Pre-flop: Starting Hands and Equity incl. Big Blind Defense
1. Introduction
In this article
- Background information: Equity and Modified Outs
- Background information: Equity and ORC (Open Raising Chart)
- Implications in various situations
This article will take a look - both at the surface and in depth- at
the concept of equity, or rather, determining the probability that a
hand will win and putting this knowledge to use with respect to your
pre-flop game equity and the ORC. This is an article that you should
most definitely read, re-read, and re-read again.
It is imperative that you have developed a solid post-flop game; there
is no sense in pressing out a minimal pre-flop edge just to lose it
with poor play after the flop.
Don't be deceived when the article refers to seemingly trivial and
obvious concepts. The difficulty lies in correctly interpreting how
various aspects interplay and putting the theory to practice. There is
rarely a single, correct answer.
To keep things from getting too abstract, we will start with an example and go on to simplify the concepts and strategy.
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2. Modified Outs
Since poker is a game of partial (or incomplete) information, (almost)
everything is a matter or probability. Playing the game can be seen as
an attempt to make sense of all these probabilities. Before we move on
to the central concept of equity, let's take a look back at Bronze
Section. Working with probabilities - on an intuitive level - has been
a part of your poker career from the very beginning.
The concept of 'modified (discounted) outs' was
introduced in the article on 'Odds and Outs.' As you will see, this
concept goes hand in hand with the concept of equity. Modified outs are
referred to as 'clean' outs, meaning outs that really will give you the
best hand. If, for example, you have a straight draw with 8 outs and
there are two cards of the same suit in the flop, you can't give
yourself all 8 outs, since an opponent will, at times, have a flush
draw. In a worst case scenario you only have 6 outs for the best hand.
You are sure to have noticed that this line of thought is also moving
on an abstract level. Your opponent won't always have a flush draw in
such a situation. Modifying, or discounting, outs is a way of taking
this possibility into account and answering the question, "How many
outs will I have on average in this situation if I were to play it out
over and over again?"
Modified outs are primarily a measuring tool used to determine the
average probability of winning a hand when you are behind at the moment
but still have outs for the best hand. After determining the
probability of winning the hand you can decide whether or not it is
profitable to stay on your draw based on the size of the pot.
Outs (number of cards) and modified outs
(probability) or only indirectly related. The similarity in name comes
from the fact that it is much easier to perform these calculations and
come to a conclusion on how to best play your hand by using this
method. The correct move is then easily determined by comparing the
odds of winning to the pot odds, as you well know.
3. Definition of equity
The concept of equity is basically a
generalization of the concept of modified outs. This was defined in the
Silver article 'Switching to Short-handed Play.'
Equity is the probability of the hand and beating given hands (or hand ranges).
Unlike modified outs, equity is generally expressed as a percentage
very hard to determine exactly. It is often not even an objective
figure, but varies depending on opponents, position, the playability of
your hand, initiative etc. (detailed explanations follow). The
'probability of winning' is, as with modified outs, the probability of
holding the best hand on the river.
We will not try to reach a more exact definition, because we can use
our definition as it is in many different contexts without having to
worry about how 'exact' it is. We are interesting in knowing, for
example, whether or not the (assumed) equity is enough to play against
the hand an opponent is likely/suspected to have. This can often only
be determined by making estimations. This obviously implies that
mistakes can be made in doing so, but trying to use the information you
do have is better than trying to make your way through the dark.
4. The relation between modified outs and equity
The concepts of modified (discounted) outs and
equity are both based on probability. The difference lies in the
original question, or rather the contexts in which the two are used.
Modified outs tend to be somewhat more precise than equity.
The following will take a look at a problem from both points of view:
When you have a nut flush draw on the flop (and assume you are behind),
you have 9 outs to complete your draw on the turn. You can then
determine whether or not it would be profitable to stay in the hand
based on the pot size and base your decision on the concept of modified
outs.
Things are different when you use the concept of equity. The line of
thought is then: If I don't complete on the turn, I have another 9 outs
to complete on the river (the danger of running into a full house or
better being ignored). In other words, the probability of completing my
draw by the river is 9/47 + 38/47*9/46 = ~ 0.35.
