Push or Fold in Heads-up

1.1. Introduction

In this article

• Pushing strong hands which have low playability
  
• 4 different pushing charts

Let's examine the following situation:

Hero finally reached the
heads-up stage of an MTT after several hours of hard work. He is already assured a sizable amount of prize money, but would win a lot more as the winner. This makes Hero nervous and desperate to avoid making a mistake, whether it's by pressing the wrong button in an online game, or making the wrong decision on a real-life table.

Hero
sits in the small blind and receives A7. The blinds are at
5,000/10,000. Hero has 260,000 in chips. His opponent Villain is an
experienced player with a big bag of tricks, and a similarly-sized stack
of 250,000.

What to do?

Hero perceives his hand as
being strong, but is afraid of his opponent holding a stronger ace or
pocket pair. Not wanting to risk his whole stack, he raises
to 40,000, ready to fold after a re-raise all-in. His opponent
only calls, though, and the flop comes with T J 9. Not really what he
was hoping for. Villain checks, Hero makes a CB of 50,000, and Villain
pushes all-in. Hero folds without a second thought.

His
situation is now considerably worse (170,000 to 340,000) and his
experienced opponent can put him under pressure even better now. Hero
frets over the raise. Would a push have been a better choice after
all?
As we will see later: Yes. Even a push with a considerably
larger actual stack would have been +EV, whereas the raise puts him
into a difficult situation in the post-flop game should he not hit
his ace.

1.2. Goal of this article

In general, the playing style in a heads-up largely depends on the
size of the blinds in comparison with the stack:

Situation
1:
If one of the players has less than 5-10 BB, a raise is
most often useless and you should either push all-in immediately or fold.
A fine example for this is Morgoth's SnG coaching video. The pushing
range moves from Ax, Kx, Qx to any two depending on the opponent.

It is difficult to come up with clear rules when to push, as
it depends on the opponent's calling range which in turn depends a
lot on the actual course of the game.

Certainly, any ace and
any pocket pair is enough with such high blinds, but many more hands
come into consideration. Eventually you will get a relatively good
feel for such situations after numerous heads-ups, and a push here
can't be as “extremely wrong” as in situation 2.

The
so-called Sage System delivers a strategy where decisions are made
via solid (albeit somewhat inflexible and abstract) rules. The further article will therefore be based on the following situation.

Situation
2:
The stacks are still relatively large, above 10 BB. Here
you have to be more careful with your pushes (the blind steals are
less lucrative compared to the danger of running into a strong
hand).

Nonetheless it is often pointless to make normal raises with various “strong”
hands. The best example is 22: in heads-up the hand is strong by all means, as
you are at least 50:50 against most hands, and combined with the fold
equity it is a good hand. The bad playability post-flops puts you to a descision though: push or fold? How do you want to continue after a flop
of e.g. T J K against Villain's bet if he called our raise pre-flop?

The same applies to aces with a small kicker: You only hit
the ace about once out of six times on the flop. Without an ace on the flop, you
will often get into a difficult situation when the opponent puts up resistance.

The main goal of this article is to set the guidelines for when to push or fold in situation 2, so when you still have more than 10BB. Many of the results still apply for
situation 1 as well though.

Notice: To fully understand
the context, it is required that you have dealt with the Independent
Chip Model strategy article from the Silver section.

1.3. Starting situation

In heads-up high cards and pocket pairs become much stronger.
While at a full table e.g. suited connectors like 45s have about the
same value as 44 pre-flop, in heads-up the pair of fours is already
considered a monster, whereas 45s loses much of its worth and is
counted among the sub-par hands.

If you are in a heads-up, the
circumstances of pushing differentiate between two fundamental cases
(e.g. compared to the situation of a button push at the bubble):

• At the most you only have one caller. The chance to face a strong hand is thus smaller.
• The Chip EV is more or less equal to the $EV.

Based on these two facts, it is easier to calculate the EV=$EV for
certain actions or situations respectively.

Thus Hero will
always be in the small blind hereafter.

