ICM by Hand (quite maths heavy)

[Longy]

It is not necessary to understand this post to apply ICM principles to your game. I know people like myself, like to know the maths behind the concepts. Understanding how to calculate icm will hopefully give you a greater understanding of why ICM works etc.


Example

I will use the hand from my 1st post

***** Hand History for Game 5982493641 *****
NL Texas Hold'em $22 USD Buy-in Trny: 33504140 Level: 6   Blinds(300/600) - Saturday, May 19, 05:34:40 ET 2007
Table Speed #1330099 (Real Money)
Seat 8 is the button
Total number of players : 4
Seat 5: Markpl111 ( 7,985 )
Seat 4: pokerjerie ( 8,410 )
Seat 8: ILike_One_9 ( 1,860 )
Seat 7: xxcrackerxx1 ( 1,745 )
Trny: 33504140 Level: 6
Blinds(300/600)
** Dealing down cards **
Dealt to Hero[    ]
xxcrackerxx1 folds.
ILike_One_9 folds.
pokerjerie is all-In.
Hero ??

Step 1- Info required

All the stack sizes (see above)
Blind level
Payout Structure - Standard sng 50/30/20%. 1st =$100, 2nd = $60 and 3rd = $40. Prizepool = $200
Action and our hole cards.

Step 2 -  The problem

We want to asses whether calling or folding is correct.




Step 3- What are stack is worth if we fold?

A simple shortcut to this is to use a free icm calculator on the internet like this one http://www.icmpoker.com/Calculator.aspx and stick in the numbers. To use this put in the payouts (to get a monetary final result)  or % payouts in the top row and the stack sizes in the column. 

Basically you assess each person chance of finishing in each position and then multiply that chance by each payout. This will give you how much your stack is worth.



If we fold the new stack size look like thus

Markpl111 7385
Pokerjerie 9010
I like one 9 1860
xxCracker1xx 1745

We get our stack is worth $67.42. This is an important number as we are now going to compare it to how much our stack will be worth if we choose to call.

Step 4 How much is our stack worth if we call

There are 2 outcomes here as we can win or lose the hand, with a very small % chance we chop but the effect of that is negligble.

(a)

We lose the hand our stack is worth nothing, so $0.

(b) We call and win:

New stacks look this
Hero 15970
Pokerjerie 425
I like one 9 1860
xxcrackerxx1 1745

Back to the icm calc and our stack is now worth $91.37.

(c) Put villian on a range.

Ok this is the poker bit, my best guess is that villian is pushing about 25%

To do this it is best to use pokerstove (http://www.pokerstove.com/)

Here are the results:

563,348,016  games     0.005 secs   112,669,603,200  games/sec

Board:
Dead: 

   equity    win    tie          pots won    pots tied   
Hand 0:    71.140%     70.74%    00.40%         398519756      2243950.00   { QcQd }
Hand 1:    28.860%     28.46%    00.40%         160340360      2243950.00   { 22+, A2s+, KTs+, QTs+, JTs, A2o+, KTo+, QTo+, JTo }

So we win this hand 71% of the time.

(d)

Now we bring the two numbers together

28.86% of the time we end up with nothing
71.14% of the time we end with $91.37

(0.2886* 0)+ (.7114*91.37)= $65

So $65 is our expectation by calling here given our range is right.

Step 5- Compare the difference between folding and calling

Folding from step 3= $67.42
Calling from step 4 = $65

So overall we lose $2.42 from calling.

Result

The correct action given we have got our range on Pokerjerie correct is to fold. As I said ICM is counterintutive, which is why people play sngs so badly at times. We are giving up around 1.22% of the prize pool here, more than the rake we paid to enter.

Sidenote:

Of course if we change his range about we get different results and this change the number we get from step 4. Breakeven point can be found by doing this.

By multiplying the new probability of winning the hand by the chipstack worth of winning the hand of $91.37 and comparing that to the worth of our stack if we fold (67.42)

In this case 91.37 * x =  67.42

Where x= The probability of winning the hand. So to find x rearrange the equation 67.42/91.37 = 0.7378.

So we have to win the hand 73.78% of the time to breakeven against opponents range. Which happens when Pokerjerie is pushing about 40% of hands.