This means that 1/3 of the pot 'belongs' to you on the flop, regardless
of what is to come (this figure being an average). Since you will
complete app. 35% of the time and win, you say you have 35% equity on
the flop. A direct conclusion from this thought process is that you
should try to make the pot as large as possible with a nut flush draw
against two opponents (assuming there is no danger of a full house or
better), since a little over 1/3 of the pot belongs to you and your
costs are exactly 1/3.
Your equity shifts dramatically on the turn. If you complete, it will
be 100%; if you miss, it will fall to app. 20%. But this doesn't change
the fact that it was 35% on the flop. This concept of equity along with
the idea of 'How much of the pot do I own?' is the basis for our theory
of semi-bluffing, as well.
Let's look at another example to get a better idea of what exactly is
meant when we talk about equity: You have AKs, your opponent TT. Who is
favored to win? Finding the definitive answer to this question is
seemingly impossible. Data bank analysis and simulations prove,
however, that a pre-flop equity advantage leads to a positive expected
value. In other words, this advantage holds, on average, all the way to
the river. The answer to the question is clearly worth looking for.
The simplest way is to perform a simulation. This can be done with programs like the Equilator. When you pit AKs and TT against each other in the Equilator, you will get a result of 54.1% equity for TT and 45.9% equity for AKs. The Equilator
basically puts the players all-in and looks to see who had the best
hand on the river - over and over again. The value we receive is, of
course, a theoretical value, since the players are all placed all-in
immediately, and is not 100% accurate, since we are playing Fixed Limit
and not No Limit.
The program tends to overrate backdoor draws, which you might have
folded on the flop. Still, it's a good reference point, or, at least,
the best we've got. We will use the equity values determined by the Equilator as our basis for equity.
Let's take our example to the next level. You can see how dramatically equity can shift on various possible flops:
- Flop 652 Rainbow: TT 72.3 %, AKs 27.7 %
- Flop A62 Rainbow: TT 8.5 %, AKs 91.5 %
This does not contradict our statement about a pre-flop equity
advantage holding to the river. What is meant is that the advantage is
retained on average. A good flop for AKs is simply less likely than a
good flop for TT. After all, AKs has to hit to win.
TT is clearly the pre-flop favorite against AKs, even though this can
change after the flop. Whatever ends up happening on the flop is
irrelevant to your pre-flop equity (remember the nut flush draw
example). The fact that the many different possible flops cause a shift
in equity is taken into account in the simulation.
If you don't have any experience with the Equilator,
play around with it a bit. You may notice, for example, that 22 is
barely favored to win against AKs (whereas TT was a clear favorite).
Look at how equity changes when your hand is suited/non-suited. There's
a lot you can learn and a little practice never hurt anyone.
5. Average equity
Putting two hands against each other in a simulation shows the basis on
which equity is calculated. You can also pit more than two hands
against each other. Of course, you never know exactly what hands your
opponents have at the table, but you can get information from the
PokerTracker stats. We'll come back to this in a moment.
This brings us to another fantastic function of the Equilator:
You don't have to enter an exact hand; you also have the option of
entering a hand range. Let's look at an example with QJs against two
random hands.
Enter QJs, random and random in the Equilator
and you will get a result of 44%. This means you will win less than
half the time on average. Does this mean you shouldn't play your hand
after all? No, in fact you should. It's a matter of how much you can
expect to win. If you were to play this situation out 100 times, you
would pay 100 bets, and win 44 * 3 = 132 bets back on average (your own
and one from each opponent).
This leads us to a new concept: average equity. Average equity is the
equity a randomly selected hand has against other randomly selected
hands. With two players the average equity is 50% for each, with three
players 33.3% for each, with four players 25% for each etc. Going back
to our example, there are three opponents in the hand, meaning each has
an average equity of 33.3%. QJs has 44% equity, which is higher than
the average equity, which is why it can be expected to be profitable.
