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 Using ICM To Improve Your Tournament Performance
Using The Independent Chip Model To Improve Your Tournament Performance
The Independent Chip Model (ICM) models the likelihood of players placing 1st, 2nd, 3rd, etc. in a poker tournament, based on the stack sizes of all the remaining players. Craig Howard developed the model and presented it in a post on the 2+2 forum in Feb., 2004. His original post can be found here. Tests have been run comparing ICM's prediction vs. actual tournament results. These tests support that there is a good correlation between ICM's model and actual performance.
Having a working model opens up a lot of opportunities. When a few remaining players decide to chop the tournament prizes rather than play to conclusion, ICM provides a mechanism for fairly distributing the prize money. Consider, for example, the situation of 3 players remaining with stacks of 1000K, 350K, and 150K with prizes of $10K, $6K, and $3K. All the players agree to a split, but how much should each player get? The player with 150K is in 3rd place, but he is certainly not going to accept simply the 3rd prize of $3K. He is guaranteed that much if he plays and loses it all the next hand. He has some chance at 2nd and some chance at 1st so he will require more than just the 3rd prize to be willing to split. ICM provides a model for computing a 'fair' distribution.
The Model Defined
A player's chance of taking 1st is equal to the fraction of the total chips held by that player.
A player's chance of taking 2nd is equal to the sum of the chances of each of the other players taking first times the fraction of the total remaining chips held by that player (remaining chips being the total chips not including the chips held by the other player that is assumed to win.)
3rd and so on are computed similarly.
For example, Consider 3 players remaining (A, B, C) with stack sizes of 1000K, 350K, 150K respectively.
Code:
Player A has a .667 chance of taking first.
1000K / ( 1000K + 350K + 150K) = .667
Player B has a .233 chance of taking first.
350K / ( 1000K + 350K + 150K) = .233
Player C has a .1 chance of taking first.
150K / ( 1000K + 350K + 150K) = .100
Player A has a .277 chance of taking second.
Player B takes first (.233) * fraction of remaining chips [(1000K / (1000K + 150K) = .870] = .203
Player C takes first (.1) * fraction of remaining chips [(1000K / (1000K + 350K) = .741] = .074
.203 + .074 = .277
Player B has a .493 chance of taking second.
Player A takes first (.667) * fraction of remaining chips [(350K / (350K + 150K) = .700] = .467
Player C takes first (.1) * fraction of remaining chips [(350K / (1000K + 350K) = .259] = .026
.467 + .026 = .493
Player C has a .230 chance of taking second.
Player A takes first (.667) * fraction of remaining chips [(150K / (350K + 150K) = .300] = .200
Player B takes first (.233) * fraction of remaining chips [(150K / (1000K + 150K) = .130] = .030
.200 + .030 = .230
Player A has a .056 chance of taking third.
Player B takes first (.233) * Player C takes second if B takes first (.130) * fraction of remaining chips [( 1000K / 1000K ) = 1] = .030
Player C takes first (.1) * Player B takes second if C takes first (.259) * fraction of remaining chips [( 1000K / 1000K ) = 1] = .026
.030 + .026 = .056
Player B has a .274 chance of taking third.
Player A takes first (.667) * Player C takes second if A takes first (.300) * fraction of remaining chips [( 350K / 350K ) = 1] = .200
Player C takes first (.1) * Player A takes second if C takes first (.741) * fraction of remaining chips [( 150K / 150K ) = 1] = .074
.200 + .074 = .274
Player C has a .670 chance of taking third.
Player A takes first (.667) * Player B takes second if A takes first (.700) * fraction of remaining chips [( 150K / 150K ) = 1] = .467
Player B takes first (.233) * Player A takes second if B takes first (.870) * fraction of remaining chips [( 150K / 150K ) = 1] = .203
.467 + .203 = .670
Continuing the example for the purposes of tournament splitting
If the tournament payout was 10K, $6K, $3K then:
Code:
Player A would receive .667 * $10K + .277 * $6K + .056 * $3K = $8.5K
Player B would receive .233 * $10K + .493 * $6K + .274 * $3K = $6.1K
Player C would receive .100 * $10K + .230 * $6K + .670 * $3K = $4.4K
Using ICM to Improve Your Game
ICM can be used to compute whether an end-game play is good or bad. When a player's stack size drops below the 10x'big blind' mark, many advocate a "push or fold" strategy. Put simply, they should either shove all their chips in pre-flop, or they should fold their hand. This puts other players, particularly the players in the blinds, in the position of either folding or calling an All-In. Obviously, poker is a complex game. But, at this point the game has devolved into two scenarios:
1) When no one has entered the pot, should a player move All-In. The people after him may call. How many people are left to act after him, what types of hands those players would call with and how well the player's hand would hold up against these likely calling hands greatly influence this choice.