Queries and comments are welcome.

Im off to pray i haven't screwed the maths up.

[boldie]

FK me..I am properly confused..will bookmark this one untill I have more time to read.

Thanks a million for this Longy..I know this is going to be one of them massively useful posts.

[Simon Galloway]

I will post a bit more later after Luton tourny, I am very interested in this, but in short, this is a perfect example to amplify thoughts on:

In $20-$100 sngs, the ICM numbers are close between call and fold.  Close enough that a slight error in range negates the % difference.  Yet in practice, QQ calls here I am guessing 80% of the time, maybe even higher.  In fact, I call here.  I understand that seat 7 and 8 are ecstatic about this news, but I am so often 70% and sometimes much higher to win this hand.  

Pokerjenies range is about as wide as it can get, a big part of thier range derived from expecting you to be unable to call and risk elimination with 2 shorties left, of course.  Longy, you've made good sense so far, but how the hell you get 25% here I don't know.   Perhaps the stack sizes are a bit too equal, if Hero has 2k less for example, then I am pretty much pushing 100% as pokerjenie with any two.

In practice, QQ calls and then loses 99% of the time to K4, or QQ folds and then doubles up each of the other short stacks in the next orbit.  Take your pick.

Is this a flaw in the ICM model  or more likely a flaw in poker theory from sng players or some hybrid combination of both?

[Longy]

I will post a bit more later after Luton tourny, I am very interested in this, but in short, this is a perfect example to amplify thoughts on:

In $20-$100 sngs, the ICM numbers are close between call and fold.  Close enough that a slight error in range negates the % difference.  Yet in practice, QQ calls here I am guessing 80% of the time, maybe even higher.  In fact, I call here.  I understand that seat 7 and 8 are ecstatic about this news, but I am so often 70% and sometimes much higher to win this hand.  

Pokerjenies range is about as wide as it can get, a big part of thier range derived from expecting you to be unable to call and risk elimination with 2 shorties left, of course.  Longy, you've made good sense so far, but how the hell you get 25% here I don't know.   Perhaps the stack sizes are a bit too equal, if Hero has 2k less for example, then I am pretty much pushing 100% as pokerjenie with any two.

I will be honest i picked 25% out as in my experience players at this level, players never push wide enough. In fact i would say less than 10% of players playing mid level sngs push over 40% of their hands from the sb here, which is the magic number for qq to be a call in this spot. This is more a read on a standard mid-low stakes sng player.I would push 100% in Pokerjeries spot as long as i knew the bb was not a complete idiot. I don't know what Pokerjerie has but i bet it wasn't 23o, though he should push this.

ICM is a model which still relies on putting people on ranges then doing the maths. It is not an a model that is based on people playing ICM poker back at you, as the joy of poker is humans are not perfect. If Pokerjeries was a competent regular I would snap call this but in this hand from memory he was far from that, so i folded.

I will expand on situations where good sng players playing ICM poker back at each other. Leads to something called Nash equilibrium, where there is point where ranges merge to a fixed point where everyone is playing close to perfect.

ICM is not flawed in the way you are describing here, the model simply analyses the ranges we give it. If we get the ranges wrong it as our mistake as a poker player not ICM's fault.

[vegaslover]

ICM is not flawed in the way you are describing here, the model simply analyses the ranges we give it. If we get the ranges wrong it as our mistake as a poker player not ICM's fault.

While I have some understanding of the maths involved in ICM, the above is probably key to me to using ICM. i.e getting the range of the players and then using ICM to push/fold spots, rather than all out aggression or simple odds/outs percentages.

[EvilPie]

This is brilliant. So the correct move here is to fold according to ICM.

I think if I was playing this I would instacall and be shocked afterwards to find that I'd made a mistake.

The only way I find a fold is if I'm playing at a higher than normal level and wanted to cash which unfortunately would sway my range. Even then I don't think I can fold QQ.

I guess the problem is do we want to get outdrawn by someone with Ax / Kx? These hands make up quite a lot of his range so the chance of being outdrawn with little chance to improve is higher than we would like given our current chances of cashing.