We can generalize and say that any hand whose
equity is greater than the average equity is profitable. This can be a
number of hands. Try it out with AKo and TT against a random hand.
When you pursue this line of thought, pre-flop play may seem to become
trivial: Just play hands with a higher equity than the average equity.
Raise when you have an advantage in equity to increase this advantage.
This is our goal, but there is no room for talk of triviality. So how
can we make the principle of equity the basis of our pre-flop play?
6. Equity and ORC
A possible first approach could be entering your hand in the Equilator,
entering random hands for the players behind you and seeing if your
equity is greater than the average equity. You can use the Equilator
to calculate your equity with KTo from MP2 against 5 random hands. You
will see that KTo has 21.5% equity, well above the average equity of
16.67% with 6 players in the hand.
But this only counts when all players behind you call, both those with
strong hands and those with weak hands. And this will, of course, not
be the case. The weak hands will fold, the strong hands will re-raise.
You will end up facing a strong hand and having a distinct disadvantage
in equity. You have to take the probability that a player behind you
will have a strong hand into account. Don't worry though, help is in
sight; the ORC.
All the hands found in the ORC are guaranteed to have an above average
equity against the opponents behind you. This advantage is real,
meaning, for example, that the possibility that there could be stronger
hands behind you has been taken into account. Following the ORC is
obviously no guarantee that there won't be a stronger hand behind you,
but it reduces the probability enough for you to have sufficient equity
on average to play your hand. The ORC is based on numerous data
analyses and a whole, whole lot of experience.
7. PokerTracker data
Since we are assuming that the hands listed in the ORC have an equity
advantage when you are first in, your action is clear: You must raise
to increase this advantage. But what should you do if an opponent (or
several) has already entered the hand? You only want to play when you
have an equity advantage, but you don't know what you're up against.
The PokerTracker data provides information on your opponents' pre-flop
playing style. The following stats are relevant in this context:
- Voluntary Put $ in Pot (VPiP) The percent of his hands he plays.
- Pre-flop Raise (PFR) How often he raises before the flop.
- Attempt to Steal (AtS) How often (%) he tries to steal the blinds.
- Total Hands Size of sample pool.
- Total Aggression Post-flop Ratio between the number of bets/raises to calls after the flop.
- Went to Showdown (WtS) How often (%) he took his hand to the showdown after seeing the flop.
- Folded SB to Steal How often (%) he defends his SB against a steal.
- Folded BB to Steal How often (%) he folds his BB to a steal.
But be warned: PokerTracker data is worthless if your sample pool is
not large enough. Tendencies can be recognized fairly quickly with
regard to VPiP and PFR, but 100 hands aren't really very many. We will
assume that the PT data used in this article is very reliable. The
stats that are not based on pre-flop actions are used to identify an
opponent's general playing style. As you will see, you play differently
against a rock than you do against a maniac.
8. Reacting to an opponent's raise
8.1. Estimating his range
Let's start by looking at the situation when an
opponent raises in front of you. You see his PFR stat, that being the
frequency (%) with which he raises before the flop. He obviously does
this with the best hands, meaning we can narrow down the hands he might
have fairly accurately.
The PFR value isn't directly useable, since almost every sensible
player plays tighter when he doesn't have good position. PT doesn't
take position into account, it only gives you the average. Since our
goal is to determine his hand range, we have to make a rough estimate
when it comes to the various positions from which the raise can be
made.
Here is one way of doing this:
- Hand range for a raise from MP2 = ~ 0,75 * PFR
- Hand range for a raise from MP3 = ~ 1 * PFR
- Hand range for a raise from CO = ~ 1.5 * PFR
- Hand range for a raise from BU = ~ 2 * PFR
You should try to estimate the range from the CO and BU positions, even
though they are in the AtS stat. AtS also takes raises from the SB into
account. If, for example, an opponent with a PFR of 16% raises from
MP2, you should give him a hand range of 0.75 * 16% = 12%.
