Examples and explanations for ICM

Introduction

In this Article

• Examples and explanations for ICM
  
• Tight vs Loose
  

Exact calculation: Push

The first two examples are all about gaining a proper understanding of the mathematics behind the independent chip model. In the first example, we will evaluate a push step by step in the same way that programs like SNG Power Tools do.

EXAMPLE

55$ SNG, 4-handed, Blinds 300/600

CO: 6000

BU: 4000 (Hero)

SB: 4000

BB: 6000

CO folds. Hero has 22. Push or Fold?

First, we estimate our opponent's calling range:

SB: 88+, A8+

BB: 88+, A8+

Now we calculate how much our chips are worth in actual dollars. To do this, we must determine the probability with which we place in each of the top three places:

It's easy for first place:

P(1st place) = Hero's Chips / Total Chips = 4000 / 20000 = 0.2 = 20%

So Hero Has a 20% chance of placing first, not taking position and skill into account.

It is a bit more difficult to make this same calculation for second and third place. We must assume in turn that one of the other three players has taken first place and then calculate the probability that hero wins against the remaining players. It goes like this:

P(2nd place) = P(CO takes 1st) * Heros Chips / (Total Chips - Number Chips CO) + P(SB takes 1st) * Heros Chips / (Total Chips - Number Chips SB) + P(BB takes 1st) * Heros Chips / (Total Chips - Number Chips BB) = 0.3 * 4000 / (20000 - 6000) + 0.2 * 4000 / (20000 - 4000) + 0.3 * 4000 / (20000 - 6000) = 0.0857 + 0.05 + 0.0857 = 0.2214 = 22.14%

The calculation for third place is messy but works analogously.

P(3rd place) = 0.257 = 25.7%

The probability of our placements multiplied by the payoff for that place and summed over the first three places gives us the true value of our chips:

EV(T4000) = P(1st place) * $(1st place) + P(2nd place) * $(2nd place) + P(3rd place) * $(3rd place) = 0.2 * $250 + 0.2214 * $150 + 0.257 * $100 = $108.91 or 21.8% ($109/$500).

Now we know our starting point (the pre-post value of our chips, before the blinds are posted). We must now compare these with the result of a push so that we can find out whether a push is worth it or not.

We assume the following: if we push and aren't called, we win 900 chips. If we are called, we must play against the caller for all our chips.

If both our opponents call with 88+, A8+, then we get the following:

P(No Call) = 78.7% EV(No Call) = 24.9% / $125

Explanation of the table:

P(Hold): This is the probability that one of the opponents holds a hand with which he will call.

P(Call): This value is different from P(Hold) because it is somewhat true that all players behind the first caller will fold, even if they hold aces. The 10% from the BB comes from P(Hold BB) * (100% - P(Call SB)) = 0.113 * 0.887 = 0.1. This is not exact, but will not affect the outcome much.

P(Win): We will win with this probability against our opponent's calling range.

EV(Win) / EV(Lose): Hero's situation when he is called and wins/loses. We calculate how many chips he will have if he wins/loses (for example, SB calls and Hero wins: Hero would have 4000 + 4000 + 600 = 8600 Chips) and then his situation would be evaluated as above.

EV(Call) = The expected value for Hero if he's called (EV(Win) and EV(Lose) weighted with their probabilities): P(Win) * EV(Win) + (100% - P(Win)) * EV(Lose).

P(No Call): how often a steal will succeed: 100% - P(Call SB) - P(Call BB).

EV(No Call): Hero's situation if the blind steal is successful. Calculated as EV(Win/Lose).

Now we have everything we need to find the expected value of a push. We weight the expected values of all the possible outcomes (nobody calls, SB calls, BB calls) according to their respective probabilities and sum them.

