Theory: An ICM Tutorial Chat. (Long)

[Insty]

I try to take a couple of days a week off from actually playing, but I usually have an IRC window open. (Gotta do something with those multi-monitors when you're not playing!) Gigglebet asked a question about ICM and since I had some spare time I thought I'd explain a few things. It turned into quite a tutorial and many people suggested I should post it here.

This is an introduction to ICM it contains the basics about ICM, what equity is, stack size considerations and starts to get into ranges. It should hopefully give you some insight into how SNGPT and SNGWiz work this is NOT a tutorial in how to use those programs. It does contain some math, but nothing you shouldn't easily be able to do with a calculator. If you follow along and actually do the calculations yourself, you should learn something.

I've stripped out most of the chat that wasn't relevant to the discussion and fixed a few typos. But this is how it happened:

<Gigglebet> okay, on this ICM calc. (http://www.chillin411.com/icmcalc.php) i put everyone with even stacks & calculated it & everyone has an ICM equity of 0.1111. what am i looking at? :\
<Insty> do you know what equity is?
<Gigglebet> no.
<Insty> do you know what a prize pool is?
<Insty> (that is a good page)
<Gigglebet> yes.
<Insty> ok.
<Insty> ok, so $11 SNG.
<Insty> lets say 10 players
<Insty> prize pool is $100 and rake is $10
<Insty> and everyone starts with 1000 chips.
<Insty> hmm you must have done 9 player, but anyway [img]/images/graemlins/smile.gif[/img]
<Gigglebet> i did. 9 player sng.
<Gigglebet> i think i need to just sit down & study ICM for a day.
<Insty> we set up a fake $11 sng
<Gigglebet> mhm.
<Insty> and we put the numbers into the ICM calc.
<Insty> and it tells us that the equity is 0.1 for everybody.
<Insty> yes?
<Gigglebet> yes.
<Insty> one of the main assumptions of ICM is all players are of equal skill.
<Insty> so everybody has an equal chance of winning .
<Insty> so everbody has a 1/10 chance of winning the tournament.
<Gigglebet> with equal stacks.
<Gigglebet> okay.
<Insty> a 1/10 chance of coming second and 1/10 chance of coming 3rd.
<Insty> so everybodies expected return from the prize pool is 1/10
<Gigglebet> alright.
<Insty> this is equity
<Insty> - how much of the prize pool do you expect to win (on average)
<Gigglebet> ah, alright.
<Insty> now, lets bust a player out.
<Insty> player 10 goes out
<Insty> now has 0 chips
<Insty> player 1 gets them all and so has 2000
<Gigglebet> ah, we're using 1000 as the chip stacks.
<Gigglebet> okay.
<Insty> (I was, but it doesnt matter what the starting stacks are)
<Gigglebet> now everyone has a .102, excluding player 1, who has .184
<Insty> done that?
<Insty> yep
<Insty> now this is interesting.
<Insty> because
<Insty> everyone else now has a bigger equity value.
<Insty> and they werent even in the hand.
<Insty> and the bigstack's equity has NOT doubled
<Insty> even though his chipstack has.
<Gigglebet> is this why playing tight is +ev. 'cause, if you allow people to bust ea. other out, your ev is increasing even though you're not in the hand.
<Insty> not exactly. but it's still interesting [img]/images/graemlins/smile.gif[/img]
<Gigglebet> ha, i tried.
<Insty> now ICM calculations with 9 people left are good.
<Insty> but not particularly useful while the blinds are still low.
<Insty> so bust a few people out and get to the bubble.
<Gigglebet> alright.
<Insty> player 1: 5000, player 2: 3000, player 3: 1000, player 4: 1000
<Gigglebet> Dang.
<Gigglebet> .376, .308, .158, .158.
<Insty> yep
<Insty> so, you remember what equity is?
<Gigglebet> expected return of the prize pool.
<Insty> yep
<Insty> so player1 will get $37.60 etc.
<Insty> (on average, equal skill..)
<Insty> now.
<Gigglebet> wait!
<Gigglebet> i want to see if i'm correct in this thought.
<Gigglebet> okay, now if the big stack ends up HU with a short stack, he should be pushing with the majority of his hands. pushing & winning - +0.036. pushing & losing - -0.4.
<Gigglebet> *-0.04.
<Insty> Gigglebet, you are getting ahead of things here.
<Gigglebet> sorry.
<Gigglebet> continue.
<Insty> the ICM calcs we have been doing
<Insty> are the "root" of ICM
<Insty> they take chipstacks and convert them to equity.
<Gigglebet> alright.
<Insty> now where this is useful
<Insty> is it allows you to work out some "what if" scenarios.
<Insty> so, let's assume we are player 2.
<Gigglebet> okay.
<Insty> blinds are 100, 200
<Insty> we are in the sb, bigstack player 1 is in the bb.
<Insty> we can work out our EV(Fold)
<Insty> just subtract the 100 chips from our stack, and add them to the BB.
<Gigglebet> so, p#2 = .304.
<Insty> yep
<Gigglebet> & p#1 = .379.
<Insty> yep
<Insty> so, write down EV(Fold) = 0.304 on a bit of paper
<Insty> now, we can also push.
<Insty> and if we do, we will either win or lose
<Insty> (lets not worry about ties right now..)
<Gigglebet> alright
<Insty> now what is our EV if we push and lose?
<Gigglebet> 0
<Gigglebet> ?
<Insty> correct [img]/images/graemlins/smile.gif[/img]
<Insty> and if we push and win?
<Gigglebet> .405?
<Insty> yep.
<Insty> now, the cards are dealt.
<Insty> you look down at Ah Kh
<Insty> BB looks at his cards and says: black 2's my lucky hand that I always call allins with every time.
<Insty> (he's an idiot.)
<Gigglebet> ha, i figured.
<Insty> so we know that if we push, we are getting called and will win 50% of the time.
<Insty> so we can use those odds to work out the EV of pushing.
<Insty> 0.5 * EV(PushAndLose) + 0.5 * EV(PushAndWin)