2) When someone has gone All-In, should a player call. Obviously, the hand will go to showdown if he does. The types of hands the All-In player may have, how the player's hand will perform against such hands and the likelihood of players left to act going All-In certainly affect this decision.
Using the ICM, and estimating the 'tightness-looseness' of the remaining players for (1) or the 'tightness-looseness' of the All-In player and the remaining players for (2), one can compare the value of making a play vs. not making a play. Thus, plays can be determined to be likely to increase your tournament winnings or likely to decrease your tounament winnings.
An Example of Calling a Raise
Three people remain in a tournament with a $5, $3, $2 payout. You are in the big blind. The blinds are 250/500 and, with the blinds already paid, the stack sizes are:
BB (you): 3500
SB: 1750
Button: 2000
The pot is 750, you are dealt a pair of 9s.
The button pushes All-In and the SB folds, should you call or fold?
If you fold, the stack sizes will be:
BB (you): 3500
SB: 1750
Button: 2750
Using ICM, your chance of 1st, 2nd, 3rd is .4375, .3517, .2108 respectively, giving you a tournament equity of $3.66.
If you call and win, the stack sizes will be:
BB (you): 6250
SB: 1750
Using ICM, your chance of 1st, 2nd is .7813, .2187 respectively, giving you a tournament equity of $4.56.
If you call and lose, the stack sizes will be:
BB (you): 2000
SB: 1750
Button: 4250
Using ICM, your chance of 1st, 2nd, 3rd is .25, .3533, .3967 respectively, giving you a tournament equity of $3.10.
Suppose you feel the button is tight and estimate that he would go in on TT+, AQs+, AKo. Firing up PokerStove, we see that a pair of 9s has a 37.260% equity against those hands. So, if you call your resultant tournament equity will average out to:
$4.56 * .3726 + $3.10 * .6274 = $3.64
If you fold, your equity is $3.66; so this move is just slightly negative.
Suppose you feel the button is maniac aggressive and will push with any two cards. PokerStove tells us that a pair of 9s has a 72.057% equity against any random hand. So, if you call your resultant tournament equity will average out to:
$4.56 * .72057 + $3.10 * .27943 = $4.15
Again, if you fold, your equity is $3.66; so this is an enormously positive call to make against such an opponent.
But Who Can Do All That Math That Fast In A Tournament?
No one. But what you can do is do the math ahead of time to get a feel for when to push and when to call. There are tools to save you all the tedious math, an example being SnG Power Tools. By running examples before hand, you can get a feel for what the right course of action is based on the stack sizes, the blinds and the table texture.
For example, stack sizes are even at 10xBB, you are on the button. Both the SB and the BB are tight and will only call an All-In with TT+, AQs, AKo. You should push with *ANY* two cards. The extra value you get from the tournaments where this steal is successful plus the value you get from the times you are called and still win exceed the tournament value lost from the times you are called and lose. But, take the same situation and assume both the SB and BB are very loose, calling with 22+, A2s+, A3o+, KTs+, KJo+, QJs. Then the push hands are 55+, A9o+, A8s+. Consider the same loose players in the blinds but now the blinds have increased such that the stack sizes are 4xBB. Now the push hands are 22+, A2+, KTo+, K6s+, QTo+, Q8s+, JTo, J7s+, T9o, T7s+, 97s+, 86s+, 76s.
This Seems Too Easy
ICM is a simplified model and it does not take certain things into account. As a poker player, you will need to learn when to apply the model and when not to apply it. ICM obviously does not take player skill into account. It also does not make special allowance for unusual situations. Consider the example of a tournament with the top 3 places paid out and 4 people remain. You are in the BB with a healthy stack. The player to your left has 1/6 of a BB remaining. He folds and the maniac on the button, who has you covered, pushes All-In. You have a pair of 6s. It is probably best to just let this go. The player to your left will almost certainly be blinded out in the next two hands, assuring you 3rd place. ICM would not show much of a difference in tournament equity if the short stack to the left had 1/6 BB or 1 BB. But, the fact that the blinds are about to hit him and he has to win both of the next two hands to stay in the match is something that ICM does not model. Consider you are in the SB and you have observed that the BB always blows All-In if it is limped to him. You are dealt AA. ICM would tell you to push. But observation tells you that limping to him and then calling his All-In raise is a better move. ICM provides general guidance, you still need to play poker.
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Originally Posted by MAX