EV(Push) = P(No Call) * EV(No Call) + P(Call SB) * EV(Call SB) + P(Call BB) * EV(Call BB) = 0.787 * 24.9% + 0.113 * 15.1% + 0.1 * 14.2% = 22.8% / $113.76

The EV of a fold is equal to our initial state (pre-post):

EV(Fold) = Prepost = 21.9% / 109.63$

Now we compare the expected values of push and fold to find out which move is correct in this situation:

EV Diff = EV(Push) - EV(Fold) = +0.8% / +$4.13

At +0.8%, a push is correct. The boundary should be around +.5% since we can assume a skill edge for Hero.

Exact calculation: Call

The following example deals with calculating the EV of a call according to the ICM model.

EXAMPLE

55$ SNG, 4-handed, Blinds 300/600

CO: 6000
BU: 4000
SB: 4000
BB: 6000 (Hero)

CO folds, BU pushes All-In. SB folds. Hero has A9. Call or Fold?

Calculating of a call is much less complicated than the calculation of a push.  The first step is to calculate the opponents range.  Here we will put him on:

BU: 22+, A2+, KT+, QJ, JT

If we call, we'll get the following values against the button:

P(Win) = 49.7%

EV(Win) = 38.8% / $194

EV(Lose) = 13.5% / $68

(See previous example for definitions)

Now we weight EV(Win) and EV(Lose) according to their probabilities and obtain the expected value of a call with A9o.

EV(Call) = P(Win) * EV(Win) + (100% - P(Win)) * EV(Lose) = 0.497 * 38.8% + 0.503 * 13.5% = 26.07% / $130.57

The calculation of the expected value of a fold is different here than in the previous example. If we fold here, we don't get to keep our starting stack, rather we lose 600 chips to the big blind. The calculation is:

EV(Fold) = EV(T6000 - T600) = 26.5% / $132.41

Now we compare expected values again:

EV Diff = EV(Call) - EV(Fold) = -0.4% / -$1.84

At -0.4% a fold is the correct move.

This is how to calculate the EV of a push or call under the independent chip model. It isn't exactly something you can do in your head. For this reason, I will not focus on the calculation of exact EVs in the following examples. Rather, I will focus on the factors that influence our decision. The calculations will be done with SNG Power Tools.

Tight vs Loose I

55$ SNG, 8-handed, Blinds 100/200

CO+4: 3400
CO+3: 1800 (Hero)
CO+2: 1800
CO+1: 2500
CO: 1500
BU: 3500
SB: 2500
BB: 3000

CO+4 folds. All opponents are unknown to us. With which hands should Hero push?

The hardest part is in the beginning - estimating our opponents' hand ranges. We can assume that CO+2 will call less than CO who will call less than the BB, since the risk of an overcall from a monster hand is greater the more players are behind you. Furthermore, SB and BB get better odds for a call than players in earlier positions.

In the end, all we can do is guess. I will assume two different calling ranges for our opponents and see what it would take for Hero to push:

Opposing calling ranges a)

CO+2: QQ+, AK

CO+1: QQ+, AK

CO: JJ+, AK

BU: JJ+, AK

SB: JJ+, AQ+

BB: JJ+, AQ+

Hero's Push-Hands: 22+, AQo+, A2s+, KTs+, QJs (12.5%)

Opposing calling ranges b)

CO+2: JJ+, AK

CO+1: JJ+, AK

CO: TT+, AQ+

BU: TT+, AQ+

SB: TT+, AJ+

BB: 99+, AJ+

Heros Push-Hands: TT+, AQ+ (4.7%).

We note: the tighter our opponents, the looser we can push. Also, we see that suitedness plays an important role in the push hands for very tight calling ranges.

I will not make an exact recommendation here for which hands to push. In practice tables as tight as a) don't turn up, so I tend toward the conservative push range. (When in doubt, keep the variance low and fold!).

Tight vs Loose II

55$ SNG, 8-handed, Blinds 100/200

CO+4: 4000
CO+3: 2000
CO+2: 2000
CO+1: 3000
CO: 2000
BU: 4000 (Hero)
SB: 1000
BB: 2000

Everybody folds to Hero on the button. The question is how our push range looks when a) the SB is very loose and the BB is very tight and b) the SB is very tight and the BB is very loose.

