Exploring how marginal chip value changes with stack size

[Galwegian]

I don't belive that ICM coincides with the brownian motion model although I have not checked this myself. ICM is a model based on the following assumptions (let t_i = number of chips that player i has and assume sum_i t_i = 1)

A) Prob(player i will win tourney) = t_i
B) Prob(player j will finish in (k+1)th position given that players i_1,...i_k have finished 1..k) = t_i/(1-t_i_1-...t_i_k)

Of course A is just a special case of B. However while there is a convincing argument that A is a good model of reality, I have not seen such an argument (myself) for B

[thylacine]

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What is ICM?


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Shuffle the players' chips. Rank the players by their highest chips. Equivalently, remove the chips from the table one at a time, eliminating a player when his last chip is removed.

Independent Chip Model calculators:
http://www.chillin411.com/icmcalc.php
http://sharnett.bol.ucla.edu/ICM/ICM.html (not as convenient, but it has some explanations)

You can write the formulas for n players by summing over the possible first place finishers, removing their chips, and applying the ICM to the reduced tournament on n-1 players. This doesn't seem to simplify. Does anyone know what the probabilities are of finishing last in a tournament where the stack sizes are 1, 2, ..., 100?

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Is it random walk (brownian motion) on a simplex with absorbing boundaries? I saw Tom Ferguson wrote an exact solution for n=3 players. What else is known?

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The Brownian motion model is different. It's simple to analyze with 3 players because conformal transformations in two dimensions are understood and preserve Brownian paths, and the Riemann maps from the triangle to the disc are even known explicitly. Conformal transformations in higher dimensions are much more rigid, and there isn't much hope to extend the solution to more players without adding ideas.

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LOL! Funny thought! ICM corresponds to (a certain discretized version of) Brownian motion on a simplex with absorbing walls, with the added feature that the simplex is constantly shrinking towards its centroid! THE WALLS ARE CLOSING IN! AAAAAAAAAARGGH!!!

So there is a continuum of models between ICM and Brownian motion on a simplex! There is no reason not to unify the models.

(Now don't someone go selling my ideas to some poker site for fifty squillion dollars [img]/images/graemlins/mad.gif[/img] )

[Galwegian]

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LOL! Funny thought! ICM corresponds to (a certain discretized version of) Brownian motion on a simplex with absorbing walls, with the added feature that the simplex is constantly shrinking towards its centroid! THE WALLS ARE CLOSING IN! AAAAAAAAAARGGH!!!

So there is a continuum of models between ICM and Brownian motion on a simplex! There is no reason not to unify the models.

(Now don't someone go selling my ideas to some poker site for fifty squillion dollars [img]/images/graemlins/mad.gif[/img] )

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Have you shown that ICM is equivalent to discrete Brownian motion on a simplex?

[Galwegian]

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The Brownian motion model is different. It's simple to analyze with 3 players because conformal transformations in two dimensions are understood and preserve Brownian paths, and the Riemann maps from the triangle to the disc are even known explicitly. Conformal transformations in higher dimensions are much more rigid, and there isn't much hope to extend the solution to more players without adding ideas.

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I have a question for you pzhon, as you seem to have studied Ferguson's paper. Does the Brownian motion model give significantly different numbers to ICM in the 3 person case?

[pzhon]

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I have a question for you pzhon, as you seem to have studied Ferguson's paper.

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Actually, I haven't, but the argument is natural for someone who has studied that part of mathematics.

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Does the Brownian motion model give significantly different numbers to ICM in the 3 person case?

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A good place to look would be Bozeman's posts here which may have coined the term Independent Chip Model, as he did some comparisons of the models even with 4 players. You can't see the graphs now, though. I can't recall seeing a large difference between the models, but the predicted probabilities were different.

http://archiveserver.twoplustwo.com/...?Number=519924

If no one has copies, I'll try to illustrate the difference between the two models later, but probably only for 3 players.