Note: this is not exactly a 50/50 but it's close enough for this example and it makes the math easier.
Homework: Find a matchup that is exactly 50/50.

<Insty> so EV(Push) = ?
<Gigglebet> .2025.
<Gigglebet> $20.25.
<Insty> yep.
<Insty> now how does this compare to ev Fold?
<Gigglebet> a difference of .1015.
<Insty> so are you pushing or folding here?
<Gigglebet> folding.
<Insty> good choice.
<Gigglebet> how easy/hard is it to implement this into playing?
<Insty> Type in the following chipstacks:
<Insty> player 1: 1000 (he donked off all his chips in the next few hands)
<Insty> player 2 (hero): 3000 (we won some blinds and got back to where we were more or less)
<Insty> player 3: 6000 (went on a total heater and won all player 1's chips)

Note: I made a mistake here, Player 3 should only have 5000 here. but the rest of the example uses these numbers so I'll keep them.

<Insty> player 4: 1000 (just folded a lot)
<Gigglebet> .154, .301, .390, .154.
<Gigglebet> obv p#3 w/ the most equity.
<Insty> yep
<Insty> now, we're in the sb again
<Insty> player1 is in the bb.
<Insty> blinds are still 100/200
<Insty> EV Fold?
<Gigglebet> .295.
<Insty> yep.
<Insty> woohoo, we've got Ah Kh again.
<Gigglebet> sweet!
<Gigglebet> rigged
<Gigglebet>
<Insty> player 1 looks at his cards and sees 2 black 2's again.
<Insty> you've got a sick read on him so he doesnt have to tell you this time [img]/images/graemlins/smile.gif[/img]
<Insty> we already know the cards.
<Insty> so he's always calling.
<Insty> you want to work out EV(Push)?
<Gigglebet> so, .407 pushing equity.
<Gigglebet> & up against the .295 of folding, we're pushing.
<Insty> howd you get 0.407?
<Gigglebet> uh-oh. bad math.
<Gigglebet> please hold.
<Insty> have you worked out the answer?
<Gigglebet> .220186.
<Gigglebet> ~.220.
<Insty> hmm.
<Insty> something has gone wrong somewhere...
<Gigglebet> must have.
<Insty> EV(PushAndWin) =?
<Gigglebet> .356.
<Insty> yep
<Insty> EV(PushAndLose) =?
<Gigglebet> .237.
<Insty> P(win) * EV(PushAndWin) + P(lose) * EV(PushAndLose) = ?
<Insty> 0.5 * 0.356 + 0.5 * 0.237 = ?
<Gigglebet> .2965?
<Insty> correct.
<Gigglebet> woah.
<Insty> now EV(Push) compared to EV(Fold) ?
<Gigglebet> with a difference of .0015. would either decision be correct?
<Gigglebet> or are we just pushing.
<Insty> well, in this hypothetical world it's +EV so we'd have to push.
<Insty> but you can see that the answer was quite different from last time.
<Gigglebet> ah, alright.
<Gigglebet> yes.
<Insty> now
<Insty> things get more interesting when you dont know what he has.
<Insty> lets say mr twoducks only ever plays 22
<Gigglebet> alright.
<Insty> back to the start of this hand, we have 3000, he has 1000
<Insty> we look at AK hearts
<Gigglebet> okay.
<Insty> he hasnt looked at his hand yet.
<Insty> it may or may not be 22.
<Insty> what is our EV if we push and he folds?
<Gigglebet> .289?
<Insty> no
<Gigglebet> why not?
<Insty> how many chips do you have?
<Gigglebet> ah, i know what i did.,
<Gigglebet> .313?
<Insty> yep. what did you do?
<Gigglebet> instead of p#1 having 1000 chips, i had him @ 2000. & instead of having 800 chips after folding, i had him @ 1800.
<Insty> ah
<Insty> ok
<Gigglebet> alright.
<Insty> so, now we know P(PushAndWinBlinds) and P(PushAndGetCalled)
<Gigglebet> yes.
<Insty> so what is P(he holds 22)
<Gigglebet> uh.
<Insty> I dont know this off the top of my head either [img]/images/graemlins/smile.gif[/img]
<Gigglebet> how do you figure this out?
<Insty> you have AKh
<Insty> there are 50 cards left in the deck
<Insty> chance he is dealt a 2 for his first card is 4/50
<Insty> chance he is dealt a 2 for his second card is 3/49