If we call and win we are almost definitely going to win the whole thing. If we call and lose we give up what was a very good chance of cashing / winning.

I love this ICM stuff purely because of situations like this.

Did you find the fold?

[thetank]

I've got a question about the term unexploitable shove. Does the adjustment of play from the big blind have to be profitable for this term to be applied?

Say I'm the small blind in the above example and you know I'm shoving 100% of hands, you can only correctly call only with TT+

If I knew you were playing perfectly and only calling with TT+, it's still correct for me to shove 100% of hands.

So does that make it an unexploitable shove?

You could if you wanted to destroy my EV from the shove by calling with a range like 22+,A2+,K2+,Q4o+,Q2s+,J8o+,J5s+,T9o,T7s+,98s
Doing so has obviously made it massively incorrect for me to shove anything less than JJ, let alone ATC.

But doing so has made a huge chunk in your own EV, and I wouldn't expect anyone to play like this.

So is the shove still unexploitable?

[Longy]

This is brilliant. So the correct move here is to fold according to ICM.

I think if I was playing this I would instacall and be shocked afterwards to find that I'd made a mistake.

The only way I find a fold is if I'm playing at a higher than normal level and wanted to cash which unfortunately would sway my range. Even then I don't think I can fold QQ.

I guess the problem is do we want to get outdrawn by someone with Ax / Kx? These hands make up quite a lot of his range so the chance of being outdrawn with little chance to improve is higher than we would like given our current chances of cashing.

If we call and win we are almost definitely going to win the whole thing. If we call and lose we give up what was a very good chance of cashing / winning.

I love this ICM stuff purely because of situations like this.

Did you find the fold?

Yes did find the fold in game though i ummed and ahhed about it for a quite well I think. Have editted the op to make that more clear that folding is what ICM is advising here given we have the ranges correct.

[Longy]

I've got a question about the term unexploitable shove. Does the adjustment of play from the big blind have to be profitable for this term to be applied?

Say I'm the small blind in the above example and you know I'm shoving 100% of hands, you can only correctly call only with TT+

If I knew you were playing perfectly and only calling with TT+, it's still correct for me to shove 100% of hands.

So does that make it an unexploitable shove?

You could if you wanted to destroy my EV from the shove by calling with a range like 22+,A2+,K2+,Q4o+,Q2s+,J8o+,J5s+,T9o,T7s+,98s
Doing so has obviously made it massively incorrect for me to shove anything less than JJ, let alone ATC.

But doing so has made a huge chunk in your own EV, and I wouldn't expect anyone to play like this.

So is the shove still unexploitable?

Inexploitable means that sb can shove a given hand and no matter which range the bb calls. It will still be a positive +ev shove if bb calls 0% or 100% or anything in between.

In this actual situation (i used sngwiz i will be honest) sb can only shove TT+ for it to be inexploitable. This is a kind of unique situation where you have to have a really strong hand for it to be inexploitable blind vs blind. As bb spite calling you with a massive range is pretty disastrous as losing the hand is pretty bad news for either of us. A showdown here just dumps a tonne of equity, to the shortstacks as it vastly increases their chances of finishing 3rd (or in fact 2nd).

The reason why sb can push so wide if bb is comptent, as bb knows that calling without a monster is disastrous for him. In fact ak is not a call against sb shoving any two here for the bb here.

So basically im saying that the term inexploitable is not really appropriate, it is simply a case of both players understanding ICM (or not) therefore giving us some extreme ranges.

[thetank]

I think I get you.

So I can only use the term inexploitable when I am talking about shoving a specific hand, and not when I am talking about what range I would shove with.

Cheers.

[david3103]

I have a real life situation which I've attempted to analyse on an ICM basis. Am I getting this right?

Final Table, £10 FO. Stack sizes (approx)

Button (Me) - 7,000
SB - - - - - - -  15,000
BB - - - - - - -  18,500
UTG - - - - - -  9,000
UTG + 1 - - - - 10,000
+ 2 - - -- - --- - 7,500
 Mid - - -- - - - -16,500
+1 - - - - - - - -  15,500
Chip Leader - - -26,000

Binds 500/1000

Top 3 paid - 50%, 30% 20%

Two callers when it got to me, one of them the chip leader who will see ANY flop if it's not raised, but will fold if there's a raise and he's ATC

I pushed. The hand previous I had called an all-in from the other short stack and we'd both shown AJ - the debate afterwards included my statement that AJ was at the edge of my range for the shove and I'd be looking for 55+ and AJs+

Is the BB's call justified with A2o?

