## Independent Chip Modeling Part 3: Pushing Decisions from the Small Blind on the STT Bubble As A Function Of Stack Size

- Independent Chip Modeling Part 1: Derivation and Sample Analysis
- Simulation #1
- Simulation #2
- Simulation #3
- Simulation #4
- Independent Chip Modeling Part 2: Positive Chip EV Doesn’t Mean Positive Monetary EV
- Independent Chip Modeling Part 3: Pushing Decisions from the Small Blind on the STT Bubble As A Function Of Stack Size
- Independent Chip Modeling Part 4: Beyond ICM
- ICM Calculator

**By Tony Guerrera**

## Introduction

Independent chip modeling (ICM) projects our finishing distribution as a function of how the chips are distributed in a tournament. As a result, our tournament decisions aren’t just a function of our cards, our opponents’ cards, and their action distributions. Instead, our tournament decisions are a function of the above *and* our corresponding distributions of our possible stack sizes. In this article, we’ll take the same hand and explore it for a few different relative stack sizes to see how your monetary EV (mEV) changes.

## The Hand

You’re in a single table tournament (STT) in which 1st pays $500, 2nd pays $300, and 3rd pays $200. The blinds are 300-600. You’re in the small blind. Action folds to you, and you have 34s. If you push all-in, the big blind will call with [AA,55]||[AK,A8]. Given that you have 34s, there are possible combinations of hole cards that your opponent can hold. [AA,55] represents (6)(10) = 60 combinations, and [AK,A8] represents (6)(16) = 96 combinations. In total, the calling distribution [AA,55]||[AK,A8] represents 60 + 96 = 156 combinations meaning that the probability that your opponent will call is and the probability that your opponent will fold is . Poker Stove calculations show that P(win) = .3103, P(tie) = .0072, and P(lose) = .6826 when your opponent calls.

When you push all-in, there are four possibilities:

1.) You push, your opponent calls, and you lose. We’ll call this PCL (push/called/lose)

2.) You push, your opponent calls, and you tie. We’ll call this PCT (push/called/tie)

3.) You push, your opponent calls, and you win. We’ll call this PCW (push/called/win)

4.) You push, your opponent folds, and you win. We’ll call this PFW (push/fold/win)

Let ICM(X) denote the mEV output by ICM when X happens, where X is PCL, PCT, PCW, or PFW. Furthermore, let P(PYZ) = P(Y)P(Z|Y) meaning that, for example, P(PCL) = P(called)P(lose|called). In that case, your mEV for pushing is given by the following formula:

I applied this formula to pushes for a wide range of different starting stacks and compared the mEV to the mEV of folding in each circumstance.

## When You’re Chip Leader And Everyone Else Has The Same Stack

The first interesting decision regarding pushing is what your EV of pushing is when you enjoy a chip lead over three opponents with roughly equal stacks. Before posting, the stacks are as follows:

Small Blind (You): 7,500

Big Blind: 2,500

Under The Gun: 2,500

Button: 2,500

In this situation, your mEV for pushing is $384.23. Meanwhile, your mEV for folding is $362.69. Against a big blind with a calling distribution of [AA,55]||[AK,A8], pushing increases your mEV by $16.18. It’s obviously tough to make generalizations after analyzing just one hand; however, this example suggests that being aggressive with a big stack against relatively tight short stacks will increase your mEV in the long run.

## When You Share The Chip Lead

With respect to this situation, there are two possibilities. Your fellow chip leader can be in the big blind, or one of the short stacks can be in the big blind. Let’s first look at a situation in which the other chip leader is in the big blind:

Small Blind (You): 5,000

Big Blind: 5,000

Under The Gun: 2,500

Button: 2,500

Your mEV for pushing is $297.07, and your mEV for folding is $294.12. Pushing increases your mEV by $2.95 in mEV. Now, let’s put one of the 2,500 stacks in the big blind to see what happens:

Small Blind (You): 5,000

Big Blind: 2,500

Under The Gun: 5,000

Button: 2,500

With one of the short stacks in the big blind, your mEV for pushing is $315.66, and your mEV for folding is $292.34. Pushing here increases your mEV by $23.32.

Even though ICM analysis suggests that both pushes are +mEV, notice that the push into the small stack carries much more +mEV than the push into the other big stack. The results from just two situations aren’t conclusive, but these results strongly suggest that you should be highly aggressive against short stacks when you share a chip lead but that you should tighten your pushing distribution substantially with fellow big stacks remaining to act.

## You’re a Short Stack and There’s One Big Stack

When you’re a short stack and there’s one big stack, two scenarios exist. The first is the following:

Small Blind (You): 2,500

Big Blind: 7,500

Under The Gun: 2,500

Button: 2,500

With the stacks distributed like this, the mEV for pushing is $218.90, and the mEV for folding is $194.81. Pushing gains you $24.09 in mEV.

The other possible situation is:

Small Blind (You): 2,500

Big Blind: 2,500

Under The Gun: 7,500

Button: 2,500

ICM analysis suggests that the mEV for pushing is $227.20 and that the mEV for folding is $191.35. Pushing here increases your mEV by $35.85. Pushing into both the big stack and the small stack carry large +mEV; however, note that pushing into the small stack carries substantially more +mEV than pushing into the big stack. This suggests that you should be more apt to target your fellow small stacks. Most of your gains in mEV in these situations come from your fold equity, and if you push all-in *every* hand, your fold equity will be shot, meaning that it’s probably best to target your fellow short stacks and only to push into big stacks when you have very good hands.

## You’re a Short Stack and There Are Two Big Stacks

One of the big stacks can be in the big blind:

Small Blind (You): 2,500

Big Blind: 5,000

Under The Gun: 2,500

Button: 5,000

Pushing in this situation yields an mEV of $209.57, and folding results in an mEV of $180.45. Pushing increases your mEV by $29.12.

When the other short stack is in the big blind, we have the following:

SB (You): 2,500

BB: 2,500

UTG: 5,000

B: 5,000

Pushing in this situation yields an mEV of $216.47. Meanwhile, your mEV is only $177.52 if you fold. You gain $38.95 in mEV by pushing against this opponent. Pushing against one of the big stacks is +mEV, but pushing against the other short stack yields even more mEV provided that he’s on a sufficiently tight calling distribution.

## The Big Picture

This analysis was by no means exhaustive; however, it does potentially suggest a few things. First, if you’re a big stack, you should be merciless against tight short stacks. Second, if you’re a big stack in the jam/fold stage of a tournament, you should only push with premium hands against other big stacks, even if you have sufficient fold equity. Third, when you’re a short stack, pushing against short and big stacks is +mEV, but pushing against short stacks is higher +mEV. Since you need to preserve fold equity for pushes to remain +mEV, you should consider only pushing with premium hands against big stacks but pushing with almost any two cards against tight short stacks.

Tony Guerrera is the author of *Killer Poker By The Numbers*

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