The Amount of Equity to Call All-In
How much equity do I need to call all-in here? This is quite a typical question in different situations and at different stages of a tournament. This article will give you formulas you can use to calculate the necessary equity, as well as outlining some frequently made mistakes.
We choose to call all-in in a tournament if an ICM EV call is bigger than an ICM EV fold.
The formula for this calculation is as follows:
Hand win% × ICM EV win + Hand tie% × ICM EV tie + Hand lose% × ICM EV lose > ICM EV fold (1)
If we haven't yet got into the money, the function can be simplified thus:
Hand win% × ICM EV of win + Hand tie% × ICM EV of tie > ICM EV of fold (1 simplified)
The formula still includes win% and tie%, but does not include hand equity. The formula for the equity calculation is as follows:
Equity% = win% + (tie% / 2)
The wrong standard formula of equity calculation is:
Hand equity% × ICM EV of win + (100 - equity%) × ICM EV of lose > ICM EV of fold
If we do not receive any payout when we lose (ICM EV lose = 0%), it can be simplified to:
Equity% × ICM EV of win > ICM EV of fold (2)
It can be written this way as well:
Equity% > ICM EV of fold ICM EV of win (3)
The last formula is of practical use and is easy to handle. Everything we need to know is EV fold and EV win against the opponent. This way we can quickly get the information concerning the amount of equity required.
Heads-up and ITM situations analysis
It was mentioned above that formula 1 is wrong, while formula 2 is not completely precise. It concerns the situation in tournaments with more than two players and different payouts, but both formulas are fine for cash games and heads-up.
Take a look at the following HU situation:
We're on the BB, and the SB raises. We can calculate EV in chips, as the payouts are always the same in this case.
Let's substitute the variable with the following values:
Hand win% × 1000 + Hand tie% × 500 + Hand lose% × 0 > 480
Hand win% × 1000 + Hand tie% × 500 > 480
If we win more than 48 percent of deals, we will get a +EV call, no matter the tie%.
equity% × 1000 > 480
Let's calculate EV push (the left hand side) if the hero wins 45 percent of the deals and ties 10 percent of the deals against the villain's range. According to the standard formula the hero with such a hand will have 50 percent equity.
Hand win% × 1000 + Hand tie% × 500
accordingly 0,45 × 1000 × 0,1 х 500 = 500 chips
equity% × 1000
accordingly 0,5 × 1000 = 500 chips
We arrive at the same result using both formulas, so we can use the easier one for calculating equity.
Analysis of tournament situations with ICM
Both formulas give the same result for HU situations, but the second formula gives an incorrect result in tournaments with ICM.
Let's consider the following situation with 3 players in a 6max tournament:
The SB raises 500 chips, but now our equity should be calculated taking ICM in consideration as well.
So, ICM EV = 55 percent in case of winning, 33 percent in case of a tie and 0 percent in case of a loss.
Please be aware that if we double our stack, our ICM EV does not double. If it is doubled, it would be 66 percent, put the real value is equal to 55 percent of the prize bank. This is the so-called non-linear effect in action.
Now you understand that the formula gives different results. Let's presume that we win in 45 percent of deals, and tie in 10 percent of deals.
Let's calculate the raise EV by formula (1): 0,45 x 55% + 0,1 х 33% = 28,05%
Raise EV by formula (2) is equal to: 0,5 х 55% = 27,5%
Notice that the result is bigger according to the first formula than the second formula, so our equity is still less than 50 percent, but the EV is bigger than the one calculated by the second, incorrect formula. This is why the tighter call range that we arrive at via the standard formula is wrong as well.
EV calculation for certain hands and ranges
The second formula is all by itself very popular, partly because such programs as SnG Wizard perform calculations based on it, and do not take cases of ties into account.
Let's now analyze a hypothetical situation on bubble in a 9max SnG.
Hero is on the BB, SB raises. How much equity is necessary to make a profitable call? Let's use the incorrect formula first for this example.
Potential results ICM EV:
EV Fold = 23,53%, EV Win = 38,33%, EV lose = 0%
Let's use those values in formula (3), then we get:
equity% > ICM EV of fold / ICM EV of win or
equity% > 23.53 / 38.33 × 100% or
equity% > 61.39%
Equity of 61% is a lot. This means that we should play very tightly in this situation is to be believed.
I can say that our EV could be much larger, and that 50 percent equity would be enough – 11 percent less than the formula tells us.
It could be hard to believe. Probably, the formula is wrong, technically, but how could the result be different by 11 percent? However, it's accurate. You are welcome to be convinced by the following example:
Aces equity is equal to exactly 50 percent against the opponent’s range (aces too). However, 50 percent equity is enough for the call which is much less than the 61 percent calculated earlier.
To see how we got this result we should use the right formula (1).
AA wins from AA in 2.17% deals, and ties in 95.65% deals.
So, win% × ICM EV of win + Hand tie% × ICM EV of tie + Hand lose% × ICM EV of lose > ICM EV of fold
2,17% × 38,33% + 95,65% × 25% + 2,17% × 0%> 23,53% или 24,75%> 23,53%
Notice that if you enter this data into SnG Wizard, you will see that it is using the wrong formula. According to its calculations you should fold aces in this spot. But as you can see from the above, it is wrong. ICMIZER uses a formula which is more accurate and as a result the provided figures are more precise.
Despite the second formula being incorrect, it can be used for equity calculation of hands that are surely worth calling, but it does not take hands whose equity amount is not enough for a call, but still have +EV, into account. Folding them can be an expensive mistake.
You can use the simplified formula in heads-up and cash games too, and having gain enough experience, you might soon be able to calculate equity on the fly.
PS. You can calculate ICM equity using my new ICM calculator.