8.2. Excursus 'range'
Before we go on we need to make sure you really understand what is
meant when we talk about ranges. What range do you need to have at
least more than 50% equity against a 12% range?
a) < 12 %
b) = 12 %
c) > 12 %
Answer a) is correct. Imagine listing every possible hand according to
strength. The best hand is at the top of the list, the worst at the
bottom. Keep in mind that there is no exact order of hands; you can
even find different versions from the same author (Sklanksy). Either
way, the exact order is not very important for our purposes.
A range is a certain section in the list of all possible hands. Your
opponent has three options before the flop: raise, fold and call. He
will raise with a hand that in the upper portion of the list (his
raising range), call with a hand in the middle portion (his calling
range), and fold hands on the lowest end (his folding range). With PT
stats you can narrow down the hands for each action relatively
accurately.
Within each range is a sub-range which is divided between the better
hands and the weaker hands within the entire range. Of course, it's
impossible to say exactly what hand an opponent has. Every possible
hand within his range is probable (assuming you are differentiating
between 26*51 different hands and differentiating between AKs and AKo.
AA is obviously less likely than AKo, but AKo increases the range more
than AA does).
If you were to play against your opponent with the same range he has,
you would be playing with the weakest hand in that range. This hand
will obviously only have 50% equity against your opponent's hand when
he has the exact same hand from the range. And this is very unlikely.
You would have less than 50% (sometimes much less) against every other
hand in his range. So why would you want to play the hand? As you can
see, your range must be smaller than your opponent's if you are to have
at least 50% equity against him. (And it should go without saying: If
playing with the same range is a poor idea, playing with an even larger
idea is an even worse idea)
8.3. Minimal (required) equity
Let's go back to our example. The real question
is, "How much equity do you need?" You obviously want to raise when
your equity is greater than 50%. But our goal isn't to raise as many
hands as possible, our goal is to win as much money as possible.
There's already money in the pot with the blinds and now it's a
free-for-all to see who is going to take it home (no one would play if
it weren't for the blinds; everyone would just sit and wait for AA).
This is why you don't have to have at least 50% equity to play.
Assuming the blinds fold and one opponent calls, there will be 7.5 SBs
in the pot after you 3-bet, of which you only invested 3. By now it
should be very clear why you don't need 50% equity to have +EV.
46% equity is usually sufficient. Since, however, this is a marginal
figure that depends on the opponent you are facing and you must react
correctly to play your hand profitably, we will take a look at how we
arrived at this figure.
If both blinds fold, there will be exactly 1.5 SBs
of dead money in the pot. But the hand isn't over with a single 3 SB
re-raise. You may have to invest again on the flop, turn and river.
Playing your hand against an average opponent will usually cost a total
of app. 6 SBs by the time you see a showdown.
The entire pot will then be 2*6 SBs (6 from each player) + 1.5 SBs (dead money) = 13.5 SBs.
In order to have +EV (and that is our only goal!), your equity must be
at least as high as the ratio of costs : winnings. In this case you
would therefore need: Equity = 6/(2*6+1.5) = 0.444 = 44.4%.
This can be written as a formula to make it easier to see how changing
the parameters effects the amount of equity you need to play your hand.
- D is the investment (both yours and your opponents')
- M is the dead money
- The pot is 2*D + M in size.
Therefore: Equity = D/(2*D + M)
Two assumptions were made in this calculation: your opponent is average
and the blinds will fold. Let's see what happens when we change these
parameters.
The players in the blinds play very well and defend their blinds accordingly.
The amount of dead money in the pot will then only be 0.75 SBs. When playing against an average opponent you would then need:
Equity = 6/(12 + 0.75) = 0.47 = 47%
If the blinds are terrible players and call with too many weak hands, the amount of equity you need decreases significantly.
If the raiser is loose-aggressive, the entire costs will increase.
The entire costs will then increase to app. 8 SBs. Against good blinds you would then need:
Equity = 8/(16 + 0.75) = 47.8%
If, on the other hand, your opponent is a rock and often folds on the
flop (or if you know you can get out of his way if he check/raises the
flop), the entire costs will only be around 5 SBs. You would then need:
Equity = 5/(10 + 0.75) = 46.5%
Our examples with the formula show: the more aggressive and the looser
the raiser, the higher the entire costs will be on average; and the
better the players in the blinds, the less dead money to be won,
meaning you need all the more equity to play. As you can see, our
assumption that 46% equity is sufficient is pretty marginal.
