When can you profitably go all-in with a hand? Which hands can you call an all-in with? Answers to these questions are provided by pushbot charts and equilibrium strategies. This article will provide equilibrium push bot charts for poker tournaments and explain how and when to use them.

Example situation in a tournament

Well, let’s say you Blackjack What Does Push Mean deposited €100 and got another €100. Now you have €200 to play with. Now you have €200 to play with. You bet €200 on colour in Roulette you basically have 49% chance to win €400 when only risking €100 of your own money. While many tournaments do permit this type of agreement, tournament rules vary as to if and when it is allowable to push bets. EXAMPLE: 'I was a slight chip leader and had a decent chance of winning first place, but I agreed to save bets. A house dealer who has come to replace the current dealer at the table, such as during a shift change. EXAMPLE: 'I will be leaving shortly. My push should arrive soon.' APPLIES TO: Land-based Venues. To award chips to the winner of a hand. EXAMPLE: 'The dealer pushed me the pot.' APPLIES TO: Online and Land-based Venues. Push/Fold is a common strategy used in tournaments in which you either go all-in preflop or fold. Push/fold should be utilized when your stack becomes short — around 15 big blinds (bb) or fewer. There are a ton of different push/fold charts available, but we will be using the program that those charts get their data from — ICMIZER. PokerNews.com is the world's leading poker website. Among other things, visitors will find a daily dose of articles with the latest poker news, live reporting from tournaments, exclusive videos.

Let’s say we’re in the late stages of a big tournament. The stacks are shallow and most action happens before the flop. We’re in the small blind with a mediocre holding (say K6) and a small stack (say 9 big blinds). It is folded to us. What do we do?

Or, let’s say we’re in the big blind with the same hand and the same stack. Everybody folds to the small blind – a good and aggressive player – and he goes all-in. What do we do now?

Pushbot charts help making decisions in situations like the ones in this example. In this article we’re going to focus on equilibrium pushbot charts which work for push-or-fold decisions between the small blind and the big blind and to some extend between the button and the blinds.

Equilibrium pushbot and callbot charts

The tables below show the maximum effective stack for a profitable push from the small blind against the big blind and the maximum effective stack for a profitable call from the big big blind according to equilibrium strategies. (The effective stack is the smaller of the stack sizes between you and your opponent.)

Equilibrium pushbot chart for moving all-in from the small-blind

Suited Cards

O f f s u i t C a r d s

AA50

AKs50

AQs50

AJs50

ATs50

A9s50

A8s50

A7s50

A6s50

A5s50

A4s50

A3s50

A2s48

AKo50

KK50

KQs50

KJs50

KTs50

K9s50

K8s50

K7s49

K6s36

K5s32

K4s26

K3s20

K2s19

AQo50

KQo50

QQ50

QJs50

QTs50

Q9s50

Q8s50

Q7s20

Q6s29

Q5s24

Q4s16

Q3s14

Q2s13

AJo50

KJo50

QJo50

JJ50

JTs50

J9s50

J8s50

J7s32

J6s19

J5s16

J4s14

J3s11

J2s8.8

ATo50

KTo50

QTo45

JTo46

TT50

T9s50

T8s50

T7s36

T6s25

T5s12

T4s11

T3s7.7

T2s6.5

A9o45

K9o24

Q9o24

J9o29

T9o32

9950

98s50

97s36

96s27

95s14

94s6.9

93s4.9

92s3.7

A8o43

K8o19

Q8o13

J8o14

T8o18

98o21

8850

87s43

86s31

85s19

84s10

83s2.7

82s2.5

A7o41

K7o16

Q7o10

J7o8.5

T7o9.9

97o11

87o16

7750

76s36

75s24

74s14

73s2.5

72s2.1

A6o35

K6o15

Q6o9.8

J6o6.5

T6o5.7

96o5.2

86o7.1

76o11

6650

65s29

64s16

63s7.1

62s2

A5o37

K5o14

Q5o8.9

J5o6

T5o4.1

95o3.5

85o3

75o2.6

65o2.4

5550

54s24

53s13

52s2

A4o35

K4o13

Q4o8.3

J4o5.4

T4o3.8

94o2.7

84o2.3

74o2.1

64o2

54o2.1

4450

43s10

42s1.8

A3o32

K3o13

Q3o7.5

J3o5

T3o3.4

93o2.5

83o1.9

73o1.8

63o1.7

53o1.8

43o1.6

3350

32s1.7

A2o29

K2o12

Q2o7

J2o4.6

T2o3

92o2.2

82o1.8

72o1.6

62o1.5

52o1.5

42o1.4

32o1.4

2250

You can shove all-in profitably:

• If you are in the small blind,
• everyone before you has folded,
• your effective stack (in big blinds) is smaller than the number given in this table.

Note: the maximum stack size considered is 50 big blinds.

Download this chart as PDF

Calculations by Bill Chen and Jerrod Ankenman (Mathematics of Poker)

Equilibrium callbot chart for calling an all-in in the big blind

Suited Cards

O f f s u i t C a r d s

AA50

AKs50

AQs50

AJs50

ATs50

A9s47

A8s41

A7s36

A6s31

A5s30

A4s26

A3s25

A2s23

AKo50

KK50

KQs50

KJs45

KTs32

K9s24

K8s18

K7s15

K6s14

K5s13

K4s12

K3s11

K2s11

AQo50

KQo46

QQ50

QJs29

QTs24

Q9s16

Q8s13

Q7s11

Q6s10

Q5s8.9

Q4s8.5

Q3s7.8

Q2s7.2

AJo50

KJo27

QJo20

JJ50

JTs18

J9s14

J8s11

J7s8.8

J6s7.1

J5s6.9

J4s6.2

J3s5.8

J2s5.6

ATo50

KTo24

QTo16

JTo13

TT50

T9s12

T8s9.3

T7s7.4

T6s6.3

T5s5.2

T4s5.2

T3s4.8

T2s4.5

A9o40

K9o18

Q9o12

J9o9.9

T9o8.5

9950

98s8.3

97s7

96s5.8

95s5

94s4.3

93s4.1

92s3.9

A8o35

K8o14

Q8o9.8

J8o7.7

T8o6.7

98o6.1

8850

87s6.5

86s5.6

85s4.8

84s4.1

83s3.6

82s3.5

A7o29

K7o13

Q7o8

J7o6.4

T7o5.5

97o5

87o4.7

7750

76s5.4

75s4.8

74s4.1

73s3.6

72s3.3

A6o22

K6o11

Q6o7.4

J6o5.4

T6o4.7

96o4.2

86o4.1

76o4

6650

65s4.9

64s4.3

63s3.8

62s3.3

A5o21

K5o10

Q5o6.8

J5o5.1

T5o4

95o3.7

85o3.6

75o3.6

65o3.7

5543

54s4.6

53s4

52s3.6

A4o19

K4o9.2

Q4o6.3

J4o4.8

T4o3.8

94o3.3

84o3.2

74o3.2

64o3.3

54o3.5

4432

43s3.8

42s3.4

A3o17

K3o8.8

Q3o5.9

J3o4.5

T3o3.6

93o3.1

83o2.9

73o2.9

63o3

53o3.1

43o3

3322

32s3.3

A2o16

K2o8.3

Q2o5.6

J2o4.2

T2o3.5

92o3

82o2.8

72o2.6

62o2.7

52o2.8

42o2.7

32o2.6

2215

You can call the all-in profitably:

• If you are in the big blind,
• everyone before the small blind has folded,
• the small blind moved all-in,
• your effective stack (in big blinds) is smaller than the number given in this table.

Note: the maximum stack size considered is 50 big blinds.

Download this chart as PDF

Calculations by Bill Chen and Jerrod Ankenman (Mathematics of Poker)

How to use these equilibrium charts?

Let’s go back to the example situations above. We’re in the small blind with K6 and a 9 big blind stack. It is folded to us. What do we do?

Checking the first table reveals that K-6-offsuit has an equilibrium push rating of 15 big blinds. Meaning, as long as our stack is 15 big blinds or smaller we can profitably move all-in.