<Insty> so 4/50 * 3/49 =
<Gigglebet> .00489%?
<Gigglebet> well, approx.
<Insty> yep
<Insty> um
<Insty> I'm not sure that % is correct
<Insty> that % is not correct
<Insty> its 0.00489 out of 1
<Insty> so 0.489%
<Gigglebet> oh, my bad.
<Insty> but it's more useful as the out of 1 number.
<Gigglebet> yeah.
<Insty> because we want to calculate:
<Insty> P(holds pocket 2s) * EV(PushAndGetCalled) + P( doesnt hold pocket 2s) * EV(PushAndWinBlinds)
<Gigglebet> EV(PushAndGetCalled) = ?
<Insty> we worked it out before.
<Insty> .2965
<Gigglebet> okay, hold on.
<Insty> hope you have a calculator handy [img]/images/graemlins/smile.gif[/img]
<Gigglebet> yeah. it's a nice number.
<Gigglebet> .2053244005.
<Gigglebet>
<Insty> that number doesnt look right.
<Insty> what was your calculation?
<Gigglebet> ah!
<Gigglebet> .00489 * .2965 + .99511 * .313?
<Insty> yep.
<Gigglebet> .2201955.
<Insty> um
<Insty> too low
<Gigglebet> .311923244005.
<Gigglebet>
<Insty> not exactly the number I got, but much closer [img]/images/graemlins/smile.gif[/img]
<Gigglebet> what'd you get?
<Insty> 0.312919
<Gigglebet> you got one of them nifty rounder-upper calcs?
<Insty> probably [img]/images/graemlins/smile.gif[/img]
<Insty> let's call it 0.312
<Gigglebet> AH.
<Gigglebet> you were right.
<Gigglebet> my math was wrong.
<Gigglebet> .312919315.
<Insty> ok
<Insty> let's call it 0.3129
<Insty> hmm I think our rounding has screwed us somewhere.
<Insty> when I did the calculation with the long numbers on my fancy calculator I got:
<Insty> 0.3127
<Insty> which sounds sensible.
<Insty> so:
<Insty> P(holds pocket 2s) * EV(PushAndGetCalled) + P( doesnt hold pocket 2s) * EV(PushAndWinBlinds) = 0.3127
<Gigglebet> alright.
<Insty> so EV(AllThatPushStuff) > EV(Fold)
<Insty> where before it was just close.
<Insty> now it is clearly obvious that pushing is way better.
<Gigglebet> wow. that's interesting.
<Insty> what is the difference?
<Gigglebet> the difference is not knowing his hold cards, obv. & the percentage that he's not holding 22 is greater than that of him holding 22.
<Insty> yeah he's hardly ever holding 22.
<Insty> this is an extreme example of the magical "Fold Equity"
<Insty> (we're getting toward the end of the lesson now [img]/images/graemlins/smile.gif[/img] )
<Gigglebet>
<Insty> What was EV(Push) - EV( Fold ) when we knew we held AK hearts and he held 22?
<Gigglebet> .1015.
<Gigglebet> i believe.
<Insty> .0015?
<Insty> <Gigglebet> with a difference of .0015. would either decision be correct?
<Gigglebet> i was looking for that, thanks.
<Insty> Ok, now we dont know what he has but we know his (extremely tight) RANGE what is it
<Insty> EV(Push) - EV( Fold )
<Gigglebet> uh.
<Insty> I'll help you out, I've known what to write down [img]/images/graemlins/smile.gif[/img]
<Insty> EV(Push) = 0.3127
<Insty> EV( Fold ) = 0.2953
<Gigglebet> 0.0174.
<Insty> bingo
<Gigglebet> so, his range is 1.7%
<Gigglebet> ?
<Gigglebet> haha.
<Insty> no no no [img]/images/graemlins/smile.gif[/img]
<Gigglebet> haha.
<Gigglebet> i was about to say.
<Gigglebet> holy mother.
<Insty> as a proportion of our $100 prize pool.
<Gigglebet> what a tight-ass.
<Insty> well he is a tight ass
<Insty> as a proportion of our $100 prize pool
<Insty> a push is worth $1.74
<Insty> before it was only $0.15c
<Gigglebet> Damn.
<Insty> now
<Insty> you have sngpt?
<Gigglebet> yes.
<Insty> all these calculations are a real bitch to do by hand
<Insty> as you have probably been discovering [img]/images/graemlins/smile.gif[/img]
<Gigglebet> okay. & yah, these calcs are crazy.
<Insty> but, we can use sngpt to do all the hard work for us.

