In a cash game a chip is a dollar — win one, you are a dollar richer, full stop. In a tournament that is a lie. You cannot cash chips; you cash finishing positions, and the prizes are fixed and top-heavy. The Independent Chip Model converts your stack into the only thing that actually pays: your expected share of the prize pool. Learn to see that number and you stop making bets that win chips but lose money.

From chips to money

ICM rests on one assumption, the Malmuth-Harville rule: your chance of finishing first equals your share of the chips in play. Take the winner out, re-normalise the stacks that remain, and the same rule gives second place — then third, and so on. Multiply each prize by your probability of finishing there, add it up, and you have your prize-pool equity: the slice of the money your stack is worth right now.

That sounds tidy, but it hides the result that matters:

Tournament equity is concave. Doubling your stack does not double your money.

Your first chips are your most valuable — they buy survival and a min-cash. Chips piled on top of an already-big stack are worth progressively less, because there is only so much prize pool to win. The direct consequence:

The chips you lose are worth more than the chips you win. Risking your stack to win an equal number of chips is a losing trade in real money, even at even odds.
A short stack is worth more than its chip share (the survival premium up to the money), and a big stack is worth less than its chip share.

That gap is "ICM pressure," and it is strongest right before a pay jump — the bubble, and every rung of the final-table ladder.

Where people go wrong

Treating chips like money. The cardinal error. A flip for your tournament life can be wildly positive in chips and badly negative in dollars.
Looking only at your own stack. Your equity depends on everyone's stacks and the payout structure, not just yours. The identical stack is worth different money at different tables.
Folding the pressure away. ICM cuts both ways. It makes a big stack's bluffs cheap — opponents cannot call off near a pay jump — so applying pressure is the same coin as avoiding gambles. The big stack should be busier, not quieter.
Mistaking the model for the truth. ICM ignores blinds, position, and skill. A two-big-blind stack is not really worth its ICM figure with the big blind about to swallow it, and a clear skill edge is worth real equity the model cannot see. Use ICM as the baseline you adjust from, not gospel.

A rule of thumb that travels: the chips you lose cost more than the chips you win — so near a pay jump, fold more, and with a big stack, bluff more.

How the ICM puzzle trains this

The ICM puzzle lays out every player's chip stack and the payout structure, then asks for one seat's share of the prize pool. You estimate the percentage and submit; we grade it against the exact Malmuth-Harville figure inside a tolerance band that tightens as the spots get trickier. The seats that fool people most — the big stack worth less than its chips, the short stack worth more — are exactly the hard ones, and your rating tracks how well you feel that gap. A few reps a day and you start to see the money sitting behind a wall of chips.

Go deeper

ICM is really just equity measured in dollars instead of chips: the same "what is my share?" question, asked of the prize pool rather than the pot. And the place ICM bites hardest — a short stack staring down a pay jump — is exactly where the push-or-fold endgame lives.