(We'll ignore the fact that every player plays differently with 5BB than with 10BB. We aren't trying to discuss a real situation, rather we want to gain insight into the impact of different styles of play.)

Let's first define loose and tight:

Loose: 22+, A2s+, A3o+, KTs+, KJo+, QJs

Tight: TT+, AQs+, AKo

What is Hero's push range against SB (5BB) and BB (10BB)?

a) Hero's Push-Hands with a loose SB and tight BB: 22+, A2+, K9o+, K4s+, Q9o+, Q5s+, J9o+, J7s+, T9o, T6s+, 96s+, 86s+, 75s+, 65s, 54s (39.4%)

b) Hero's Push-Hands with a tight SB and loose BB: 33+, A7o+, A5s+, KQs (14.8%)

A very clear difference. When the calling station has more chips, steals become too expensive and we must reduce our push range a great deal. A loose player with a lot of chips is a danger that should not be underestimated. In contrast to cashgames, he hurts not only himself with bad calls but also his opponents.

5BB Push vs 10BB Push

a) Push with 5BB

55$ SNG, 6-handed, Blinds 100/200

CO+2: 6000
CO+1: 1000 (Hero)
CO: 4000
BU: 2000
SB: 4000
BB: 3000

CO+2 folds. With what should Hero push?

b) Push with 10BB

55$ SNG, 6-handed, Blinds 100/200

CO+2: 5000
CO+1: 2000 (Hero)
CO: 4000
BU: 2000
SB: 4000
BB: 3000

CO+2 folds. With what should Hero push?

This example should demonstrate the relation between the chips you risk and the chips you gain by stealing. We will assume the same calling range for our opponents in both a) and b), which is not true in reality but will make our example more tractable.

Opposing calling range:

CO: 99+, AT+
BU: 99+, AT+
SB: 88+, A8+
BB: 88+, A8+

a) Hero's Push-Hands with 5BB: 22+, A8o+, A5o, A4o, A2s+, KTo+, K2s+, QTo+, Q4s+, JTo, J7s+, T7s+, 97s+, 87s, 76s, 65s, 54s (31.8%)

b) Hero's Push-Hands with 10BB: 99+, AJ+ (6.3%)

So we can see the influence of stack size on the profitability of a push. In reality, our opponents will adapt to our stack (and therefore, our range), but it is still important to be aware of this effect.

Ultrashortstack on BB

55$ SNG, 8-handed, Blinds 100/200

CO+4: 4000
CO+3: 2000
CO+2: 2000
CO+1: 3000
CO: 2000
BU: 4000
SB: 2400
BB: 600 (Hero)

CO+4 raises to 600. All folds to Hero in the BB. Hero has 43. Call or Fold?

Let's assume CO+4 will raise 77+, AT+, and KQ. Then the EV of a call from Hero would be -0.0% / -$0.21. But Hero should call in spite of this! If he folds now, he'll have to go all in on the next hand, and probably with a worse hand.

Here, I recommend making ever call that is +chipEV even if it is -$EV.
chipEV(Call 43s): +0.1% / +12.9 Chips

SB vs Ultrashortstack on BB

55$ SNG, 8-handed, Blinds 100/200

CO+4: 4000
CO+3: 2000
CO+2: 2000
CO+1: 3000
CO: 2000
BU: 4000
SB: 2400 (Hero)
BB: 600

All folds to Hero in the SB. The BB is a good players and knows that he must call loosely here. What is Hero's push range?

Before we consider what we can push with, we should consider what the BB will call. If he assumes that we push with these hands: 22+, A2+, K2+, Q2s+, Q6o+, J7s+, J9o+, T8s+, 98s, then he will have a +chipEV call with a range of ca. 80%. I think 80% is realistic for the BB in this situation, so this should be a fine estimate for our purposes.

Our push range under these conditions: 22+, A2+, K3o+, K2s+, Q7o+, Q4s+, J8o+, J7s+, T8s+ (44.2%)

If we want to keep a minimum edge of .5%, the push range falls well under 50% even though we get odds of 1:1.4 and we are against a nearly random hand. Whether we should retain our .5% edge in this situation is questionable. We're only risking a quarter of our stack and so we don't need to be quite as sure.

