The ICM is not all its cracked up to be

[dethgrind]

Sorry for being a jerk about it.

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I've been curious if anyone has done an analysis on how accurate the ICM is using tournament data. Do you know of any such investigations?


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No, I haven't seen anything like this, though I'd be very interested as well.

[zephyr]

This thread got a little out of control from my perspective. I think I may have delved a touch too deep in some of my criticism/comments, however, I just want to reiterate again that the point of my original post was to:

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I'm not trying to critize the ICM as a method of analysis, but am just pointing out the fact that it is only a rough estimate.

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I agree that the ICM is the best thing we have right now. Since I've done a fair amount of criticism on it I think that I need to do something to add to this area. With regards to ideas for improvement, I think that the biggest gains would come from doing an experimental investigation of CEV -> $EV, and then compare it to the various analytical means that we have of going from chips to $. I'm beginning that experiment now, but as I'm especially busy this time of the year, I'll welcome any data that people would like to send me.

In my field of study, aerospace engineering, if we consider any type of real world fluid flow the analysis cannot be done analytically. The problem arises in the non-linear Navier Stokes equations which to this day have no general solution. Thus, most aerodynamic problems are solved using experimental correlations, with some simpler ones now being solved using computational fluid dynamics. My point is that when problems cannot be solved purely mathematically, it's often a very good idea to attack them from an experimental point of view.

In terms of my analysis I'm thinking of this format:

The instant that the fifth place finisher goes out (down to 4):

Record stack sizes,
Record blind sizes,
Record current position,

Then record what place you come in.
I also think that the players approximate ROI should be considered.

Of course a much more indepth analysis could be done, but I think this will serve fine to begin with. The analysis could probably be done immediately as I'm sure there is no shortage of tournament histories floating around.

Any thoughts?

Only my opinion,

Zephyr

[dethgrind]

I really like this idea.

It'll be necessary to define a way to lump together similar stack size scenarios. Otherwise the amount of data you'd need to collect would be too huge. I mean, how many times have you run across this exact situation: blinds 100/200, you're on the button, stacks are 1250/575/3060/3115?

This will require some good scripting skills as well.

Good luck, and let me know how I can help

-Sean

[JNash]

Hi Zephyr & Dethgrind

I just came across the ICM concept for the first time and started to research old posts. I came across the debate between the two of you on whether in heads up play the probability of winning equals the fraction of total chips a player has.

I happen to agree with Zephyr that the S&M proof is unconvincing. In fact, the entire theorem may be wrong--although I can't disprove it myself either. Here goes...

One of the central assumptions is that the two players are equally skilled--an assumption I want to keep. However, if you follow the proof, you'll see that it uses the argument that a player with 1/4 the total chips needs to double up twice to win (so far so good). I agree that WITH EQUAL STACKS, the probability of two equal-skilled players winning must be 50/50. However, it is not clear to me that a player with 1/4 the total chips has a 50/50 chance of doubling up to where he has 1/2 the total chips--even if the two players are equally "skilled". For this to be true, it would require that the probability of doubling up is independent of the relative stack sizes of the two players.

Now we can simply assume that this is so, in which case the S&M proof goes through without any problem. However, my personal intuition and experience is that the big stack enjoys an advantage simply because he has more chips (even if he is playing against his identical, equally smart twin). My heuristic argument for why this is so is that the big stack can (playing optimally) bluff and steal the blinds more than the small stack. This leads to what I once posted under the title"S-Curve Hypothesis"--that the probability of wining in heads-up play as a function of fraction of chips looks like a logistic-type function which goes through 50%/50% (i.e. the probabilities must be equal when the stack sizes are equal and you have equal skill). The curve must be a reflection around this mid-point since the game is zero-sum.

Mine is not a theorem--it's a conjecture which I can't prove. I do know however that to disprove it requires more work than the S&M argument. The random walk references that dethgring gave also don't apply since they assume the probabilities in the Brownian motion are constant. If you assume that the probabilities change depending on where you are along your random walk, the proof would no longer work.

One possible way of tackling this problem would be to prove that the optimal strategy in heads up play is independent of stack size. If that is the case, then my intuition is wrong and the "conventional wisdom" on the freezout calculation is indeed correct.

I'd very much appreciate your thoughts on this...or references to materials that may shed more light on this.

P.S.: Can you give me the definitive reference to the theory and assumptions that underlie the ICM? Thanks!

[pzhon]

The independent chip model is much more than the assumption about what happens heads-up. It is more than an assumption about the probability of winning a multiplayer freezeout. The ICM is an assumption about how frequently each player will finish in each position, and while it is not perfect, it is a good start.

It doesn't look like you are objecting to the ICM's more subtle projections for finishing 3rd, for example. It looks like you are objecting to far more fundamental issues.

I have argued repeatedly that it is a theoretical advantage to have a small stack in a multiplayer game. You get more information and you get to use the big stacks' fear of each other. Small stacks should gain chips on average. So, the assumption that the probability of winning in a multiplayer NL game is proportionate to stack size is wrong. However, I don't think it is so wrong as to make the ICM unusable.

As for the probability of winning heads-up, the only real question should be the effect of position. If you are playing a game like stud where position is determined by a card, then there is a strategy for each table limit which at least breaks even against any opposition. Following this strategy (paying attention to the size of the smaller stack, not who has it) means you will not lose chips on average. Since at the end of the game, you have all of the chips or none, not losing chips on average means you will win with probability at least as great as the fraction of the chips you hold. This doesn't work precisely for Hold'em because of position, but unless the blinds are huge, it shouldn't make a big difference.