In the second scenario we’re in the big blind with the same hand and the small blind moves all-in. Now we just check the second table and see that K-6-offsuit has an equilibrium call rating of 11 big blinds. Meaning, as long as our stack is 11 big blinds or smaller we can profitably call against the small blinds all-in.

What the hell are “equilibrium pushbot strategies”?

[su-custom_gallery source=”media: 1205″ link=”lightbox” width=”350″ height=”500″ title=”always” class=”alignright”]

No let’s examine how those pushbot and callbot charts above are derived.

Imagine a very simple Texas Hold’em game. Two players play heads-up, the small blind can decide before the flop whether to push or fold. If he pushes, the big blind can now decide whether to call or not. There are no post-flop decisions. Situations like this occur often in later stages of poker tournaments when the stack sizes are small and everybody folds to the blinds.

This simple push-or-fold game can be solved mathematically and optimal strategies can be given for both players. The optimal strategies form a balance for both players (an equilibrium). This means that none of the players can unilaterally improve their strategy.

Determining these strategies is a bit time-consuming and works (in a nutshell) like this: For each possible effective stack size, you check the range with which the small blind can profitably push if the big blind would always call. Then you check with which range the big blind can call profitably against the small blinds range. Now check again which range the small blind can push with, if the big blind only calls with the range found in the previous step. Repeat this until the ranges do not change anymore and then you have found the equilibrium strategies for the given stack size.

This way you can check for all stack sizes which hands can be pushed profitably and which hands can call an all-in profitably. Bill Chen and Jerrod Ankenman did this in the book Mathematics of Poker. The results of these calculations are charts above.

How to work with the equilibrium pushbot charts?

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For each hand, the charts show the maximum stack size (in big blinds) for profitable pushes and calls according to the equilibrium strategy. The pushbot chart applies when you’re in the small blind and it is folded to you, the callbot chart applies when you’re in the big blind and the small blind open shoves.

For example, if you have J2 in the small blind, you can go all-in with an effective stack of 4.6 big blinds or less profitably. Or if you have T9 in the big blind and the small blind open shoves, you can call profitably with an effective stack of 12 big blinds or less.

Equilibrium pushbot charts from the button

The Equilibrium pushbot chart can also be used from the button. The rough approximation is as follows:

Equilibrium pushbot chart button rule

You can go all-in profitably from the button if your stack is smaller than half the equilibrium pushbot ranking for the hand you are holding.

There is no button rule for calls against a button shove in the big blind.

When can you apply equilibrium strategies profitably?

You should not start pushing your stack from the small blind just because your hand has an equilibrium pushbot ranking of 36 big blinds and you also should not blindly defend your big blind just because your stack is slightly below the equilibrium callbot ranking. The charts above give you a rough idea, which hands you can safely push with and which hands you can safely call an all-in with if your opponent is a good and aggressive player.

Conditions for the equilibrium push-fold-charts

• Late preflop position: The equilibrium pushbot chart only works from the small blind (and to a lesser extend from the button).
• Or Big Blind: The equilibrium callbot chart only works if you’re in the big blind.
• No other players: Both charts assume all other players have folded.
• No Antes: Calculations for the charts assume no antes. With antes the maximum stack size for profitable calls and pushes increases considerably.
• No ICM: No tournament specific mechanisms are considered for those charts. “Profitable” throughout this article means profitable when looking at chip-EV. In many tournament situations you have to call much tighter and can push much looser than the charts indicate.
• Skilled opponent: The charts assume your opponents are skilled, aggressive players that will call or shove with reasonable ranges. If the tightest player at the table shoves into you from the small blind, you better don’t call with K9 and a 20 big blind stack, just because the equilibrium chart says so. You fold because his range is much more narrow than any equilibrium strategy suggests.

Like the Sklansky Chubukov rankings, the equilibrium rankings help you to develop an idea which hands are good enough to merit an all-in and which hands are good enough to call an all-in against an aggressive opponent.

How do equilibrium rankings and Sklansky Chubukov rankings differ?

Another approach to explore profitable shoving ranges are the Sklansky Chubukov rankings.

While the Equilibrium strategy assumes your opponent has a realistic calling (or pushing) range, the Sklansky Chubukov strategy always assumes the worst case, namely that the opponent always calls when he has a better hand than you (or at least gets sufficient odds).