[pzhon]

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ICM corresponds to (a certain discretized version of) Brownian motion on a simplex with absorbing walls, with the added feature that the simplex is constantly shrinking towards its centroid!


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That's a good point, but there are quite a few extra good features of the ICM. You can use the ICM to obtain the probabilities of other finishes relatively easily. The probabilities can be determined by the analysis of the discrete model, not just as a limit. (You can define it by the limit, but all of the intermediate terms are equal, which is remarkable.)

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(Now don't someone go selling my ideas to some poker site for fifty squillion dollars [img]/images/graemlins/mad.gif[/img] )

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After the US legislation, they were only willing to pay 15 squillion. Sigh.

[trojanrabbit]

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Does the Brownian motion model give significantly different numbers to ICM in the 3 person case?

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This table shows different stacks for 3 players and shows the predicted finish distribution of player A in the diffusion model and ICM.

For the most part they are pretty close, but it looks like ICM is slightly optimistic about short stacks. The diffusion model is probably a more "realistic" model, but its complexity makes it essentially unusable for more than 3 players. ICM does a good approximation and is a lot easier to calculate.

Tysen

[WRX]

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THE WALLS ARE CLOSING IN! AAAAAAAAAARGGH!!!

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Hey, I understood that part.

Having seen the discussion of this issue, which is undoubtedly fascinating in the abstract, and having seen the comparison of some results of the diffusion model and the ICM, I have some questions. In what sense can one model or the other be said to be a "better" representation of a poker game? Are we able to draw any conclusions as to what model most closely mimics the results of play between players of equal skill, in a no-limit hold'em tournament with an escalating blind structure?

There are several obvious respects in which "coin-flip" models fail to correspond to how a poker tournament is decided:

(1) They assume no skill differential.

(2) They disregard the variety of ways in which chips can be won or lost--e.g., bets of various sizes, two-way pots, and multi-way pots.

(3) They disregard the blind structure.

Pikachu proposed some other interesting models here. It strikes me that each of these (with the exception of the decay model, which seems completely unrealistic) has something to teach us. It strikes me that there are many random decision models that can give us insights, but that none of them corresponds very closely to how a poker tournament is actually decided.

So what model is "best," and why?

[Red_Diamond]

Hey guys, maybe someone can figure out a little problem of mine. In Harrington's WB problem #39 he goes a bit into eq and figuring out places of finish.

Now I've coded an ICM and the data for third and fourth place does NOT match up with his values, though his columns do add up to one.

Maybe someone can take a look at his Prob. of Finish table and tell me where he is getting his numbers from. Some approximation method? I never could make sense out of how he tallied up those last 2 columns, the explanation seems a bit too vague.

[pzhon]

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Start with just the target player's stack. Add the other players' stacks one by one, maintaining a joint distribution of the location of the player's highest chip and the player's rank. At the end, ignore the extra information about the location of the player's highest chip, i.e., sum over the possible locations.

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Something was been bugging me about this method. While some descriptions of the ICM allow non-integer stack sizes with no alteration, this one does not, and the complexity blows up when the stacks are large. It also obscures the fact that increasing all stack sizes in proportion will preserve the place probabilities.

I thought there should be a continuous limit, with density functions replacing the probability vectors and integrals replacing the sums. However, the correct limit as the stack sizes are increased in proportion is still discrete! The expected location of the second place player's highest chip is bounded as the stacks increase, and the limit is a geometrically decreasing infinite series of probabilities.

As the stack sizes increase proportionately to infinity, the joint distribution of the rank and the location of the highest chip stabilizes. For example, after inserting one stack, the probability that the player is second and has his highest chip in location n>1 is p^(n-1)(1-p), where p is the proportion of the chips owned by the first player. It still might not be obvious that the probabilities do not change when the stacks are all doubled, but the limiting distributions are clearly the same, and the insertion method can be used to compute the ICM probabilities for irrational or very large stacks.