Push)=1)=52%
• pFold(p(Push)=0.5)=42%
• pFold(p(Push)=0)=24%

This obviously means that if your opponent pushes none of the hands that he wants to continue playing, you need him to fold in only one of four cases. If he always goes all-in, the EV of your hand is the same as any other hand and you need him to fold about 50% of the time.

Here, the question is, how often your opponent would play bet/fold, how often he'd go for a bet/call, and how often he'd consider a bet/3-bet line. If Hero doesn't have position on his opponent in a similar situation it can even have a positive effect: it's even more unlikely that your opponent will go all-in because he supposedly does not need to protect his hand and will surely be able to get all his money in on the river.

We will continue to use numerical values rather than a range. Anybody who is interested in this topic should play around with possible ranges themselves to decide whether the move should be used in play or not.

Continued Bluffs

An important concept, which can be expressed mathematically, is the following: if a hand develops in such a way that a bet is profitable in itself, all actions that lead to the bet are +EV as well.

Hence, a continuation bet on the flop can have a negative EV itself as long as it leads to profitable betting opportunities on the turn often enough. The reason simply is the individual values in the case differentiations in the formula of the EV: if a sufficient amount (or all) values become positive as a result of your bet then the overall EV will be positive, regardless of your hand.

So called bluff outs are an important factor. They already existed in example 3 despite not being mentioned explicitly. The 9 on the turn could have completed an OESD. Hero does not hold the OESD but the fold equity on the turn should be slightly higher for a 9 compared to a 2 for example. Again we will work with an example:

SH, Hero with 100BB

Pre-flop: Hero is UTG with XY
Hero raises 4BB, 2 folds, BU calls 4B, 2 folds

Flop: (9.5 BB) 5, 3, 5 (2 players)
Hero bets 6BB, CO calls 6BB

For a bet with EV = 0 the required fold equity is 39%. Hero's hand will not play a role in this analysis. Depending on Villain's pre-flop calling range and the frequency, with which he will bluff, 39% fold equity might not be realistic hence the bet itself will be -EV. This is how the game continues:

Turn: (21.5 BB) X (2 players)
Hero bets 14BB, CO calls 14BB

River: (49.5 BB) X (2 players)
Hero bets 28BB

The river-bet is slightly smaller than the flop bet, in relation to the pot. Hero needs 36% fold equity. The question is: can Hero expect this to be a realistic value?

The answer to this question will heavily depend on the behaviour of our opponent. Regardless, it is possible to examine the probabilities, for the CO's hands of different strengths:

Assuming he will 3-bet TT+ pre-flop, he can have trips/full-house or 9s up. Even 99, his strongest hand on the flop, will unfortunately be a bluff catcher by the river. Except for a hero-call he has no reason to call three barrels.

CO example range here will be 22-99. How often does he have a strong hand? There is one possible combination for 55 and three for 33. Against this there are 6*6 = 36 possible combinations for a simple two pair. In only one out of ten cases does he hold a a hand on the flop that is strong enough to call/raise the river. Nine out of ten times he holds a hand on the flop that will be a bluff catcher by the river if neither the turn nor the river helps him.

This is closely linked to the fact, that this flop is very hard to hit and it is incredibly unlikely that Villain has a strong hand. Hero however, can represent a much wider range (he could be betting TT+ for value).

Obviously CO's range will get smaller towards the river, but there still are some players against which a third barrel is profitable (although cannot be balanced if played with any two). The key issues are the following:

• How often will Villain fold weak hands on the flop and turn?
• How often will he pull out the hero-call on the river?

In relation to this it is worth looking at the probabilities for different cards on the turn and river. For example it could be advisable to only bet the turn (and river) in case a scare card comes. The probability can be calculated quickly.

Let's say scare cards are J+. The probability of hitting one of the 16 cards we want to see is 16/49 = 0.32.

The probability of hitting one of the scare cards by the river is 16/49 + ( 1-16/49 ) * 16 / 48 = 0.55!

A similar scenario could be constructed for a monochrome board. The approach would rely on the fact that your opponent will only very rarely have a hand that can stand against your value-betting range, or the possibility to play cards aggressively that you haven't hit (keyword: bluff-outs).

G-Bucks

The last part of this series will be dedicated to so called G-Bucks or Galfond Bucks. As far as I know the term was first used in an article by Phil Galfond in Bluff Magazine in April 2007, in which he coined the term. We are talking about a particular form of analysis based on the calculation of a very particular EV.

In showdown situations G-Bucks look at the EV of the particular hand of your opponent compared to Hero's entire range. Phil Galfond does use slightly different calculations in his article but he reaches almost the same conclusion. The only question is whether combinations of Hero's range, that are blocked by his opponent's hand, should be taken into account explicitly.

This form of analysis is similar to the well known Slansky-Bucks who always look at the EV rather than the result. Because the deviations from the EV will disappear in the long run, this analysis is free of those deviations. In one particular hand however, the quality of the result is limited because a basic concept of poker gets violated: players play hand ranges, not individual hands. The perfect analysis would compare the EV that the ranges of the players involved have against each others range.

We have worked in this direction when we tried to give individual players hand ranges in our calculations. This leads to a big problem: we make assumptions. In any situation you can only say "I think the range is this" - but you never know.

