Think they totally underestimate the value of a big stack. If you take some gamble to double up at the final table or close, the chips you gain may not be worth as much as the chips you lose. However, if you win, you often have the tournament lead or close.

There are big ICM effects in certain situations like if there are several medium stacks close together and certain microstacks, then there are really bad problems for the medium stacks to gamble allin. However, I don't think the formulas are accurate in indicating how bad a flip is for you.

yeah thats right

ICM is the most accurate way to estimate the current value of your stack if all skill was equal. This and only this is ICM.

Obviously a lot of other things come into consideration while playing and decided what plays to make when.

ICM is a model using assumptions (equal skill, chance of winning equal to chip%). If you think the assumptions are wrong, you can often easily adjust the model to account for that. For your example (big stack is worth more), just use a bigger $EV than the one based on ICM for that result and use that in the calcs. So for a simple scenario of calling an allin, just replace $EV_win with your own, better value in the following equation that determines whether the call is +$EV:

x*$EV_win+(1-x)*$EV_lose > $EV_fold (x=equity vs villain’s range)

You can do similar stuff in other cases: use a bigger $EV when you think you have a skill edge, or when blinds are such that people are forced all-in the next hand, etc.

Of course, finding that better, more realistic $EV value will always be a bit of guesswork.

No. I happen to be an expert on ICM so I will explain if anyone is curious. I will try to be as non-mathematical as possible, but it is inherently a mathematical subject.

[Given some payout structure,] equity by definition can be predicted through a chip flow (transition) function. What that means is: in any tournament, if you know how everyone will play, and you know the payout structure of the tournament, you theoretically know everyone's equity at any given point in time.

ICM, as a model, classically expressed equity by saying that the probability of a tournament ending up with finishing order sigma was equal to some formula* (at bottom.) This more or less (with respect to equity) was an implicit statement on (chip) flow, even though it was not phrased in this way. However, ICM gives more weight to big stacks without any information on chip flow, which is probably part of the reason ICM has stood up and other models have not, because tournaments do generally allow for the bigger stack to have an advantage.

Lets give an example. Lets say a tournament is 3 handed with stacks of Alice=3000, Bob=2000, and Charlie=1000. If flow were really simple (for example, say the transition function is that one random person gives another random person one chip), then you can show coming third will be proportional: Charlie comes 3rd twice as often as Bob, three times as often as Alice, and Bob comes 3rd 3/2ths as often as Alice. And so you get that the chance that Alice comes 3rd is 2/11, or about 18.18%.

Some models think this way, and for good reason. For example, if this was a winner take all freezeout with players of equal skill, then infact Alice does come 3rd 18% of the time more or less (the disadvantage of a larger stack affects things, etc. so it is not exact obviously.) If this is a satellite paying 2 places, then Alice would come third far less often.

Now, ICM says that the chance that Alice comes 3rd is 15%. Big difference. But more importantly, it does not change this advice based on the payout structure. With respect to equity, one can think of this as: it has a fixed, implicit flow in mind. So of course ICM is going to be very flawed depending on the format -- it is going to warp significantly whenever actual chip flow in the tournament deviates strongly from the fixed way it thinks.

For almost all cases though, this doesn't really matter, because you are not making decisions so precisely, but rather guesstimating everything anyways. For some cases or formats (especially in SNGs), it does matter.


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*ICM prediction of finishing order sigma is

Oh, with regards to your question, its unclear. For example, say you have a 7 player tournament where you have 6000 chips, and everyone else has 1000 chips. What is the chance you come last? ICM says 6!6!/12! = 1/924, or about 0.11%. The other model applicable to a cash game freezeout says 1/37, or about 2.70%. So maybe ICM gives too much weight to the big stacks in atleast some spots. At the very least, its unclear - partly because ICM does not take into account the (payout) structure, and partly because we don't know.

Can you explain how ICM does not account for payout structure?

My understanding is you take the probabilities of each finishing order times the payout for finishing in that position to come up with an expected equity for each player. These numbers certainly change if the payout is 1/0/0 or 1/1/0

Edit:
obviously you state "given some payout structure", etc., but I'm confused by "partly because ICM does not take into account the (payout) structure"

That was a really good post AWice thank you. Didn't understand it all the first time around but I will be coming back to try again a few more times for sure.

Wow am I reading this right, a general chip flow model has big stack finishing in last 25 times more than ICM?

Alex if you and 6 clones of yourself were playing out a SNG with blinds at 50/100 under general MTT payout structures and this chip counts how often do you think the big stack would actually get 7th?

Sorry I did not explain this well enough. I will try to give a 3 player example, but it will work in general.

Suppose you have 3 players with stacks of A, B, and C. ICM predicts six different probabilities, of the events (A,B,C), (A,C,B), (B,A,C), (B,C,A), (C,A,B), (C,B,A), where (A,B,C) means A=1st, B=2nd, C=3rd. Namely, these probabilities are AB/((A+B+C)(B+C)), etc.

Of course, the *equity* of A is going to change as you change the underlying payout structure. But these six probabilities don't change. This is what I was referring to. So it only "takes into account payout structure" in a simple sense.

However, the probabilities in real life ARE different depending on the structure. For example, say the stacks are 5k,1k,1k,1k,1k,1k. Clearly, the chance that A comes last place [(B,C,D,E,F,A) + (B,C,D,F,E,A) + .... , all 120 combinations] is dramatically different in a DON than in a WTA. But ICM still predicts this chance of coming last to be the same.


