this forum has two reasons. 1: to teach players who don’t know what ICM is what it is and how to use it. 2: two questions i don’t know myself about it i ask to the other players that do know to teach me.
for the people who don’t know what ICM is. it means independant chip model, and it is a formula that proves that lost chips are worth more then won chips by changing the chip value to money value. it is used to know how much equity you need to make your play profitable. you don’t have to actually make this formula after every hand you play (players won’t gonna like you if you do 
) but understanding it will help you to make more good decisions.
here an easy example:
you are in a 6max sng with a buy in of 50. the 1st place gets 70% (210) and the 2nd place gets 30% (90)
there are three players left and player A has 500 chips. player B has 300 chips. player C has 200 chips
the first thing you have to do is calculating the first place equity.
the only thing you need to do is this: (player chips/total chips=winning odds) * (1st place prize)
so player A has 500 of the 1000 chips in game so 500/1000= 0,5. the first prize is 210. so
210 * 0,5=105. this means player A has 105 1st place equity. (0,5 also means 50% chance to win)
player B has 300/1000. so it’s 0,3 * 210= 63 (0,3 also means 30% chance to win)
player C has 200/1000. so it’s 0,2 * 210= 42 (0,2 also means 20% chance to win)
the calculation of the second place is much more difficult to do, but still very doable if you get how.
this is what you have to do:
(((player A second = ((B winning odds * percent of A’s chips to (total chips - B’s chips) * 2nd place prize))+((C winning odds * percent of A’s chips to (total chips - C’s chips) * 2nd place prize)) ))).
same formula counts with the other second places.
player A second place equity: player B has 30% winning odds. A has 500/700 chips, which = 0,714285714285714285 etc. so i just call it 0,714. the 2nd prize is 90. so when ptting it all together you get 30% (is 0,3) * 0,714 = 0,2142 (i’l just say 0,214) * 90= 19,26…player C has 20% winning odds. A has 500/800 chips, which = 0,625. the 2nd prize is 90. so when putting it all together you get 20% (is 0,2) * 0,625 = 0,125 * 90 = 11,25. now put the 2 together, 19,26 + 11,25 = 30,51. this means player A has 30,51 second place equity.
player B second place equity: A has 50%. 300/500 = 0,6. so it’s 0,5 * 0,6 = 0,3 * 90 = 27… C has 20%. 30/800= 0,375. so it’s 0,2 * 0,375 = 0,075 * 90 = 6,75. so 27 + 6,75 = 33,75
player C second place equity: A has 50%. 200/500 = 0,4. so it’s 0,5 * 0,4 = 0,2 * 90 = 18… B has 30%. 200/700= 0,2857142857142857142 etc. (so i’ll just say 0,286). so it’s 0,3 * 0,286 = 0.0858 (i’ll just say 0,086) * 90 = 7,74. so 18 + 7,74 = 25,74
now to put it al together:
player A has 5000 chips. this means he has 105 + 30,51 = 135,51.
so he has 5000 chips but it’s the same as 135,51 money equity
player B has 3000 chips. so 63 + 33,75 = 96,75.
so he has 3000 chips but it’s the same as 96,75 money equity
player C has 2000 chips. so 42 + 25,74 = 67,74
so he has 2000 chips but it’s the same as 67,74 money equity.
now that everything is clear i’ll give an easy ICM example (no blind values to keep it easy):
assume you are player C. player A pushes all-in and player B folds. now it’s your turn. if you only look at chip value, you only need more then 50% equity to get +EV. so let’s just say your and your opponents hand have an 55/45 chance. this means normally calling is +EV. but now look at it again with ICM involved.
- when you call and lose, you lose 67,74 equity
- when you call and win we get a new amount of chips
player C (you) 4000. player A 3000. player B 3000.
so C has 0,4 * 210 1st place equity which is 84
so A and B has 0,3 * 210 1st place equity which is 63
now the second place again:
C second place equity: A has 0,3. 400/700 = 0,571428571428571428 etc. (or 0,571). so it’s 0,3 * 0,571 = 0,1711 (or 0,171) * 90 = 15,39. since B has the exact same stack as A we can just double it. so 15,39 + 15,39 = 30,78.
A and B had the exact same stack so this is the second place equity for both: C has 0,4. 300/600 = 0,5. so 0,4 * 0,5 = 0,2 * 90 = 18… AorB has 0,3. 300/700 = 0,428571428571428571 etc. (or 0,429). so it’s 0,3 * 0,429 = 0,1287 (or 0,129) * 90 = 11,61. so A and B has 18 + 11,61 = 29,61.
so when putting it all together:
C (you) get 4000 chips which is 84 + 30,78 = 114,78
A and B have 3000 chips which is 63 + 29,61 = 92,61
this means that with the call of the all in. you would lose 2000 chips if you lost and won 2000 if you won. which is exactly the same. but the looking at the ICM. you know now that losing 2000 is a real loss of 67,74. and winning 2000 is a real win of 47,04. so this means your 55% chance hand is actually -EV. so +chip EV may sometimes also be real -EV.
i hope this formula will help 
…
now that the explanation is done, i will ask my own questions:
since i also have a few things i don’t know about ICM, perhaps people that got this already can help me with my questions.
1: as you can see my formula is assuming there are 2 people ITM. the thing i like to know is what formula you have to apply when using 3 or more in the prize pool.
2: even while this is a big formula i used and it took me a few hours to do the math and post it all. this is actually one of the easiest examples possible. there are things like more players, difficult chip stacks (like 27319, 81624, 66392 etc.), blinds, position, more prize pools and perhaps many more stuff. so for example: things like A 2843, B 9235, C 11382, D 1980, E 25683, F 8855, G 4583, H 3399 with 5 ITM spots. this would by using this calculation probably weeks to months to find out. so my question is: are there some algorithms to make it easier to count difficult ICM? (not specificly for the hardest stuff possible but an useful algorithm for any of those things would probably help already).