That's why Phil Galfond eliminates the opponent's hand range from the analysis and hence creates a form of analysis with complete information: We look at our own range (which we know) compared to our opponent's hand (which he did obviously have). G-Bucks are the EV of the known range against the known hand. Here is a simple example:

SH, Hero mit 20BB

Pre-flop: Hero is MP with A, K, UTG with 7, 7
UTG raises 4BB, folds, Hero raises 20BB, folds, UTG calls 16BB

It does not matter why Hero only has 20BB, or why UTG plays the way he does, it's just an example. The result for Hero will rarely be +-0 minus rake. In all other cases it will be +21.5BB or -20BB. It is clear that any exact analysis will lead to a value between both extremes. The EV of the all-in is:

EV=-0.523 * 20BB + 0.477 * 21.5BB = -0.2BB

The tendency is for the play to really be +-0 or even slightly negative. Does that make it wrong to go all-in? My feel as a poker player tells me, probably not. To calculate the G-Bucks Hero needs to give himself a range. It shall be TT+, AQ+. Against this, 77 only has 37% equity. Hence the G-Bucks for Hero come to:

0.63*21.5BB – 0.37 * 20BB = 6.145BB

Hero makes a +G-Bucks move. The average result of UTG against Hero's range can also be looked at with a result of 0.37*25.5 – 0.62 * 16 = -0.645BB.

He is still making a small mistake against Hero's range. If Hero did not at call or only rarely push AQ, his mistake would become much worse. For a possible range of TT+, AQs+, AK for Hero his value would be 7.8BB compared to UTG's -2.3BB.

G-Bucks analysis does take cases into account where the result is heavily influenced by Hero holding one extreme of his hand range. G-Bucks analysis can not however, take the following into account:

If UTG holds a hand like AA he will obviously win G-Bucks, because of his extremely strong hand and Hero will lose G-Bucks. Regardless, the result is often more useful for theoretical analysis than the simple EV of two hands against each other. Especially when looking at balancing, G-Bucks do a good job.

SH, 100BB Stacks

Pre-flop: Hero is BB with Q, K, BU with A, T
BU raises 3BB, SB folds, Hero calls 2BB

Flop: (6.5 BB) A, J, 5 (2 players)
Hero checks, BU bets 4.5BB, Hero raises 15BB, BU calls 10.5BB

Turn: (36.5 BB) 3 (2 players)
Hero checks, BU checks

River: (36.5 BB) 8 (2 players)
Hero bets 29BB, BU calls 29BB

We want to analyse the river-bet. Hero could play every street differently but his line isn't absurd. The question is whether Hero makes a sensible bluff on the river. The answer depends on Hero's range which he will play this way pre-flop, on the flop, turn and river.

We will look at two scenarios:

• a) Hero plays all gutshots with suited broadways (9 combinations) and all his flush draws (78, 89, T9, QT, KQ, KT = 6 combinations) this way.
• b) Hero plays 50% of his gutshots (4.5 combinations) and all his flush draws (6 combinations) this way. Additionally 33% of the time he will play 55, AJ and 0.5 combinations A5s (discount because of pre-flop action) the same way (1 + 3 + 0.5 combinations).

In case a) BU gets beat by 6 hands and beats 9 himself. Because he calls 29BB into a 36.5BB pot for a showdown the result for him is:

9/15*(36.5+29)BB – 6/15*29BB = 27.7BB

BU wins a lot of G-Bucks and does not make a mistake in any form of analysis when he calls on the river. Hero doesn't have a flush draw often enough to make a profitable bluff if he decides to bluff all his gutshots. Hero himself wins G-Bucks because of the dead money in the pot (8.8BB, so a lot less than BU). The goal should be to make it a close decision for BU (here he has a risk-reward ratio of almost 100%).

In case b) things change. Hero has played a total of 15 combinations of which BU only beats 4.5 not 9. Hence the G-Bucks for BU are:

4.5/15*(36.5+29)BB – 10.5/15*29BB = -0.65BB

In this situation BU has a slight negative G-Bucks call and a much more difficult decision against Hero's range. As you can see, G-Bucks are particularly useful to have a close look at your own balancing.

A final word about G-Bucks. What they can not do in this simple form, is take into account implied or reverse implied odds. It can be the right decision to fold the potentially best hand, despite a chance of winning G-Bucks on one street, if it is likely that you will lose a large amount on the streets yet to come (out of position with a hand that is difficult to play or against a very strong opponent with big implied odds).

Conversely, it can be profitable to play against G-Bucks on one street. For example if your opponent is bad, you position is good and your hand is easy to play or if the implied odds are simply much bigger than possible reverse implied odds.

Summary

This was the last chapter of the last part of the series "Mathematical Concepts for NL-Holdem". In three parts, alongside several mathematical basics, you have learned about a few calculations relating to No-Limit Holdem. You have been introduced to examples in different scenarios which have been calculated in detail.

This way, I hope that I have motivated some people to do some analysis on their own or at least think about the results in this article. There are many situations where the simple maths from this article can lead to interesting results that can directly change your game for the better.

LINKS
Related articles: Mathematical Concepts for No-Limit Holdem (1) - EV & Ranges Mathematical Concepts for No-Limit Holdem (2) - Combos & Odds Advanced Mathematical Concepts for No-Limit Holdem (3) - Foldequity & EV Advanced Discussion Forums: Discuss the content of this article in the Forums Videos: Video series: Mathematical Concepts for No Limit Holdem 1-4