Let's return to our example with the raise from MP2 (PFR 16%). Since he
is raising from MP2, you put him on a range of 12%, which is roughly
made up of 77+, A9s+, KTs+, QTs+, JTs, ATo+, KJo+.
It's impossible to say for sure if he has this range or another range
of 12%. He might be the type who raises A8s, but not QTs. It's
ultimately a matter of speculation, but such deviations will usually be
pretty minor.
And this is where one of the Equilator'sfantastic
functions becomes very useful. You can easily determine the range of
hands to 3-bet with. You can even say how much min. equity you wish to
have with your range. All you have to do is enter your opponent's
range, enter the min. amount of equity you wish to have against his
range, and then let the Equilator do the rest. The result is the range with which you can 3-bet to try to isolate the pre-flop raiser.
Once you have Equilator
up and running you might want to try varying the amount of equity you
wish to have to see how this effects your range. With which hands can
you re-raise when the blinds or weak or your opponent a rock/maniac
etc.? This is one way of getting a feeling for when you can (and
should) play hands that aren't in the charts (keep in mind that the
charts exist to help deal with such situations in the first place). The
Approx charts are good. You can find them in the Forum or download them
as .pdfs below:
There is a 3-betting chart along with the ORC in
the guide to switching to short-handed play. The 3-bet chart is based
on the assumption that your opponent is raising according to the ORC.
His PFR stats would then be:
- MP2: 13.1 %
- MP3: 17.1 %
- CO: 24.6 %
- BU: 37.7 %
- SB: 62.2 %
The chart is also based on the assumption that the raiser and blinds
are normal players, meaning you want app. 46%+ equity. You can then use
the Equilator to see how you should react to tight/loose opponents. All you have to do is change the min. amount of equity you wish to have.
9. Small Blind defense
An exceptional case can be made for making an isolation raise as part
of your Small Blind defense, since you were already forced to
contribute to the pot and your total costs will therefore be less. We
can use the same formula: Equity = D/(2*D + M).
When defending your SB (and assuming the BB will fold), you therefore need:
Equity = (D – 0.5)/(2*D + 1)
You have already paid 0.5 SBs and there is only 1 SB in dead money. When facing an average opponent, you need:
Equity = 5.5/13 = 42.3%
This is why we could say that 43% equity is sufficient for 3-betting
from the SB - at least in theory. Putting the theory to practice,
however, showed that constantly playing with a sub-optimal hand from
the worst position is not profitable. This is why you really need at
least 46% equity to defend your SB with a re-raise. You already know
how to use the Equilator to determine the range that gives you this much equity.
10. Big Blind defense
Things are a bit more complicated when you are the Big Blind. You've
paid a fair amount already and isolation is no longer an issue (there
aren't any opponents left), so you can consider straying from your
standard 'raise or fold' line and think about just calling. You should
obviously raise if your equity is 50%+, but you can also call with
weaker hands since you are getting such good pot odds.
We make another assumption for our formula and assume our total costs
will be D = 4 SBs on average (it must of course be less than in the
previous examples since we are talking about calling from the BB and
not 3-betting). The costs will definitely be 1 SB less, since you were
already forced to invest that amount. This leads us to the following:
Equity = (D – 1)/(2*D + 0.5) = 3/(2*4 + 0.5) = 3/8.5 = 35%
35% is the minimal amount of equity needed and does not depend on your
opponents as it did in the previous examples. The blinds don't matter
anymore (we are hoping the BB is pretty good), and the difference
between a rock and a maniac isn't very important either, since the pot
is small and you have an informational advantage and can exploit it
with lines like check/raise and check/fold.
The BB defense chart is based heavily on having a minimal equity of
35%. There is, however, one more correction to be made. It's a matter
of how easy or difficult it will be to play the hand after the flop,
which leads us to our next section on 'playability.'
