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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(Hold)	P(Call)	P(Win)	EV(Win)	EV(Lose)	EV (Call)
SB	11.3%	11.3%	41.5%	36.3%/ $182	0.0% / $0	15.1% / $75
BB	11.3%	10%	41.5%	34.2% / $171	0.0% / $0	14.2% / $71

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.