With a minimum edge of .1%, we can push with these hands: 22+, A2+, K2+, Q2+, J2+, T5o+, T2s+, 96o+, 95s+, 87o, 85s+, 75s+, 65s (70.1%)

The latter range in this situation (minimum risk for us if we're called) is more sensible than the conservative range we first assumed. One risk we probably do take is to destroy our table image. Worse opponents could misunderstand our situation, which could later lead to -EV calls for us.

Late Position vs Ultrashortstack on BB

55$ SNG, 8-handed, Blinds 100/200

CO+4: 4000
CO+3: 2000
CO+2: 2000
CO+1: 3000
CO: 1600 (Hero)
BU: 4000
SB: 2800
BB: 600

All folds to Hero in the CO. Again, we seek Hero's push range.

We're in an interesting situation. We have a small fold equity (since BB will again call around 80% of the time), but we risk playing for our whole stack against the BU or SB.

In this kind of situation we can put tighter calling ranges on BU and SB (insofar as their heads are in the game) since they must know that we cannot push with trash (since we will surely be called by BB):

BU: TT+,AQ+
SB: 99+,AJ+
BB: 80%

Hero's Push-Hands: 66+, ATo+, A7s+, KQo, KTs+ (11.6%)

We are far from an aggressive push here. The fold equity that is so massively profitable with worse hands is not present here and we would be risking our entire stack.

Missing fold equity and two opponents make a clear impact on our push range, which would normally be much looser from the CO.

The -EV UTG-Push

55$ SNG, 6-handed, Blinds 200/400

CO+2: 1500 (Hero)
CO+1: 6000
CO: 3000
BU: 2500
SB: 4000
BB: 4000

Hero is UTG with 3.5BB and has T6. Push or Fold?

Let's first estimate the opposing calling range:

CO+1: 88+,A8s+,ATo+
CO: 77+,A8+
BU: 55+,A2+
SB: 22+,A2+
BB: 22+,A2+,KTs+,KQo,QJ,JTs

Then Hero could push with: 22+, A8o+, A7s+, KQs (13.7%)

A push with T6 would have an EV of -0.5% / -$2.60.

In spite of this, Hero should still push. The ICM calculation is mistaken in this example since it cannot account for some decisive factors. If Hero folds, he'll have to pay the big blind in the next hand and will lose 400 chips and his fold equity. A fold would be even more -EV than a push. In this light, the whole situation is -EV for Hero and all we can do is choose the lesser of two evils and push.

It's worth noting if you see a player making such a push. In this kind of situation many players soften their push ranges and with proper notes on their ranges you can exploit this better in the future.

Note: here we see the limits of the ICM. It is not a flawless or infallible system but it is still the best method for recognizing +EV situations if you know what you're doing.

Bubble - +ChipEV, but -$EV

55$ SNG, 4-handed, Blinds 300/600

CO: 5000
BU: 5000
SB: 5000
BB: 5000 (Hero)

2 folds, SB pushes All-In. Hero has AT. Call or Fold?

Here is the biggest difference between tournament and cash games. In a tournament it is often correct to fold even if you have the superior hand. We will examine this special case with ATo.
I put SB into the following range: 22+, A2+, K2+, Q2s+, Q6o+, J7s+, J9o+, T8s+, 98s
(aggressive players will push with more!)

If we call with ATo, we win 58.8% of the time against his range and will score an additional 1200 chips in dead money:
chipEV Call: +7.4% / +1482.9 Chips

Our problem is that we will be out of the bubble in 41.2% of cases and won't make any money at all. For this reason, the $EV is negative for a call:

$EV Call: -0.5% / -$2.25

This is probably the most important realization for every SNG player. A call with a better hand can be -EV. This effect can occur in earlier phases (5-6 handed) of the SNG under certain circumstances, however it is the most important criterion for almost all decisions when 4-handed.

