The Independent Chip Model is a model that output's a player's overall equity in a tournament, given the stack sizes of players at the table.

An assumption underlying the calculations in the model is that the probability of a player coming 1st is equal to the proportion of chips the player has at the table.

Taking heads up as a simple example, a player with 1000 out of 9000 chips has a 1/9 chance of placing 1st, while a player with 8000/9000 chips has a 8/9 chance.

This linear relationship between chip proportion and probability of winning seems reasonable and simple, however is there a theoretical basis for this assumption?

For example, could it be a non-linear relationship, where someone with 8/9 of the chips may have a greater than 8/9 chance of winning (and someone with only 1/9 of the chips may have a lower chance of winning than 1/9)?

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    Those are your exact odds of winning if every play went all-in every hand regardless of their cards, which seems like a reasonable assumption when eliminating skill as a factor. – Paul Dec 29 '15 at 23:00
  • Thanks Paulpro, this was my first thought, having tested it for simple heads up, but I wasn't sure how it applied with multiple opponents. – Kenshin Dec 30 '15 at 10:18

There is a theoretical basis behind this assumption. Since the odds of a person winning a tournament is the inverse of the odds of everyone else losing, gambler's ruin comes into play. Intuitively, if you have 100% of the chips you'll win 100% of the time; if you have 0% of the chips you'll win 0% of the time; if you have 50% of the chips against an even opponent, you'd expect to win 50% of the time and this suggests a linear relationship in chip value (though it's not a proof in itself).

I've seen (but can't find it now) where Monte Carlo simulations have been done in the following way: make up some theoretical stack sizes, start taking random amounts of chips from any random stack and add it to any other random stack--continue doing this until all but one stack goes to 0. The results are right in line with what ICM would suggest with the linear model.

In real life situations, it's a bit of a stretch to assume that everyone is exactly equally skilled. The model also doesn't take into consideration factors such as chips to blind ratio (a small stack with 2 big blinds plays out a lot differently than one with 50 big blinds), the blinds and antes themselves, or strategic shifts that may come about from the pay structure -- for example a small stack under the gun in a satellite with one remaining elimination might be thinking differently than an equal sized stack who sits on the button. Some of those things and more are addressed in this paper, but generally speaking, it works out easier for ICM models to be able to use this simple linear relationship with very basic assumptions.

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  • I understood your first paragraph, but I was imagining a sigmoid curve which could also fit. Thanks, the monte carlo idea is exactly the kind of thing I was looking for. It would be great if I could find this or perhaps I could try it myself. – Kenshin Dec 30 '15 at 10:16
  • This isn't the simulation that I had in mind, but refer to zbicyclist's answer here for one example. – Dr.DrfbagIII Dec 30 '15 at 14:48

Yes there is theoretical basis. A user with 1000 chips would have to lose exactly twice as many chips (and be eliminated from the tournament) as a player with 500 chips.

It could be a lot of relationships. Luck and skill have lot to do with the final outcome. But linear is how many chips you would have to lose and that is what eliminates you from the tournament.

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  • But chips are usually won and lost in an exponential fashion rather than linear. E.g. If i'm small stack on 400, I'll probably double to 800 then to 1600. Rather than go from 400 to 600 to 800 etc. – Kenshin Dec 29 '15 at 12:02
  • Usually won and lost in an exponential fashion - small stack is a very small fraction of the chips moved. So you would base the model on small stack? Base the whole model on the small stack and small posts? Small stack is as likely to go to zero than double. If not called they don't double up. Small stack faces fixed sized blinds. The winner is never a small stack. – paparazzo Dec 29 '15 at 15:55
  • it may well be linear it might not be, but your post isn't enough to conclude money is lost and won in a linear fashion at the poker table. I'm looking for more rigor behind the assumption, such as the Monte Carlo simulations Dr Drfbagll suggests exist. – Kenshin Dec 30 '15 at 10:14
  • What ever simulation you want to run but if you base it on small stacks it is not going to give you a good answer. And I sill hold with it takes twice as long to lose twice as many chips is solid theoretical basis. – paparazzo Dec 30 '15 at 10:20

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