2.1. Calculation

Hero has the stack S1 and plays against Villain with stack S2,
with S being the smaller of the two stacks. Hero pays the SB,
Villain the BB, both receive their cards and Hero ponders about
either pushing or folding.

If Hero folds, his Expected Value,
which we will call EV, is obviously

(0) EV =-SB.

(In
comparison to his stack prior to paying the blinds he exactly loses
the SB he had to pay.)

If Hero decides to push, the following
might happen:

(1) Villain folds
(2) Villain calls and
loses
(3) Villain calls and wins

How do we calculate the EV
now for Hero's push?

It is made up of the three “parts”
EV1, EV2, EV3, according to the three possible outcomes (1) (2) and
(3) combined:

(1) Villain has a certain calling range (more
about the observation of different calling ranges later on). All
hands outside of this range will be folded by Villain against our
push. The probability for a fold, so for a hand outside of his
calling range, shall be F.

Therefore: EV1= F*BB

(Hero
receives the opponent's BB and his own SB. His initial stack S –
prior to paying the SB – thus increases by BB)

(2) Villain
has with the probability C=1-F (C stands for Call) a hand of his
calling range. Against this we have a certain win probability W.
If Hero has e.g. 88 and faces TT, we can roughly calculate: W=19%

EV2=C*W*S

(When Hero gets called and wins, he
receives S chips from Villain.)

(3) Villain has against Hero a win probability of (1-W). Thus the
following equation applies:

EV3=C*(1-W)*(-S)

(When
Hero gets called and loses, Villain receives S chips from Hero.)

For
the total EV of the push holds true: EVt = EV1+EV2+EV3

We
can put this on a level with the EV of the Alternative Fold EV and
receive:

(*) -SB = F*BB + C*W*S + C*(1-W)*(-S)

Now
we can solve via the value S/BB and receive:

(**) S/BB =
(F+0.5) / [C*(1-2*W)]

For a given calling range we can
calculate C and W relatively easy:

• C is the amount of hands from the Villain's calling range, divided by the amount of all possible hands. The later is 50*49/2 = 1225, as Hero already has two of the 52 cards.
• W
  can be calculated with a program like Poker Calculator or PokerStrategy.com Equilator, by including every hand of Villain's calling range in accordance to Hero's hand. Thus W always depends on Hero's hand of course.

If we included these values in (**) (for F we insert 1-C), we
would receive a ratio of S to BB. Thus a value how big S, expressed
in BB, can be at the most for the push not to become -EV. Or put
another way: When the actual stack is smaller than this value, then
the push is +EV.

Example:

Hero holds ATo. He
assumes that Villain's calling range is 66+, AT+ and A8s+.

First
we figure out the amount of hands in Villain's calling range:

As
Hero has an A and a T, there are only three different combinations
for AA and TT respectively, for any other pocket pairs 6 different
ones, so overall 7*6+2*3 = 48 pocket pairs.

The aces
AK, AQ and AJ, for which it shouldn't matter if they're suited or
not, give 12 different combinations, as the A in Hero's hand already
eliminates four of the general 16 possible combinations. For AT, the
T in Hero's hand thwarts another three combinations. A9s and A8s
finally add another three combinations respectively.

Overall
the calling range has 7*6 + 2*3 + 3*12 + 9 + 2*3 = 99
hands. From this we get

F = 1 - (99/1225) = 91.92%

For
W we have (with help from PokerStrategy.com Equilator or similar):

W =
35.5%

Included into the equation (**) it results into:

S/BB =
60.6

If the opponent's stack or your own so the actual stack, is smaller or equal to 60.6 BB, then Hero is
able to push +EV. The results depend on your own hand as well as on
the opponent's calling range.

Equation (**) goes towards
infinity, if 2*W goes towards 1 or F goes towards 1.
This makes sense since: If we have a winning probability W of at least 50%
against Villain's calling range, we are able to push with any big
stack. This is also true if he folds with a probability F=1, so every
time.

The situation W >=50% would be the case if Villain
calls with every hand and Hero has an above-average hand (everything
above Q5) or when Hero has AA. The situation F=1 happens for example
when the opponent is no longer at the table. Unfortunately, it'll
never be that easy.

