Comment here: Taipan168's ICM calcs: Calling AI after Raising/Pushing Over Raise

Background

I'm not going to cover the theory underlying ICM as this has been covered much more comprehensively elsewhere (Link here), so these posts are intended to be a step-by-step guide to the following situations that cannot be calculated using SNGPT (note that SNG Wiz allows you to calculate the second, but not the first of these):

a. You open raise and somebody pushes over, should you call?
b. Somebody open raises, should you push over?

Hopefully this guide will help players answer their own questions when they find themselves in situations 1 or 2 above. I hope that this will become clearer once I have covered it, but ideally the only question we should be discussing on these forums is whether the hand ranges we have assigned to opps are correct. The calculations themselves should be relatively uncontroversial and become relatively quick and easy with a bit of practice.

Of course we will never be able to do these calculations whilst playing in a game but by analysing hands after the event we can hone our own instincts such that these decisions will become intuitive, if not necessarily second nature.

a. You open raise and somebody pushes over, should you call?

This is a very common situation that occurs in most SNGs. Obviously there will be many situations where the answer is obvious, so let’s take a slightly trickier example (with apologies to the poster of this hand):

PokerStars No-Limit Hold'em Tourney, Big Blind is t100 (4 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

SB (t3912)
BB (t2194)
Hero (t3455)
Button (t3939)

Preflop: Hero is UTG with 9, 9.
Hero raises to t300, 2 folds, BB raises to t2294, Hero ????
Hero’s reads were that the BB is relatively tight. So, what are the steps that we need to take here:

1. Calculate how many chips Hero will have under the following three scenarios:
- Hero folds to the push
- Hero calls the push and wins
- Hero calls the push and loses (in many cases this means that Hero is out)

2. Use the ICM calculator (link here) to calculate the value (measured by % of the total prize pool) of Hero’s chips under each of the three scenarios above. Don’t forget to adjust the other stacks if the other players have posted blinds or called.

3. Calculate what % of the time Hero needs to win to make % of prize pool (call) = % of prize pool (fold)

4. Using Pokerstove, compare the required win % to Hero’s actual win % against Villain’s likely range.

Worked example

Here are the workings for the example above:

1. Chips under various scenarios:
- If Hero folds he will have 3155 chips
- If Hero calls and wins he will have 5699 chips
- If Hero calls and loses he will have 1261 chips

2. Value of these chips calculated as % of the prize pool (remembering that even if Hero has 100% of the chips these are only worth 50% of the prize pool)
- Fold (3155 chips) = 24.2% of prize pool
- Call and win (5699 chips) = 36.1% of prize pool
- Call and lose (1261 chips) = 12.1% of prize pool

3. % of time Hero needs to win to make calling (taking into account the possibility of both winning and losing) worth at least the same % of the prize pool as folding:

24.2% = (x * 36.1%) + ((1-x) * 12.1%) where x = probability of winning

Solving for x (if you suck at algebra because like me you haven’t done it for 15+ years, you can use goal seek in Excel to solve this), x = 50.4% so Hero needs to be at least 50.4% to win against Villain’s likely range to make this call +EV.

(Note the calculation becomes easier if Villain has Hero covered because the term starting with (1-x) won’t be there)

Alternatively, we can restate the formula as:

x = (Fold - Lose) / (Win - Lose)
x = (24.2 - 12.1) / (36.1 - 12.1)
x = 12.1/24
x = 50.4%

4. Now the hard part – figuring out Villain’s likely range. Hero’s read is that Villain has been tight. Another relevant factor is that it’s the bubble, Hero has Villain covered (so if Villain pushes, Hero calls and Villain loses he is OOTM) and Villain has a >20x BB stack so isn’t yet completely desperate, a reasonable range might be TT+, AK+. In this spot I actually prefer to use a slightly looser range because if calling the push is –EV using a looser range it will be even more –EV using a tighter one. So let’s use 88+, AQ+ as our range here.

Using Pokerstove, Hero’s 99 is 41% to win against 88+, AQ+. 41% is less than the 50.4% that Hero needs to be to win against Villain to make the call +EV so this is a clear fold:

Code:
        	equity (%)  	win (%)	tie (%) 
Hand  1:	41.3296 %  	40.43% 	00.90%      { 99 }
Hand  2:	58.6704 %  	57.77% 	00.90%      { 88+, AQs+, AQo+ }
Interestingly, even if we widen Villain’s range to 66+, AT+ Hero is only 49.4% to win against this range so calling is STILL –EV.

b. Somebody open raises, should you push over?