Note: SNGWiz might be able to do this but I don't know how to use it as well. Post a reply if you know how to do this with SNGWiz.

<Insty> fire it up.
<Gigglebet> 1 step ahead of you.
<Insty> ok, choose Structure
<Insty> from the edit menu
<Insty> and select Party SNG
<Gigglebet> got it.
<Insty> set buyin to $10+1
<Insty> players to 4
<Insty> blinds 100/200
<Insty> Hand AKs
<Gigglebet> got it.
<Insty> put in the chipstacks we had from before: 1000, 3000, 6000, 1000
<Insty> select the sb position radio buton
<Insty> and for the BB call range put: 22
<Insty> hit compute, and watch the magic [img]/images/graemlins/smile.gif[/img]
<Gigglebet> i am pushing.
<Insty> what does the EV Diff number say?
<Gigglebet> ev diff. = +1.7% or $1.70
<Insty> what number did we calculate?
<Gigglebet> 1.74!~
<Insty> my sngpt sas Ev Diff. +1.7% / +$1.74
<Gigglebet> ah, mine does too. just didn't look close enough.
<Insty> heh ok [img]/images/graemlins/smile.gif[/img]
<Gigglebet> my bad.
<Insty> now, nobody is going to be only playing 22
<Gigglebet> right.
<Insty> and we're not always going to get AKs [img]/images/graemlins/frown.gif[/img]
<Gigglebet> yah, whatsupwiththat!?
<Insty> poker is rigged obv.
<Gigglebet> obv.!
<Gigglebet> okay, so insty.. summarize what you've taught me.
<Gigglebet> or do you intend on doing that in your post on 2+2?
<Insty> Why dont you summarise what I've taught you? [img]/images/graemlins/smile.gif[/img]
<Insty> I'll probably put a summary on the 2+2 post.
<Gigglebet> ...
<Insty> In this lesson we've larned:
<Insty> what equity is.
<Gigglebet> How to calculate EV(Push) & EV(Fold).
<Insty> that if someone is stupidly tight, pushing can be good.
<Insty> the result of things is more dependent on chip stacks than on the cards you hold.
<Gigglebet> yes,
<Insty> The next lesson will be about ranges.
<Insty> But I'll copy this chat, tidy it up a bit and post it.
<Gigglebet> Insty, I messed with Mr. Tight's range & turned him into Mr. Loose, calling w/ ATC & it's still profitable to push.
<Insty> yep.
<Insty> try going back to our first example though.
<Insty> when mr twoducks had the 6k stack
<Gigglebet> wasn't it 5k?
<Insty> 6 3 1 1
<Insty> it has to add up to 10.
<Insty> hmm
<Insty> yeah 5 [img]/images/graemlins/smile.gif[/img]
<Insty> oops
<Gigglebet> heh. [img]/images/graemlins/smile.gif[/img]
<Gigglebet> against ATC, folding is more profitable.
<Insty> crap we put another 1000 chips in the pot for the 2nd example [img]/images/graemlins/frown.gif[/img]
<Insty> dammit
<Insty> oh well, homework is to redo the calcks for the right number of chips [img]/images/graemlins/smile.gif[/img]
<Gigglebet> pushing is only profitable if i can put the BB on calling with 25% of his hands.
<Insty> Gigglebet, you'll find though, that as the 5K BB gets tighter, pushing actually becomes profitable.
<Gigglebet> Yep.
<Gigglebet> I'm messing with the ranges.
<Insty> even if you have a crappy hand.
<Gigglebet> 25% & below is profitable.
<Gigglebet> Well, with AKs.
<Insty> the lessons are good.
<Gigglebet> Insty, is that the end of my lesson for today.
<Insty> yeah
<Insty> I've forgotten where I was going with it [img]/images/graemlins/smile.gif[/img]
<Gigglebet> Alright, I'm looking forward to the post on 2+2.
<Insty> so I should stop for a while
<Gigglebet> Heh, it's all good.
<Gigglebet> You've taugh me so much in the past hour or so.
<Gigglebet> Thanks.