The ICM calc gives him a value of 14.56 before the hand
if he folds (and everyone else does the same) his value drops to 13.83

My stated range is actually tighter than the one that Pokerstove shows to include 55 but even if he has me on that range at 26.8% he's losing 56.65% and winning 43.35%

results of which are either
a new ICM value of  9.49
or 20.44

(.5665*9.49) + (.4335*20.44) = 14.23
which is .33 below his value if he folds?

Apologies, but the only way I can get to grips with this is to run it a few times. Is that difference small  enough to ignore?
If my range isn't extended to the pokerstove version of anyone playing 55 then he's even less likely to beat me.



I'm really trying to get the maths here, but where do I send the 50p?

[Longy]

I think I get you.

So I can only use the term inexploitable when I am talking about shoving a specific hand, and not when I am talking about what range I would shove with.

Cheers.

Yes im dealt aq on the button with 10bbs mid game this is an inexploitable shove, for example.

[thetank]

My tuppence worth david3103...

A2o looks like a bad call there from the big blind. Marginal at best.

I'd also say that your stated range for shoving that spot is far too tight. If you're pretty sure the limpers will pass and A2o is near the bottom of BBs calling range then this is a primo spot to increase your stack by 50% uncontested and you should slap it in with most cards 74s, J7o etc.

[Longy]

I have a real life situation which I've attempted to analyse on an ICM basis. Am I getting this right?

Final Table, £10 FO. Stack sizes (approx)

Button (Me) - 7,000
SB - - - - - - -  15,000
BB - - - - - - -  18,500
UTG - - - - - -  9,000
UTG + 1 - - - - 10,000
+ 2 - - -- - --- - 7,500
 Mid - - -- - - - -16,500
+1 - - - - - - - -  15,500
Chip Leader - - -26,000

Binds 500/1000

Top 3 paid - 50%, 30% 20%

Two callers when it got to me, one of them the chip leader who will see ANY flop if it's not raised, but will fold if there's a raise and he's ATC

I pushed. The hand previous I had called an all-in from the other short stack and we'd both shown AJ - the debate afterwards included my statement that AJ was at the edge of my range for the shove and I'd be looking for 55+ and AJs+

Is the BB's call justified with A2o?

The ICM calc gives him a value of 14.56 before the hand
if he folds (and everyone else does the same) his value drops to 13.83

My stated range is actually tighter than the one that Pokerstove shows to include 55 but even if he has me on that range at 26.8% he's losing 56.65% and winning 43.35%

results of which are either
a new ICM value of  9.49
or 20.44

(.5665*9.49) + (.4335*20.44) = 14.23
which is .33 below his value if he folds?

Apologies, but the only way I can get to grips with this is to run it a few times. Is that difference small  enough to ignore?
If my range isn't extended to the pokerstove version of anyone playing 55 then he's even less likely to beat me.



I'm really trying to get the maths here, but where do I send the 50p?

Right then i get the same numbers as you using the ICM calc, fwiw if you enter 50/30/20 you will get the % of the prize pool not the monetary value. For the monetary value actually put in the payouts in £'s. You probably realised this anyway.

I think you have misused pokerstove I get this.

---
 159,244,272  games     0.005 secs    31,848,854,400  games/sec

Board:
Dead:  

   equity    win    tie          pots won    pots tied   
Hand 0:    72.610%     71.22%    01.39%         113415897      2211148.50   { 55+, AJs+, AJo+ }
Hand 1:    27.390%     26.00%    01.39%          41406078      2211148.50   { Ac2d }

So (.72610*9.49) + (.27390*20.44) = 6.89 + 5.598 =12.49

Then we compare to the number if he folds, not the number before the hand.

Which is 13.83

So 12.49 -13.83= -1.34

So he makes a -1.34%  of the prize pool in calling, this is pretty bad with this many players left in my experience.