ICM: why chips aren't dollars at the final table

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.

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The formula

For n players with stacks s₁, s₂, …, sₙ and total chips S = Σsᵢ:

1. First-place probability for player i:
   P(i = 1st) = sᵢ / S

2. Second-place probability for player i:
   P(i = 2nd) = Σⱼ≠ᵢ [ P(j = 1st) × sᵢ / (S − sⱼ) ]

3. Third place and beyond follow the same pattern: remove the first-place winner, re-normalise, and apply the rule again.

ICM equity for player i =
P(i = 1st) × Prize₁ + P(i = 2nd) × Prize₂ + … + P(i = nth) × Prizeₙ

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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.