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Many players tense up when their stack reaches 15 big blinds and below. In truth, you should relax; with a stack this short, poker just became really easy to play. Instead of having to figure odds and read players, all you have to do is decide whether to shove or fold. At this stage of a tournament, I can give you as close to a poker 'system' as there is available.

11-15 Big Blinds

First in with this size stack you're too short to raise and then fold, but too deep to risk your whole stack on poor holdings. Stick to very premium hands in early position -- throw away AQ offsuit and KQ suited under the gun, and stay away from the smaller pairs. In late position, you can open it up a great deal, but if there are no antes, don't get too crazy. With the right type of opponents, you can make a small raise or limp with Aces or Kings if you are near certain that it will be raised behind you so you can go all in when the action gets back to you. But if there's any doubt, you're better off just pushing your chips in the middle with your big pairs.

1. In this hand the solver arrived at an equilibrium strategy in which villian checks back his entire range. So I put that decision node into locking mode and forced it to shove hand type that I believe that player would shove with, such as flush, 2PR, etc. Then I ran the solver again to see how all the hands in my range should react.
2. Every position at the poker table has a certain range of starting hands that can be profitably shoved at a given stack depth. Generally, these stack depths are at 20 big blinds or less.

Ten Big Blinds and Under

Poker is real easy now. With this stack, you can play unexploitable poker. Using Independent Chip Modeling and the Nash Equilibrium you can solve every situation with your stack size and hand versus however many random hands are left to act behind you. Don't know what ICM and Nash Equilibrium are? Professional card player, Chris 'Fox' Wallace and myself did the heavy lifting for you and put our charts up at​ pushfoldcharts.com. Go to pushfoldcharts.com and study the charts—or buy one to take with you to the card room. They give you all the information you need on what cards to push with and what cards to fold.

If you were to shove here you still have the big stack to act (who could call you) and our original raiser who might have a hand he’s willing to live and die with – meaning your fold equity is poor and against the top 15% of hands your holding of KJ offsuit is a 58-42% underdog! 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. When I node locked villian’s flop play to a shove, the exploitative play with AhQh was to muck those cards fast enough to leave a vapor trail. Here are the ranges I entered into the solver. Our flatting range contains hands that might go into a 4-betting range in a higher stakes tournament.

Though the charts give you the unexploitable ranges, there are still some adjustments you should be making, listed below.

The Six-Big Blind Rule

When you reach six big blinds, it can become correct to raise even wider than the charts suggest. This is because six big blinds is about as low as you can go and still expect people to fold to your raises. Once you fall below that number, people begin to call very loosely—often with any two cards—and the potential profit from people folding (which is the most important part of these small stack calculations) drops to nothing.

Under Five Big Blinds

You're almost certainly going to get called when you go all in with a stack under five big blinds. Due to the pot odds and the fact that there are still a few opponents who will fold their trash incorrectly, it's still correct to push with most of your hands here. But the knowledge that you are likely going to get called can still affect your choices. If you have a true trash hand (seven-deuce offsuit springs to mind) and the next players on the blinds are very weak you might fold and hope to get it in first next hand. If there are no antes and you can look at a bunch more hands for free, you might fold. But if there are antes and you are first in, your best option for most hands is to push and pray.

Table Image

Another good reason to play very tight early in a tournament is so that when you get to the late stages people will give your raises respect. As above, the potential profit from people folding—called Fold Equity—is key to successful push/fold strategy. If you can convince your opponents that you only put all your chips in the pot with a nut hand, your fold equity, and your entire tournament equity, goes way up.

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

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?

Shoving Hands Poker Rules

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

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

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.

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What the hell are “equilibrium pushbot strategies”?

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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?

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.

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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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