This one is a bit tricker, because we have that lovely thing called fold equity that we have to estimate. Apologies to Gingerwizard for using his hand:

PokerStars No-Limit Hold'em Tourney, Big Blind is t100 (8 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

MP2 (t1230)
CO (t2578)
Button (t2522)
Hero (t1275)
BB (t875)
UTG (t1700)
UTG+1 (t1930)
MP1 (t1390)

Preflop: Hero is SB with T, T.
2 folds, MP1 raises to t300, 3 folds, Hero ????
Hero’s reads were that MP1 was very tight this being the first hand he had played in about 30 hands. Hero had played MP1 previously, and MP1 had demonstrated a very good understanding of positional poker (ie. if he’s raising in MP1 then he’s got a big hand). I'm going assume that BB always folds to make the calculations less complex.

So here are the steps:

1. Assign hand ranges with which Villain would open raise and call a push.

2. Calculate how many chips Hero will have under the following three scenarios:
a. Hero folds to the raise
b. Hero pushes and opp folds
c. Hero pushes, opp calls and Hero wins
d. Hero pushes, opp calls and Hero loses

3. Use the ICM calculator (link here) to calculate the value (measured by % of the total prize pool) of Hero’s chips under each of the four scenarios above.

4. Here’s the slightly tricky part. We need to figure out for each of the hand ranges in 1. above the chance of opp folding or calling a push. There are 6 ways to be dealt each pocket pair and 16 ways to be dealt each unpaired hand combination. Hopefully the example will make it a bit clearer.

5. Use Pokerstove to figure out Hero’s chance of winning against opp’s call push range

6. Calculate the overall % of the prize pool that Hero would have if Hero pushes (scenarios 2b, 2c and 2d above)

7. Compare the % of the prize pool Hero would have under the push scenario in 6 to the fold scenario in 2a.

Worked example

Hopefully the worked example will make it a bit clearer.

1. Opp is tight, so let’s say he open raises with 77+, AJ+. If Hero folds everything but 99+, AQ.

2/3. Chips and % of the prize pool under the four potential scenarios:
a. Hero folds to the raise – 1225 chips worth 9.5% of the prize pool
b. Hero pushes and opp folds – 1675 chips worth 12.6% of the prize pool
c. Hero pushes, opp calls and Hero wins – 2650 chips worth 18.8% of the prize pool
d. Hero pushes, opp calls and Hero loses – 0 chips, out (0% of the prize pool)

4. Chance of opp folding to or calling Hero’s push:
- Opp is raising pocket pairs 77-AA which is 43 possible hands (42 hands 77, 88, 99, JJ, QQ, KK and AA and 1 hand TT since Hero also has TT) and unpaired hands AJ-AK which is also 48 possible hands so he is raising 91 hands
- Opp will call a push with pocket pairs 99-AA (31 hands - remembering there is only one TT) and unpaired hands AQ-AK (32 hands) so he will call with 63 hands and fold 28 hands
- So the chance that opp will fold to a push is 28/91 = 30.7%

5. Using Pokerstove, Hero’s TT is 44% to win against the hands that opp will call with (99+, AQ+):

Code:
 	equity 	win 	tie 	      
Hand 0: 	44.255%  	43.30% 	00.96% 	     { TT }
Hand 1: 	55.745%  	54.78% 	00.96% 	     { 99+, AQs+, AQo+ }
6. Hero’s overall share of the prize pool if Hero pushes:
- 30.7% of the time opp folds and Hero’s stack is worth 12.6% of the prize pool
- (69.3% * 44%) = 30.5% of the time opp calls and Hero wins so Hero’s stack is worth 18.8% of the prize pool
- (69.3% * 56%) = 38.8% of the time opp calls and Hero loses so Hero is out.

So Hero’s % of the prize pool by pushing is (30.7% * 12.6%) + (30.5% * 18.8%) + (38.8% * 0) = 9.6%

7. So, Hero has 9.6% of the prize pool if he pushes and 9.5% if he folds, so it is a very marginally +EV push over against an opponent who is this tight. In this situation though, if we think that we are better on average than the other players on the table, I tend to fold assuming I have opp's range right because I prefer not to take +0.1%-0.2% EV opportunities since ICM assumes every player is of equal skill (that's the subject of a separate discussion ). That said, there is an argument that EV is EV so we should take any opportunity that is +EV, so IMO pushing or folding here are both fine.

However, if opp is a bit more LAggy and raises 77+, AT+, KJ+ and calls with the same 99+, AQ+, the EV of pushing is 10.2% (I won’t go through the calculations again!) this is still close but it is a good call.

Hopefully the above made some sense to somebody, apologies that it was a bit longer and more involved than I would have hoped!