2.2. Results

The results in the following three charts are rounded to one
decimal place and segmented like this:
First off there are 4 individual charts for the different types of hands with which a push is possible: pocket
pairs, Ax, Kx and connectors + one-gappers.

In the different
columns you then find the value which S, the smaller stack in BB,
must not exceed for a push to be +EV for a respective opponent
calling range.

With all cards, except the pocket pairs, it is
also differentiated between suited and offsuited. Logically, a suited
hand has a higher value in the chart than the offsuit version.

The
different calling ranges are

• (SV)
  A "Super-Villain" who only calls if his call is +EV.
• (15%)
  A tighter villian who calls with the following range: 44+, A8+, A5s+, KQ, KJs
  A hand of this range is held by Villain with about 15% probability.
• (22%)
  A looser Villain who calls with the following range: 33+, A5+, A2s+, KT+, K9s+, QTs+
  A hand of this range is held by Villain with about 22% probability.
• (29%)
  A very loose Villain who calls with the following range: 33+, A2+, K9+, K7s+, QT+, Q9s+, JTs
  A hand of this range is held by Villain with about 29% probability.

The category (SV) is thus the worst case scenarios so to speak. In
reality, it won't happen though as our opponent doesn't know our
cards. Still we should regard this category as lowest limit for our
push: If it is even +EV against this calling range, it's certainly so
against every others as well.

For the blank cells the push is
+EV regardless of the stack size (S can have any value). From this
results that Hero has a win probability of greater or equal than 50%
against Villain's calling range in these situations (compare with
notice near the end of chapter 2.1).

Instruction on using the charts:

We have a hand in the small blind and don't really know if we are
able to push with it +EV. We guess the opponent's calling range. If
we're unable to do that, we use the value of the column (SV) and
compare this value with the value of the actual stack S expressed in
BB. If S is smaller, we can push +EV.

The actual stack in BB
is as usual the smaller one of both stacks prior to paying the
blinds.

Example:
Our stack is 150,000, the
opponent's 200,000. We pay the SB of 4,000, the opponent the BB of
8,000. This means that the actual stack S in BB is 150,000/8,000 =
18.75.

We receive A6o. We are unable to evaluate the opponent
and we are under time pressure, so we take the value 28.6 of the
column (SV). As S in BB is smaller than 28.6, we're able to push +EV.

2.3. Remarks about the charts

You can clearly see that as opposed to the A-hands in chart 1, the
stacks need to be smaller in order to push with K-hands, which is logical as a
king in heads-up is by far not as strong as an ace. Furthermore, the
hands of the category Connectors + One-Gappers are better suited
against the tighter player type (15%). Against (15%) a hand like 98
(S=28.9, or 36.5 when suited, resp.) comes off slightly better than
A2 (S=28.8, or 36.4 resp.); against (22%) and certainly against (29%)
A2 is clearly ahead (concerning the maximum stack size for a +EV
push).

If you compare the values in the different columns, so
the different calling ranges, you notice: (SV) is always the smallest
value. The value of (SV) is really the lowest estimate. Even if we
revealed the cards and the opponent was able to perfectly calculate
the win probabilities and pot odds, a push with this stack size would
still be +EV.

The other values are related depending on the
strength of the hand:
For relatively weak hands (e.g. K5o) (29%)
is smallest, at a certain strength of the hand (e.g. A2o) the value
(29%) is higher than (22%), and finally for strong hands (e.g. A7o)
(15%) is the smallest of the three values.
That's because with
weak hands, we're clearly behind all three calling ranges if we get
called, and thus the push benefits more from the fold equity, than in
the case of a stronger hand. The fold equity is highest against the
(15%) range and lowest against (29%) though.

Of course a push
is, even when being +EV, not always the best decision. With monsters
like AA, KK or QQ, a slowplay with relatively big stacks is often
more profitable, and with hands with a good playability like AK, KQ,
KTs etc., and any hands from chart 4, a raise is often more
profit-yielding as well.

