For instance, let's say I am in place #10 in a tournament. There are 120 players left, already in the money. Place #10 pays $100, place #1 pays $1500, and #120 pays $20 dollars.

Now, I am in a situation where I am facing a big bet to call I need to go all in: if I call and win, I will be propelled to the place 1° and if I lose, I will bust out of the tournament. Now, this means that if I lose, I move from place #10 ($100), to place #120 ($20). SO it is a loss of $80. And If i win, I move from place #10 ($100) to place #1 ($1500), for a gain of $1400.

$1400 > $80

How is this not a situation where the chips won are of greater value than that of chip lost (contradicting what icm dictates, namely that in a tournament chips won are of lesser value than chips lost)?

You should not calculate with ICM with 120 players left! Its not like you will get 1st place price if you win that all in! Just use normal chipEV until you are down to finale 2 tables ish

Winning the hand might put you in 1st place, but it doesn't guarantee you'll win $1500 (there's a long way still to go). You'll just improve your chances. Losing the hand does guarantee you'll only get paid $20.

Just imagine it's the first hand of a tourney. If you double or even triple up, it doesn't mean you've booked the win, does it? You've still got to outlast everyone to get the top prize.

EDIT: Here's a basic example of how ICM works.

Situation: 4 players are ITM, with $50 for first place. The chip leader has four times the stack of the guy in second place and two thirds of all the chips in play, and the guy in fourth has a relatively tiny stack, and yet the leader's stack is only valued at $42.52. He can't lock up the $50 for first until he's busted everyone. The shorty, by contrast, is looking very likely to only get $10 for fourth place, but at this point in the game, the ICM value of his stack is $14.48. In short, the chip leader's stack is worth less than first place money, because he's not guaranteed to win the tournament, and the shorty's stack is worth more than 4th place money, because he's not guaranteed to be the next player to bust.


Calculation link: http://www.icmpoker.com/icmcalculator/#cqMY

I doubt there are ICM calculators for 120 players. Some day would be. Now play by chip equity. Never can get better price for chip than chip equity unless the price structure is reversed (worse place is paid higher) which does not happen.

I think your main problem might be with this actual statement as for icm it is not correct!

Some may say "chips won are of lesser value than chips lost" but this is only true in some instances. Icm does not create or destroy any wealth so if by gaining or losing chips someone gets an extra $ boost then someone or sometimes many are losing exactly this same $ boost amount. So if there is a player where "chips won are of lesser value than chips lost" then at the table others have the opposite chips won are of greater value. The total $value available is always the total prize money available, this doesn't change if you use an icm value for chips.

If you are in first place gaining more chips may not benefit you to the equivalent of the chipev amount, but if you are in last place icm may ascribe to you an amout above chip ev value for them.

Icm is just a model of chip value at some point in a tournament but is is quite complicated and takes some effort to learn.
In your post you are implying a chip value model where everybody gets the exact payout for their current position and this is quite often wrong and generally not too much use as a model ... although ...
this model could be pretty useful if this was the very last hand of a timed tournament, Hero is in 10th and by winning the hand comes first, losing comes 120th, folding comes 120th. After this hand everyone gets paid out according to current position. Using this model we could calculate how tight we should call.

Winning = $1500
Folding = $100
Losing = $20

So from this we could work out what hand equity you need to call (using your model) to see what is at least break even compared to the no risk fold option. (Rather than equity I'll just use win probability and ignore any blinds to make it read easier.)

To break even:
(w*1500) + ((1.0 - w) * 20) = the return if you take the bet.
Here w = win prob. and so (1.0-w) equals lose prob.

To break even you need to get back at least $100 you would have if you folded so:
(w*1500) + ((1.0 - w) * 20) = 100
1500w + 20 - 20w = 100
1480w = 80
w = 80/1480 = 5.4% (phew!, if I calculated this correctly you don't need much hand equity.)

So if you call with any hand that has 5.4% chance of winning you make money on this final hand. Your model is useful for this one strange situation.
If this was just a chipev situation you would only call with any hand that had a 50% win probability (nb, 50% if no blinds in pot). Folding hands where you have 20% 30% 40% is losing you lots of value here.

Icm works in an equivalent way, you get a decent estimation of how much stacks are worth and then plug these into an EV calc to come out with a break even hand range.

Sometimes people say "icm is too tight, you should always just go for the win" but I think they are wrong. Icm estimates stack size value well and sometimes it will actually tell you to push or call like a lunatic compared to the chipev, it only seems tight in some contexts and then nearly always correctly so. The context is how the stacks are arranged. If you are the chip leader at a very big bubble with a safe micro stack present you can push like a hungry monkey after bananas, but if you a middle stack with the chip leaders to follow you have to be very careful and might have to fold decent looking hands. This can make sense after you study icm and situations.

Icm is a little complicated but if you plug good quality ranges into icm tools they should generate good ranges. If you put crap ranges into the tools they come out with rubbish.

(Don't get confused by always using 'icm' tools set to icm. Early on icm is very small and chipev is close enough. To use icm you have to set up lots of information about other tables and all payouts - this would be better but very hard work when chipev gets about the same result. On final tables always use icm. You have to make adjustments for close to FT situations.)