Theory: Hands where ICM doesn't work

[marand]

When 2 stacks are really short it's often very important who will be forced to play a hand first. Just using ICM with a tool like SnGPT will give bad calling ranges.

Lets assume the following situation:
Blinds 200/400

CO 3400, folds
BTN 400, folds
SB 9100, pushes
BB 600.

Hero is in the BB. Should we fold or call?

If we fold we will have 200 chips left and be all-in next hand in the SB. Therefore we will be forced to call in the BB with any two cards. We will risk a 3-way pot in the SB next hand so even folding our worst hands is a mistake.

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Now let's change the situation slightly:
Blinds 200/400

CO 3400, folds
BN 400, folds
SB 9100, pushes
BB 601

Hero is in the BB. Should we fold or call?

Now things have changed... if we fold this hand and the next hand we will have 1 chip left and BTN will then be all-in when he is hit by the BB.

How often will the other shorty bust out if hero folds the next two hands?
Villian will either have to push next hand from CO and risk getting a call by both BTN and BB (we can't call in SB). Or he can fold and take his chances in the BB the hand after that. In each situation he risks having to play the hand 3-way, but it can also be a HU hand.

I would estimate that there is about 60% chance that the other short stack busts out before we do if we fold the next 2 hands. If he doubles up we will almost always finish 4th.

So in order for Hero to call in the hand. How many % must we win when we call?
I would guess that we need to win about 60% of the time when we call in order for it to be a good call. SB will be pushing close to 100% of his hands so we should call with something like top 15%. (66 A8 KJ A6s KTs)

If we would just have used ICM with SnGPT it would say that we should call with 86% of our hands in both hand 1 and 2. That is obviously not correct here. To test it just plug in 601 chips for BB and 401 chips for BTN (he would still be all-in before us, but SnGPT cannot handle when a player has <= the size of the BB).

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Is 100% in the 1st hand and 15% in the 2nd hand reasonable calling ranges?
Is the estimation that we need to win 60% of the time we call in the 2nd hand reasonable?

Thoughts?

[Shillx]

Hand 1 is ATC. Our equity if we fold can be approximated as

EV = .2(.5^2 + .5^4 + .5^6...) = ~6.7%

Because we have to go all-in 1st, we have a 2/3 chance of busting before the other dude. This assumes that we alternate going all-in against one other player. It might not be a fair assumption but there are too many scenarios to play out and this one should approximate it well enough.



It is tough to give an answer for hand 2 because we don't know what will happen when someone is all-in. In general you should try to outlast when given the opportunity. If we assume that people will play HU for their tourney life then our fold equity can be approximated by

EV = .2(.5 + .5^4 + .5^7 + .5^10...) = ~11.3%

Since we are left with 1 chip, we have to fade two all-ins for every one that the other guy does. If we decide to call and end up winning, he will have to go AI twice to survive so our equity (e) needed to call this push can be modeled by

.113 < .2(e*(1 - .5^2) + e*.5^4 + e*.5^6...)
e > ~ 70%

I hope this all makes sense.

e: If antes are in play, you can simplify the fold EV to ~.10 and you only need about 62% equity to call this push.

[dwigt]

Should the big stack be pushing ATC here? If BB calls 100% and big stack pushes 23o we have:
65% of the time: 3400/400/8500/1200
35% of the time: 3400/400/9700/0

If big stack folds 23o it will be:
3400/400/8900/800
but the next hand he can push ATC and worst case scenario is the 400 stack pushed first but the pot will still be sweetened by 600.
3000/400/9500/600

And here BB could push ATC again for
win: 2800/0/10100/600
lose: 3000/800/9100/600

It seems to me with a big stack, a medium stack, and two very small stacks the big stack shouldn't be pushing very weak holdings since folding keeps the bubble alive.

[BradleyT]

[ QUOTE ]
It seems to me with a big stack, a medium stack, and two very small stacks the big stack shouldn't be pushing very weak holdings since folding keeps the bubble alive.