The values in the columns of the
category (SV) correspond more or less with the so-called Sklansky –
Chubukov rankings.

If we assume Villain's calling range to be
somewhere in between those categories, we can also use the average
value or pick the smaller value alternatively. That way, we play safe
and round down our pushing range.

Example:
We have 44
and assume that Villain's calling range is a bit tighter than (15%).
If we use the value 41.3 from the (SV) column, we're on the safe
side, but we can also pick a rough average value between (15%) and
(SV), about 50. This corresponds with an opponent who is even
tighter, or calls more profitably against us in his opinion, than
with the (15%) calling range, yet not quite as perfect as with the
imaginary SV.

3. Some thoughts about pocket pairs

With a pocket pair we can mostly push, unless we want to lure the
opponent into the pot with a slowplay. This is only advisable with
real monsters (AA, KK, QQ, maybe JJ) though because:

• The probability for one or several overcards on the flop is very high for pocket pairs which are smaller than QQ. If there are overcards, it is often difficult to continue as we assumed the stacks to be relatively big, and we are often behind. We can expect to pay a lot more with only a few outs to the showdown.
• Our chance to improve the hand with community cards is much smaller than the opponent's (unless he also has a pocket pair). Therefore, the opponent is in the advantageous situation of being paid off even more if he hits, but getting away cheaply if he doesn't.

In addition: If we already pushed quite often previously, a raise
could become more suspicious than a push, and Villain might fold a
hand like A7 against our AA, even though he usually would have called
an all-in.

A push is thus the best option in most cases and
you notice in chart 1 that the stacks have to be very big already for
a push to no longer be +EV.

4. Conclusion

Naturally, it's not possible to memorize all the chart values.
Online players have the advantage of being able to print out the
charts and quickly check them in a critical situation.

If you
are unable to evaluate your opponent's calling range due to lack of
time or not enough revealed hands, it's always safe to pick the value
from the column (SV), even if you do end up missing a few +EV
pushes.

In conclusion let us examine the transition of these
values onto other situations.

Cash games:
The
chart values can be used in cash games without any restrictions, as
Chip EV = $EV certainly applies. Thus you are able to push
from the SB, provided that every player in front of you folded,
according to the values, so e.g. with an actual stack of 53.6 BB with
AT.

Most players won't feel good about this though, as you
usually don't want to risk $100 (= 50 BB) in a $1/$2 cash game.

But
on long term, this move is profitable in any case, even more so as
the opponents will often muck dominating or tying hands (in the
example of AT those would be AT, AJ, 22, 33, 44, 55...).

Tournaments
apart of heads-ups:
In MTTs and SnGs, the situation outside of the
heads-ups changes, as usually Chip EV = $EV no longer applies.
Depending on the situation, the chart values thus have to be lowered
for a push to still be +$EV, the higher the discrepancy between Chip
EV and $EV is. More about this can be found in the article on ICM. Eventually
there will be a second part of this article, in which the charts will
be adjusted with ICM situations.

We want to specifically warn
about using these charts for pushing at the bubble though.

Hero
is on the button:
If you are
not in the SB, but first to act on the button, you might consider a
similar calculation. It will become more complicated though as now
two players might call. A good approach for such a situation in a
cash game is to divide the chart values by 2. For tournaments you
have to pay heed to the ICM again.

Hero is in the
BB:
Here the situation
depends a lot on how we evaluate the opponent's limping. If we assume
that he either raises or limps with any two, as he already did this
in the previous 10 hands, we are able to push similarly to the values
in the charts. Actually we can even do it for somewhat bigger stacks,
as we won't only get 1.5 BB, but 2 BB. This then somewhat compensates
the small probability of an opponent's trap.

If the
opponent usually raised aggressively before and now suddenly only
limps, you have to be careful. The increased danger of a trap could
perhaps let the push become -EV with the chart values.

5. Appendix

Here now a list of probabilities with which the charts were
calculated in chapter 2.3. As explained in the derivation of the
equation (**), C designates the probability of Villain holding a hand
of his calling range, and W our win probability against Villain's
calling range.