Bubble - Midstack - Call against Bigstack vs Call against Midstack

a) Call against Bigstack

55$ SNG, 4-handed, Blinds 200/400

CO: 7000
BU: 3000
SB: 7000
BB: 3000 (Hero)

2 folds, SB pushesAll-In. Hero ?

b) Call agianst Midstack

55$ SNG, 4-handed, Blinds 200/400

CO: 7000
BU: 7000
SB: 3000
BB: 3000 (Hero)

2 folds, SB pushes All-In. Hero ?

The situations a) and b) differ more than appears at first glance. The reasons are the following:

If Hero calls and loses, he will make no money at all, regardless of whether it's against the big -or midstack. But if he wins, in case a) he will have attained a good position but will not be ITM. In case b), however, victory assures hims a place in the money.

To make clear the consequences of a call from Hero:

Hero has 3000 chips before the hand, which is equivalent to 18.6% or $93.

Call in case a) EV Win = 28.7% / $143, EV Lose = 0.0% / $0

Call in case b) EV Win = 32.2% / $161, EV Lose = 0.0% / $0

A win in case b) pays off by around $18 (!!) more than in a). For us, that means that we can call much less against the bigstack than against the midstack.

Let's try it out. Assuming the big- and midstacks push the same hands against us:

SB: 22+,A2+,KT+,QJ,JT

Then we can call profitably with the following hands:

a) against the bigstack: TT+, AKo, AQs+ (3.8%)

b) against the midstack: 66+, AT+ (8.9%)

As we predicted, our calling range is much looser against the midstack.

Note: In a real scenario our opponents would push a bit looser so we could call looser still against the midstack than we have listed here.

Bubble - Bullying as the Bigstack

55$ SNG, 4-handed, Blinds 200/400

CO: 7000
BU: 7000 (Hero)
SB: 3000
BB: 3000

CO folds, With what hands can Hero push?

As we saw in the last hand, it is hard to call against a bigstack as a midstack. SB and BB will be very tight here and will call Hero with roughly the hands that we called with in case a) of the previous example.

SB: TT+, AKo, AQs+
BB: TT+, AKo, AQs+

Hero can push 100% against these ranges! A push with 32o still has an EV of +1.0% / +$4.86!

Let's assume that SB and BB are good players and know that Hero is pushing with any two cards. Then they can call a bit looser:

SB: 66+, AJo+, A9s+, KQs
BB: 55+, A9o+, A7s+, KJo+, KTs+, QJs

But even against these ranges a push with 32o would be a marginal +EV (+0.1% / +$0.55). So we see: a bigstack can do what he wants on the bubble and shortstacks can't do anything about it, even if they know what he's doing.

You might present the idea that it is even more +EV to push under 100% since you would be more respected and called less, thus reducing the variance. This topic is beyond the scope of the ICM, however, and so will be addressed elsewhere.

Bubble - Bigstack vs Bigstack

55$ SNG, 4-handed, Blinds 400/800 A25

CO: 3000
BU: 3000
SB: 7000
BB: 7000 (Hero)

2 folds, SB is a complete maniac and pushes every hand. With which range should Hero call?

I've chosen such an extreme example on purpose to examine the special relationship between two bigstacks on the bubble Hero knows in this case that SB will push 100%. With what can Hero profitably call in such a situation?

Call-Hands: 99+ (2.7%)

This seems too tight to be true, but it is with just this range that Hero can call +EV. Even with AKs, a call is -EV!! (EV Call AKs = -0.4% / -$2.14) This is because even though Hero will often be ahead (He would win a call 67% of the time with AKs), but calling would still risk loosing his whole stack and then being OOTM. A fold, however, practically assures a spot ITM (there are two stacks with just 4BB left in the tournament!).

Now that we know how tight the BB must be here, let's see what a push from the SB looks like if Hero calls only with 99+:

EV Push 32o: +1.9% / +$9.37

(Even if BB would call with a much looser range, such as 66+,AT+, a push with 32o would still be +EV!)

The SB can push profitably without even looking at his cards. This and previous example show just how dominant a bigstack with initiative can be.