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I forgot to mention, of course ICM has some other problems, it doesn't take into account where the blinds are, or that shorter stacks have a mathematical advantage over bigger ones, or inflection points where eg. 8bb has more edge than 11bb, etc. I kind of left those aside but those are very real problems, but I think many are already familiar with this. For example say you are playing a DON on bubble where its 300/600/60, the stacks look like this: everyone 2.5k, the guy about to be big blind has 300, you are to his left with 100. ICM will screw up really bad.

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I mean, in that carefully constructed case. When stacks aren't as extreme, the situation is different. Also, the chip flow model I gave as an example (one person at random gives another random person 1 chip) is generally applicable to a cEV structure. Of course in tournaments the chip flow is different and generally favors the big stacks for obvious reasons. You can't say that the arbitrary flow example that I gave is somehow the right one for tournaments, in fact its pretty much guaranteed to be not. ICM is still very useful overall.


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The important takeaway is that it isn't very known how accurate ICM is. (Yes there was this paper, but I don't think it shows that much)

huh?
can you explain this a little better for a dumb guy like me?
i dont see the relation between the $EV number (stacksize related, we just asume we have more chips than we have (skill edge)) and the equity vs his range
and what would be a proper $EV number to take? $EV does not really scale alot when we have tons of chips

Alex, great posts, thank you. Can you expand on the influence of the payout structure on the probabilities of the different finishing distributions? If I understood correctly, you’re saying that the chip flow (transition) function depends on the payout structure. I’m guessing this is because people will play differently depending on the payout structure (which seems logical to me), but I don’t understand why this necessarily has an influence on the resulting finish distributions when people have equal skill and thus adapt their playing style to the payout structure in the same way. For example, the bolded in the examples below is not clear to me (it’s very possible that I’m missing something completely obvious). I kinda get it intuitively in (extreme) satellite bubble situations (A has 10000000000 chips, all the rest has 100; actual chance of bubbling is ZERO and not some calculated number ~zero), but I don’t see this effect when comparing different regular MTT payout structures.

I’m not sure I really understand your question, but I just meant that, if ICM gives you a certain $EV (X) for a given stack size distribution at the table, you just use a (slightly) bigger number than X when calculating required equity or fold equity or stuff like that, if you think the $EV that ICM gives you is too low. Of course, estimating this higher $EV is always a bit of guesswork.

thats what i meant
whats the point in doing a calc if you just guess the most important variable?
how do you know how big your edge is/ how big the difference in $EV is compared to the std $EV number you get with the std approach?

As I understand it:

- ICM estimates equity through a formula which is considered to approximate actual results better than a simple linear model (chance of winning = % of remaining chips)

- there is nothing which proves that the formula represents actual equity

- there is no embedded theoretical basis for justifying the relative weight which ICM gives to stack sizes, i.e. nothing which claims to explain why an x times larger stack has y times the equity. (I can't see how anyone can know the theoretical relative value without solving poker first.)

- essentially it is an arbitrary (bit strong I know, but just to make the point) stab which turns out to be better than a straight line.

- the model makes no attempt to explicitly account for factors which we know to be relevant such as skill differences, who is next in blinds, effect of prize structure on value of stack size differences

- the model deviates from empirical results (but not as much as a straight line does)

- empirical evidence suggests ICM tends to underestimate the value of large stacks (as betgo suggests)

Yeah indeed it becomes fuzzy math at that point; I just wanted to say that you can adapt the model if you think the output is too conservative. Also, it's not completely guessing because you start from the calculated ICM $EV value: when your ICM $EV is $4k it's probably not realistic to use $8k; $4.2k or something like that could then be a decent improvement over ICM.

so just make the $EV calc and say "ffs i get it in ingame anyway"
to me this is about the same as giving us some random? higher $EV value

Also, icm calculations are based on the current hand without looking at positional advantage/disavantage, future implictions for different type of stack distributions and possible implicit collusions or factors that hinder someones range in the future (for example 'hitting the blinds' as a medium to shortstack or going for chipleader to relatively clear 2nd etc). All this and above will lead you to agree that ICM is a relatively good model but yes experience and ur brain will lead you to adjust to it at some lvls, its all pretty tricky to master tho haha

There is an ICM effect, but if everyone plays correctly by ICM even with the same skill level, then the big stack has a lot more value than the ICM formula calculates. It seems like no one has been able to do a simulation or whatever to find equilibrium stack size values. Possible to look at stack sizes and average actual dollar finishes, but would need to get stack size data and on tournaments where chops were not allowed.

It's fine to say it is the best there is and you can adjust the values. However, the using the values leads to ridiculously nitty results. Like people here will say r/f AKo when if you one the hand, you would have the tournament lead. There are MTTSNG situations where Wiz says call KK+ or whatever. This just reinforces the tendencies of some nitty regs. People try plugging things into some software and they think they are getting correct results.

at the end of the day its all about winning the flips

or they do everything on feel and have even more big holes in their game

No, using mathematical calculations that are way off is not a better approach than using good judgement.

ok i will take this wrong statement in my mind and put it far away in the back of the corner so I will never find it ever again.

There was a thread a little while ago where people were saying to r/f AKo to a overshove from the chip leader with 12 players left. Is it better to go by feel or formula here?

If your judgement gives a better answer than the formula, then you should use your judgement.

ur thought process is just completely wrong. You use models and calculations to get a sense of equity and spots and thus get leaks out of ur game, after this u finetune it with ur brain. Not completely throw models out the window.