Equilibrium Pushbot Charts for Poker Tournaments

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.

Let’s look at an example. 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 K♥6♦) 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 make 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 extent 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 blind according to equilibrium strategies. (The effective stack is the smaller of the stack sizes between you and your opponent.)

How to use the equilibrium charts?

Let’s go back to the example situations above. We’re in the small blind with K♥6♦ 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”?

Now 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, and 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 the charts above.

How to work with 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 J♥2♦ 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 T♦9♦ 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:

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.

Like the Sklansky-Chubukov rankings, the equilibrium rankings help you to develop an idea of 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 exploring 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 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 6♦5♦ 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.