[/ QUOTE ]

Let's look at some ICM numbers in a 10 player $10


Starting ICM
Player 1 9000 0.6667 0.2689 0.0583 $42.57
Player 2 3000 0.2222 0.4706 0.2424 $30.08
Player 3 800 0.0593 0.1387 0.3703 $14.53
Player 4 700 0.0519 0.1218 0.3289 $12.82

Those small stacks are worth a bit of money. Even though when you "absorb" one the $12.82 or $14.53 gets distributed amongst 3 players you still come out with a $1.49 gain for taking out the T700 stack -

We beat the T700 player
Player 1 9700 0.7185 0.2506 0.0309 $44.06
Player 2 3000 0.2222 0.5812 0.1965 $32.48
Player 3 800 0.0593 0.1682 0.7725 $23.46

(Player 3 should be thanking you very much)



When we lose to the T700 stack it's not too bad for us -

We lose to the T700 Player
Player Chips Prob 1st Prob 2nd Prob 3rd Equity
Player 1 8300 0.6148 0.2855 0.0868 $41.04
Player 2 3000 0.2222 0.3944 0.2813 $28.57
Player 3 800 0.0593 0.1184 0.2475 $11.47
Player 4 1400 0.1037 0.2017 0.3844 $18.92

We only lose $1.41 in equity. The other stacks make up the large majority of equity gained by player 4. He gained $6.10 and our $1.41 is 23% of that so the other two stacks gave him 77% of his equity gained.



Now let's keep the short stacks alive so we can steal 400 chips from the middle stack -

We steal
Player 1 9400 0.6963 0.2480 0.0506 $43.27
Player 2 2600 0.1926 0.4642 0.2630 $28.82
Player 3 800 0.0593 0.1532 0.3628 $14.82
Player 4 700 0.0519 0.1345 0.3235 $13.10

We gained $.70 and the next steal (9800/2200) we gain $.69. Two steals have netted us $1.39 which is almost the same amount gained by eliminating the T700 player.

I would think the stealing plan would be better because there is a drastically lower chance of getting called by the medium stack than one of the small stacks - unless of course you play the $6.50's then the medium stack is calling an all in with AJo risking his $27.07 to gain $6.26 because his cards are "pretty".

However when you go for the stealing plan one of the shortstacks may decide enough is enough and take a stand - so in game you probably gain a little less than $1.39. But as shown above losing to a small stack while stealing isn't a catastrophy as the majority of their equity (75%) is gained from the other players.

So the moral of the story is - being the big stack rules.

[SitNGo Wizard]

In an attempt to address this particular limitation of ICM, I have recently added a new feature to the SitNGo Wizard I call Future Game Simulation (FGS). For every possible outcome of the current game, FGS creates a simulation of what may happen in next few games.

This is a beta release of FGS so I can guarantee I still have some bugs to work out, but this looked like a perfect test so I plugged these two hands in to see how much difference FGS makes. The best way to illustrate is using game charts. The following charts plot the small blinds push range versus the hero’s call range.

The first chart uses plain old ICM without FGS. It makes almost no difference whether hero has 600 chips or 601 chips.


The second chart gives hero 600 chips and uses FGS.


The last chart gives hero 601 chips and uses FGS.


With 600 chips hero must call with any 2 cards. With 601 chips hero should play much tighter than plain old ICM would suggest (note that the scale of the Y axis is different in the last chart). I have not taken the time yet to verify the calculations, but at first glance they look reasonable.

[marand]

Thanks for your analysis and math. It took me a while to understand your math, but I think I got it now. According to your calculations we must win 70% of the time when we call in hand 2.

If he is pushing 100% of his hand then we can call with 99+ but should fold 88 and AK. For me this seems like a too tight range in this situation.

Your calculations assume that we will alternate playing a HU pot and that each time it's a 50% chance that the short stack will bust out. Correct?

If we call in the BB we will have 1202 chips. If the other shorty doubles up he will have 800 chips and after he folds his SB he will be down to 600. We can survive another orbit and he will be forced all-in again.
So basically if we win the first hand he has to win his next two hands. Therefore I don't agree that we have a 2/3 chance of busting out first if we call.

Also with a decent hand like 66 or AT we will have a good chance of winning our first hand (about 63%).

Another thing to consider is that if we double up and shorty busts out we will have a slightly higher equity than 20% ITM since we have more chips. (This doesn't change things as much as him having to win two all-in in a row though.)

[Shillx]

Yeah you will have more like 23% equity so that drops it down a fair amount. So 60% equity w/o antes and 50% w/ antes seems about right to me. SNGW guy puts calling correct at about 53% and that seems reasonable.

The thing is that you have to offset this by the times that the guy has to play a 3-way pot (or keep the bubble alive) when you fold. We're just assuming that he will bust 50% of the time but who knows if that will be the case. You also have to consider that ICM probably overshoots your equity when you call and win so you might want to adjust for that.

[marand]

Thanks for your post. This is very interesting to me. I think the graph for hand 1 looks good.