Meaning, you can push much looser using the equilibrium strategy, since this strategy takes into account that the opponent also folds some better hands. Suited connectors in particular gain significant value when using the equilibrium strategy: They have a good equity against the typical calling (and also pushing) range, which is why they are comparatively strong hands. Take 65 for example. The hand has a Sklansky Chubukov ranking of only 3.1, meaning you can only push with 3.1 or fewer big blinds. But according to the Equilibrium strategy you can push profitably with up to 29 big blinds – a huge difference.

Relevant Resources

• Sklansky Chubukov rankings
• Pushbot trainer
• Equilibrium charts with ante (HoldemResources.net)
• Nash equilibrium in poker explained (poker VIP)
• Mathematics of Poker (Amazon)

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I was playing $15 Pai Gow poker recently and banking. (I always bank). There were one or two other people at the table. A couple turns went by without incident. Then a woman sat down and began playing two spots. She was of Middle Eastern/Balkan extraction, I think -- I only mention it because her communication skills were not exemplary.
When it becomes my turn to bank she reduces her bet to $20 (she was betting greens mostly). She plays the (horrible 7% edge) bonus bet too. She tells me she will 'play for a push' and I say okay. I get three pair, and play it the only way possible: one pair up top, two pair in the back. I win all the hands.
All of a sudden she starts demanding her bet back from me. I say I played the hand the only way I could have. She says that I should pay her regardless of whether I win or lose. I decline to pay her and fortunately she does not make a scene but mutters to the dealer and pit boss some spurious things and play continues.
Now, what would you understand 'playing for a push' to mean? As I understood it, if I had a hand that could be played more than one way, then I would set the hand the way that would most likely result in a push. I did _not_ understand it to mean that I would pay her if she lost and she would pay me if she won. That's why I bank -- I put my money at risk to have the chance at winning other people's money.
Has anybody ever encountered this practice before? The other players seemed to agree with her. The pit boss and dealer (wisely) remained silent.I don't see any merit to her complaint at all. 'Playing for a push,' means to balance the two hands, even if it lowers the combined expected value a bit. For example, splitting up a low two pair with an ace kicker. Splitting up the two pair has a high chance of pushing, but retaining the two pair has the greater expected value.
In the case with three pairs, if you played the highest pair in front, that would be playing for a push, and obvioulsy the right thing to do.I've never heard that either, but, in the context you describe it, I'll take a stab:
You had three pair. Sure, the logical / best play is to put your highest pair on top.
But, in my ignorant concept of 'play to push', you would keep your two best pairs together, and then SPLIT the lowest pair, putting two lousy singles on top. That way, you're probably not going to win, but you'll probably not lose either.
Personally, I would have told her to pound salt!
FYI: A pit boss once told me that 'House way' is NOT to win the most, but to lose the least. Very subtle difference.

(PS, I HATE when other players sit out while I'm banking... they'd rather give their money to the house than to me?)

I don't bank, but never sit out when another player banks. After all, I'm there to win. If some sucker wants to play banker, I'll be just as happy to take his money as the house's. If I didn't think I was going to win, I wouldn't be in the casino. (I know I'm not winning every hand, but you get my point.)Thanks for your comments. I thought I was in the right. Apparently there is some weird 'pact' that some players have where playing for a push means you settle up afterwards so nobody loses. Of course I will make it clear next time that when I bank, I intend to bank, period.
The subtext of the whole incident is that this woman just wanted to play the bonus without risking anything else. I was loathe to say this in the original post, but she is a total degenerate who bets huge on the bonus and spreads to two spots with bets of $25-$100s. Needless to say she is a valued customer of this particular casino...

I don't bank, but never sit out when another player banks. After all, I'm there to win. If some sucker wants to play banker, I'll be just as happy to take his money as the house's.

Heh. I'll be glad to be that sucker!When I am at the casino, I am there to play the 'house'. Players take it too personally when you beat them, and for me, I'd rather take a walk to the washroom than participate in a 'bank'. And if I choose to bank, I don't make any deals with anyone.

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