Bubble - Calling as Shortstack

55$ SNG, 4-handed, Blinds 300/600

CO: 10000
BU: 4000
SB: 4000
BB: 2000 (Hero)

2 folds, SB pushes. With what hands can Hero call?

Let's say the SB has a range of 70%, which is not unrealistic for this situation. Then Hero can call with 22+, A2+, K2+, Q7o+, Q3s+, J9o+, J7s+, T8s+ (44.5%).

To understand what this means, compare that to Hero's calling range if he had the same number of chips as the SB:

CO: 8000
BU: 4000
SB: 4000
BB: 4000 (Hero)

Assuming SB pushes with the same range as above (70%):

Call-Hands: 66+, A8o+, A7s+, KJo+, KTs+ (14.3%)

The calling range of a shortstack on the bubble is substantially looser than that of a middle stack. The reason is that in that case, Hero cannot hope for another way to get ITM than to win an all-in.

On the other hand, the SB can steal aggressively against a shortstack (even 100% against a tighter big blind), since he's only risking half his stack if he's called but will win a quarter of his stack if the BB folds. (What we said in the example "Ultrashorstack in the BB" about calling with ~3BB in the big blind no longer applies here because we're on the bubble!)

Bubble - Push as Shortstack

55$ SNG, 4-handed, Blinds 300/600

CO: 10000
BU: 2000 (Hero)
SB: 4000
BB: 4000

CO folds. Which hands should Hero push?

First, we must figure out what hands will call us. It must be clear to us that we have little fold equity against the big blind because his odds for a call are too good. The SB will be more careful since the BB is still behind him.

Let's estimate our opponents' calling ranges:

SB: 22+,A2+,KT+,QJ
BB: 22+,A2+,K5+,QT+,JT

According to Power Tools we can push with the following range (min edge .5%):

Push-Hands: 22+, A2+, KTo+, K7s+, QJo, QTs+, JTs, T9s (27.0%)

There is again the question though of whether it's worth retaining the .5% edge. We are the shortstack on the bubble - we won't win by waiting. IN the next hand, the bigstack in the big blind will call almost everything. This means that we're in the last hand in which we will have fold equity. For this reason, it's advisable to forgo the edge and even to make a -EV push.

A 40% range would be even EV; the optimal range is probably between 60% and 70%. Even though our opponents have very loose calling ranges and we're on the bubble (where a coinflip is even -EV!), we can and must push aggressively. The reason is as simple as it is illuminating: we are the shortstack and have next to nothing left to lose. We will not get ITM unless it's through our own initiative so we just have to take risks.

Comparison: Bubble - 3handed

a) Call on the Bubble

55$ SNG, 4-handed, Blinds 400/800 A25

CO: 5000
BU: 5000
SB: 5000
BB: 5000 (Hero)

2 folds, SB pusht All-In. With what hands should Hero call?

b) Call 3handed

55$ SNG, 3-handed, Blinds 400/800 A25

BU: 5000
SB: 5000
BB: 5000 (Hero)

BU folds, SB pushes all-in. With what hands should Hero call?

In both cases, Hero has exactly his share of the total chips (1/4 4handed, 1/3 3handed). What interests us is how his calling range is different from 4handed to 3handed if SB pushes with the same range both times.

Push-Range SB: 22+, A2s+, A3o+, KTs+, KJo+, QJs

a) 4-handed Call-Hands: 99+, AQs+, AKo (4.2%)

b) 3-handed Call-Hands: 66+, AT+ (8.9%)

If we've survived the bubble, we can call much looser again because we no longer risk being OOTM. Once you're ITM, you should focus on winning the tournament instead of trying to weasel your way into second place. This is because the difference between the first and second places is much larger than between second and third.

Summary

Requisite to successful play according to the ICM is the capability for feeling the situation. Whether a push or a call is profitable depends on the opposition and their playing style, the distribution of chips and your own image.

This article as a dozen examples that you should study carefully to understand the mode of thought behind the ICM and the criteria that are pertinent to making decisions. For answers to further questions, you should turn to the forum.