Homework: Redo the calculations in the second hand using the "correct" chipstacks. (1000, 3000, 5000, 1000)

Questions? Comments? Want a lesson? Think I got something wrong? Want to give a better one?

You can find us: #stt on DALNET

Blackize wrote some instructions:
"Google mIRC and download the client. When you select a server to join, use dalnet. Then a box should pop up asking what room you want to join and at that point you just type #stt and click join."

I have some thoughts about the calculations that programs like SNGPT and SNGWiz do, these are not only ICM and should probably be called something different: Ranged Equity Calculations (REC) maybe? And some thoughts about stack sizes and hand ranges.

But that's another lesson...

Insty.

[Kevin8423]

Yaaaa its posted. Anyone who has basic questions about ICM, or how to go about learning to use it in game should go through this with the tools used. I think this has the potential to help a lot of people if they put in the time to go through it.

[ymu]

Nice lesson. You're a good teacher. [img]/images/graemlins/smile.gif[/img]

It'd definitely be good to have another term that emphasises ranges in the ICM calc. There's been a lot of threads recently that seem to implicitly assume that calling ranges are a fixed thing - no reads necessary.

Opponent Dependent Ranged Equity Calculation (ODREC) or summat ...

[TheFishhh]

Homework #1: 22s vs. 22s, 33s vs. 33s, etc. [img]/images/graemlins/smile.gif[/img] I will play around with actual hands later.

Homework #2: EV (Fold): .302. EV (Push & win): .364. EV (Push & lose): .243; thus EV (Push): .3035, because .5 * .364 + .5 * .243 = .3035. Resulting in a difference of .0015. & since we're living in a hypothetical world, it's +EV, & therefore we're pushing.

I hope I passed. [img]/images/graemlins/smile.gif[/img]

[sbj]

Nice post. Thanks for sharing.

[BradleyT]

In the scenario given (3000,1000,1000,5000 and we're pushing into a 1000) IMO anything 0 or above is a no-brain push. You can't go bust, when you lose there's still another stack half the size of yours and one equal to yours, and winning gets you ITM with 20BB vs chipleaders 25BB. And if big stack has been stealing my blinds because of the bubble it makes a +0 decision that much easier.

[mildeng]

I'm still learning this too, in the beginning of the tourney, everyone's equity = .1. When Player 10 lost to player one, how does the math go for the new ICM numbers?

[Insty]

[ QUOTE ]
I'm still learning this too, in the beginning of the tourney, everyone's equity = .1. When Player 10 lost to player one, how does the math go for the new ICM numbers?

[/ QUOTE ]

I'm not sure I understand the question. Try typing the numbers into the ICM calculator that is linked and having a look.

Hint: They will always add up to 1. (100%)

If you want to know how to actually do the ICM calculations, come to IRC and ask so I can get another chat to post [img]/images/graemlins/wink.gif[/img]

[holy32]

very nice post.

Maybe add this to the FAQ as the quick ICM intro ?

[montagu]

Insty, would you be so kind as to also post the lesson on ranges, once you have done that one. I have found this post massively informative, thanks alot.