It looks like you are arguing that having more chips heads-up means you can bluff people and steal more. For heads-up play, this may be true in practice (betting 10k chips may look impressive even if your opponent only has to call 2k), but it is wrong in theory. Perhaps you have noticed the way people deviate from theoretically correct play and you exploit it, but that doesn't mean a big stack should have a disproportionate advantage against a good opponent.

[JNash]

Thanks phzon

To clarify my point, I wasn't commenting on the ICM. I just wanted to express my view that the conventional freezeout calculation for heads-up play (which, as dethgrip said is embraced by many very smart people) may in fact not be exactly correct. The proof depends on the assumption that the probability of doubling up is independent of the relative stack sizes--which to me is far from obvious.

In your reply, you express the belief that small stack size is actually an advantage--I am sure you meant relative to the value you would get by simple chip proportionality, not as an absolute statement. If your statement is true, then you'd get a different shaped S-curve. If f(c) is the probability of winning as a function of fraction c of total chips, your assumption about short stack advantage would imply that the function is concave from 0 to 0.5, and convex from 0.5 to 1. My theory of large stack advantage says it's convex from 0 to 0.5 and concave from 0.5 to 1. The traditional S&M theory says that it's precisely linear.

The linear approximation used by S&M is undoubtedly the best first-order approximation, and I do not have any better formula to propose myself, but I just wanted to challenge the statement that "it has been proven" that the proportionality theorem is correct. I am unable to prove mathematically that my "large stack advantage" conjecture is correct, and your argument for small stack advantage is also not rigorous and based only on qualitative reasoning. [Perhaps there are elements of truth in both of our views which approximately cancel each other out, so that the linear approximation is actually very close to correct.] Please note that I am not concerned with whether actual players behave one way or the other, I want to know what the game-theoretically correct answer is.

[pzhon]

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In your reply, you express the belief that small stack size is actually an advantage

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The context of that statement was multiplayer play, not heads-up.

In that post I also outlined a proof that chip value is linear heads-up for games without a predetermined position.

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I am unable to prove mathematically that my "large stack advantage" conjecture is correct, and your argument for small stack advantage is also not rigorous and based only on qualitative reasoning.

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My arguments for heads-up play are rigorous. What is nonrigorous about my arguments for the small stacks' advantage in multiplayer play is whether there is essentially collusion against the small stacks that somehow exceeds the small stacks' advantages. It is clear that the small stacks benefit when a big stack folds after calling a small stack's push. See this post from June for a numerical example of how a small stack may benefit from proper actions of big stacks.

Do you honestly believe the large stack should play differently heads-up than the small stack? If so, can you give an example?

[zephyr]

I thought that this thread had long since perished, but I guess I'm not quite correct. It is interesting to see this point being discussed further, though.

As pzhon mentioned I think that the key is that the blinds, and position aren't considered when concluding that %chips = chance of finishing first.

In poker the blinds are a part of the game itself, and thus cannot be ignored when finding solutions to that game. Likewise, we cannot ignore say the river, and come up with an exact solution that applies to the a game that has a river. With no blinds, there is no game.

Hence, any solution that we come up with where we assume that the blinds and position are negligible, does not actually apply to the game. Of course, the effect of the blinds and position may be very small and thus the solution we find may very closely approximate the game. On the other hand though, the effects of the blinds may be large, and thus our solution a very bad approximation.

Intuitively, I believe that the actual relationship between %chance of finishing first and stack size is some type of a curve that passes through 0.5, 0.5. I'm unsure of the shape of the curve, however. It may be linear, it may not.

Only my opinion,

Zephyr

[dethgrind]

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The random walk references that dethgring gave also don't apply since they assume the probabilities in the Brownian motion are constant. If you assume that the probabilities change depending on where you are along your random walk, the proof would no longer work.


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You're right. If the probabilities are stack size dependent, everything goes to hell. Even Bozeman and Thomas Ferguson's fancy pants diffusion stuff won't work. (That seems to be the post where Bozeman started calling it the ICM).

I first read about the method in GTaOT, "Settling Up in Tournaments: Part III".

If you want to read some more about this topic, I recommend checking out the RGP archives on google. Look for stuff by Tom Weideman, Barbara Yoon, and William Chen.

[JNash]

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If you are playing a game like stud where position is determined by a card, then there is a strategy for each table limit which at least breaks even against any opposition. Following this strategy (paying attention to the size of the smaller stack, not who has it) means you will not lose chips on average. Since at the end of the game, you have all of the chips or none, not losing chips on average means you will win with probability at least as great as the fraction of the chips you hold.

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If what you say here is true, then the optimal NL heads-up strategy is independent of the relative stack sizes, an important part of the proof.

Now, I would be very interested in the proof of the above statement--and in knowing what the optimal heads-up strategy is. Any references for this?

I have two reasons for believing that the probability of coming in first depends not just on the fraction of chips:
1)It seems to me that the strategic options of the small stack become severely limited when his stack is small relative to the blinds. I would think that he would have to play a wider range of hands (i.e play looser) than the big stack since he is so close to being blinded out. So the optimal strategy would seem to depend on stack size relative to the size of the blinds. (This argument is independent of position, since you'd get to be both the big and small blind about the same number of times over the remaining hands. Unless you are literally going to be blinded out in the next 1-2 hands, in which case position would matter too.)

2) As you said, in holdem the blinds do enter the picture, and I believe cannot be ignored. If you are heads-up with 20% of the chip total (opp has 80%), and the blinds are 1%, I think the small stack has a better shot than when the blinds are 10%. I think this can probably be proven with an element-of-ruin argument. This would suggest that even if the optimal strategy is independent of stack size, the probability of coming in first depends not just on the fraction of chips but also on the size of the blinds.

Please understand, I don't have a proof, just a hunch, and I am posting to learn more...