11. Playability
11.1. Definition
A hand has high playability when the correct way of playing the hand is
the same as the optimal way of playing it. Playing correctly means
making the right decision in the situation at hand, playing optimally
means playing as if you could see what your opponent has. Having high
playability basically means you will more or less know where you stand
with your hand on the flop.
EXAMPLE:
KQo has 52% equity against an ORC raise from the BU, A9o 54%. But what
will happen after the flop? If you hit the flop with KQo you will
usually have top pair with a good kicker. Things aren't as clean cut
when you have A9o. If you hit the A you will have to worry about
running into an ace with a better kicker; if you hit the 9 you will
have to worry that he might have a better pair. Both hands might even
be the best hand even if you miss. (Both KQo and A9o have more than 50%
equity, a clear raise - see below for more)
Connectors have high playability, especially when they are suited. A high card combined with a low card has poor playability.
11.2. Playability and equity
How does playability affect equity? It would be very helpful to be able
to quantify this influence. Here is one attempt: Compare 2 hands that
have the same EV as a random hand, namely A2o and 98o (the EV of both
hands is -0.09 according to PokerRoom EV stats - this is based on
empirical data analysis).
A2o has app. 55% equity against a random hand, but
very poor playability according to our definition. 98o only has 48%
equity, but has relatively good playability. The EV is equal despite
the 7% difference in equity.
We can conclude that very high/very low playability (compared to the
average) can account for a difference of +/- 3.5% in equity.
This is a huge difference and has consequences when it comes to putting
the theory to practice: A2o has 35.6% equity against an open raise from
MP2, but it has very poor playability. You have to devaluate your hand,
after which folding is clearly the best option. A hand like 65s, on the
other hand, which only has 34.1% against an open raise from MP2, has
high playability and can therefore appreciates in value, meaning it can
land in the 35% range. You can find other examples in the Blind Defense
Chart among others.
11.3. A strategic consideration
Let's take one more look at the example above in which the playability
of KQo and A9o were compared. Both have more than 50% equity, so
raising can't be completely wrong. The key difference lies in the
playability of the two hands. The flop is much more important when you
have KQo than when you have A9o, since you will usually know pretty
well where you stand with KQo once you've seen the first three cards.
Instead of raising pre-flop, which basically commits you to
contibetting, you can just call and see what shows up on the flop. If
it's no good, you can fold and save yourself 2 SBs. If you hit, you can
check/raise, after which the pot will be just as large as it would be
after a 3bet/call and bet/call on the flop. The profit you miss out on
by not raising with more than 50% equity is more than compensated for.
12. How to play against limpers
You still play according to the principle that you
only want to play (with play meaning raise) when you have an equity
advantage and when your hand is listed in the ORC. The second condition
(ORC) guarantees that you also have an equity advantage against the
other players behind you.
There are, however, two differences between playing against a limper
and playing against a raiser. First of all, there will usually be less
dead money in the pot, since the blinds will defend against a 2-bet
much more often than against a 3-bet. You also have to assume that you
need app. 50% equity against the limper.
Determining your equity against his range is difficult, since you never really know what he is limping with. The idea that VPiP-Range – PFR-Range = Limp-Range
is very simple, but it's not very accurate. We aren't interested in a
percentage, we want a concrete range. And this depends in part on
knowing what hands are in his raising range.
Does he usually just raise with high cards or does he also raise with
small pocket pairs? There are also a number of possibilities when it
comes to weaker hands he would limp with. Does he limp with (nearly)
every ace, or does he prefer limping with (suited) connectors? When you
try to establish his limping range you will find that there is a grey
area on both ends (where his folding range ends and where does his
raising range begins). This obviously has an impact on your raising
range against him. Unfortunately, you can't expect to find any exact
answers; as we've said so often, poker is a game of partial
information.
The Approx charts are a great starting point. One of the intentions of
this article was to provide a theoretical basis for you to use when
experimenting on your own. The Equilator
is a great tool for such experiments. You could, for example,
experiment with how you can use the Approx chart in relation to cases
where the same percentage ranges consist of different types of hands
and how this effects your own range.