The second graph is the interesting part. If we assume that SB will push with something between 90 and 100% of his hands then we should call with 34-38%.
That is a calling range with hands like:
22 A2 K5 Q9 JT K2s Q7s J8s T9s

That's a lot wider range than I would have expected, but the more I think about the situation the more I think it might be correct. If we double up the first hand then the other shorty must double up twice in a row if we decide to keep folding after that.

From looking at your graph and comparing it to some PokerStove win% it seems like we need to win about 53% of the hands when we call. Is that correct? Can you get that information from SitNGo Wiz?

[ QUOTE ]
For every possible outcome of the current game, FGS creates a simulation of what may happen in next few games.

[/ QUOTE ]

How many hands in advance does it calculate?
Does it consider things like 3-way pots?
For example if we fold. Next hand will be:
CO 400
BTN 9500
SB 201 (HERO)
BB 3400

CO can either call or fold. If he pushes then BTN can either fold, call or push. If BTN calls then the hand will most likely be 3-way. (BB will check most of his hand.) If BTN decides to push then BB will most likely fold unless he has a good hand. So then the pot will be HU. (In all scenarios we will fold unless we pick up a monster like AA).
If CO folds his hand then BTN will most likely steal the blinds resulting in:

CO 10100
BTN 1 (HERO)
SB 3000
BB 400

If CO calls here, then SB will almost always call and the pot is 3-way. If CO either pushes or folds, then the pot will almost always be HU. (We will always fold, even with pocket Aces.)

How does SitNGo Wizard assign probabilities for each scenario? Is it possible to view the decision tree and assign your own ranges for what hands the big stack will push/call with?

FGS seems like a big step forward compared to regular ICM in these situations, but in order for it to be correct, the probabilities for each scenario must be realistic.

Is the FGS beta available to the public yet? Thought I didn't need Wiz since I have had SnGPT for ages, but maybe I have to buy it after all [img]/images/graemlins/smile.gif[/img]

[marand]

[ QUOTE ]
Yeah you will have more like 23% equity so that drops it down a fair amount. So 60% equity w/o antes and 50% w/ antes seems about right to me. SNGW guy puts calling correct at about 53% and that seems reasonable.

The thing is that you have to offset this by the times that the guy has to play a 3-way pot (or keep the bubble alive) when you fold. We're just assuming that he will bust 50% of the time but who knows if that will be the case. You also have to consider that ICM probably overshoots your equity when you call and win so you might want to adjust for that.

[/ QUOTE ]

Yes, there are lots of things to consider in this situation. I think I have generally been calling too tight here. Most often probably tighter than my first estimate of 15% in my original post. It's confusing with two different reasons that want to give you such a drastic different calling range.

1) I have 2/3 of my stack in the BB already.... should call 100%
2) Other shorty is about to die... I should fold everything except KK+

There is nothing more frustrating than to fold down to 1 chip and then both big stacks fold leaving the other shorty with 600 chips when he was supposed to be all-in !! [img]/images/graemlins/smile.gif[/img]

Do you think the calling range from Wiz seems reasonable here?

[marand]

From my last two hands, let's back up 4 hands and see how it looked then.

Blinds 200/400
CO 4000, folds
BN 1000, folds
SB 7300, pushes
BB 1200.

Hero is in the BB. What range should hero call with?
Hero will be the first one all-in if big stacks keeps pushing every hand so he must either make a stand now or fold down to 600 chips and call in the BB next time. (Unless he picks up a good hand and can play it from SB, BTN or CO.)

I would guess that Hero should call with about 50% of his hands if SB pushes close to 100%. Reasonable? What does FGS say?

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Similar example but hero has 1 more chip.

Blinds 200/400
CO 4000, folds
BN 1000, folds
SB 7300, pushes
BB 1201.

Hero is in the BB. What range should we call with?
Here we can out last the other shorty and make him play a hand first. There are 6 hands before he is all-in so there is time for both short stacks to pick up a good hand. Not quite panic mode, but that extra chip is still very nice to have in case neither short stack picks up a good hand.

I would have guessed that we should call with about 15% here also. But after thinking about the situation some more I would now guess it's closer to 25% calling range. I really am not sure at all though. I think this is a very difficult situation.

What is a reasonable calling range here for hero? What does FGS say? Can it calculate this far ahead? Is the decision tree huge?

If we assume that SB pushes 100% then ICM would suggest that we call with 35%. (Both with 1200 and 1201 chips.)

I think this calling range is too tight with 1200 chips and too loose with 1201 chips, but I find it difficult to know by how much.

Thoughts?