ICM - what it is and what it does
By request, I'ma try to lay down a more in depth explanation of ICM and what it could mean for your poker play. As a disclaimer, I've never used ICM to analyze my game, I'm not really working on the math of my game at this point - I don't feel the stakes I'm playing at are skillful enough to merit too much effort there (playing 10+1). That said, math rules, bow to its might.
ICM stands for Independant Chip Model...don't worry about the name, not worth it. First we need to look at what the model is trying to...model. It's one of many attempts to try to equate tournament chips to cash. Like the rest of them, it's not perfect...there's way too many factors involved to get an exact answer to what your stack is worth - player skill, style matchup, despiration by very short stacks, tilt...stuff like that is just too arbitrary and complicated to put into a model. So...assuming equal skill, enough skill that straight style isn't much of an effect, noone's very short stacked (like 3xBB) or tilting, ICM does pretty well at equating tourney stacks to cash.
The idea behind the model is that every chip is a ticket to a lottery. To figure out who gets first place, you pick a ticket at random and that person wins. Take their tickets out and draw again for 2nd, and repeat that for 3rd, ect 'til your out of the money. Now, we all know that's now how poker works, but remember, this is a model...a guess. Let's look at an example.
3 players, stack sizes: 5k 4k 1k
Each person's chance of winning 1st is their stack/10k, their chance of getting picked in the first lottery. To figure out who'd come in second given who comes in 1st, their chance is stack/(sum of remaining stacks). A whole bunch of multiplication later, we end up with our estimated equities.
1/2 - chance of 5k winning
2/5 - chance of 4k winning
1/10 - chance of 1k winning
If 5k wins:
4/5 - chance of 4k in 2nd
1/5 - chance of 1k in 2nd
If 4k wins:
5/6 - chance of 5k in 2nd
1/6 - chance of 1k in 2nd
If 1k wins:
5/9 - chance of 5k in 2nd
4/9 - chance of 4k in 2nd
Let's go with a $100 pot, to make things easy on me, with standard payout of 50/30/20
So what we got is:
Equity of the 5k stack:
(Chance of 1st) + (Equity if 4k wins) + (Equity if 1k wins)
(1/2 * 50) + 2/5 * (5/6 * 30 + 1/6 * 20) + 1/10 * (5/9 * 30 + 4/9 * 20)
(1/2 * 50) + 2/5 * (25 + 10/3) + 1/10 * (50/3 + 80/9)
25 + 2/5 * 28 1/3 + 1/10 * 25 5/9
25 + 11 1/3 + 2 5/9
Equity of the 4k stack:
(Chance of 1st) + (Equity if 5k wins) + (Equity if 1k wins)
(2/5 * 50) + .5 * (4/5 * 30 + 1/5 * 20) + 1/10 * (4/9 * 30 + 5/9 * 20)
(2/5 * 50) + .5 * (24 + 4) + 1/10 * (13 1/3 + 11 1/9)
20 + .5 * 28 + 1/10 * 24 4/9
20 + 14 + 2 4/9
Equity of the 1k stack:
(Chance of 1st) + (Equity if 5k wins) + (Equity if 4k wins)
(1/10 * 50) + 1/2 * (1/5 * 30 + 4/5 * 20) + 2/5 * (1/6 * 30 + 5/6 * 20)
(1/10 * 50) + 1/2 * (6 + 16) + 2/5 * (5 + 16 2/3)
5 + 1/2 * 22 + 2/5 * 21 2/3
5 + 11 + 8 2/3
Which makes some sense - everyone's got at least $20 equity 'cause they're ITM now, and the 5k is slightly better off then the 4k, and both are way better off then 1k.
For a good excersise that'll point out an interesting aspect of this model, figure out the stack equities for 5k 2.5k 2.5k (easier, since the 2.5ks will be the same (please don't ask why, if you don't know why I don't know how you got this far)). After that you'll have the right to laugh at VQ for oversimplifying the model. (a reference to this post)
So when does this become useful? Well, like usual, when trying to figure out the $EV of a move. If you can roughly determine your % chance of winning a hand, you can figure out your stack equity if you win and if you lose, weight those by the % chance to win, and figure out if that's better then if you don't make that action (this is most useful when considering pushes 'cause future action doesn't happen). This can result in some non-intuitive results, especially on the button.