ICM on a Final Table: Working Out Chip Equity by Hand

The independent chip model (ICM) converts tournament chip stacks into a dollar value by estimating each player's chance of finishing in each paid place and multiplying by the prizes. It needs only one rule, applied over and over: a player's chance of finishing first among the players still in contention equals their share of those players' chips. With three players, the whole calculation fits on a sheet of paper, and working it once shows why a big stack's chips are worth less per chip than a short stack's.

What the model assumes, and where it comes from

The method is usually called the Malmuth-Harville model. David Harville published it for horse racing in 1973, in "Assigning Probabilities to the Outcomes of Multi-Entry Competitions" in the Journal of the American Statistical Association (vol. 68, no. 342), as a way to get probabilities for second and third place from win probabilities alone. Mason Malmuth applied the same structure to poker tournament settlements in the essay "Settling Up in Tournaments" in Gambling Theory and Other Topics (Two Plus Two Publishing).

The model makes three assumptions:

1. Chance of first place equals chip share. A player holding 30 percent of the chips in play has a 30 percent chance of winning the tournament.
2. Lower places follow the same rule after removing the winner. Given that a particular player finishes first, each remaining player's chance of finishing second equals their share of the chips held by the remaining players. Third place follows the same way, and so on.
3. Nothing else matters. Skill, seat position, the size of the blinds, who has the button and how many hands are left before the next level are all outside the model.

The second assumption is the part that does the work, and it is a modeling choice rather than a mathematical fact. It treats the remaining field as if the eliminated winner's chips had simply vanished, so the others' relative shares are unchanged.

A three-player calculation in eight steps

Inputs for the worked example:

• Player A: 5,000 chips
• Player B: 3,000 chips
• Player C: 2,000 chips
• Total: 10,000 chips
• Prizes: $500 for first, $300 for second, $200 for third ($1,000 in total)

The procedure:

1. Write each player's chip share. A = 5,000 / 10,000 = 0.50; B = 0.30; C = 0.20.
2. Set first-place probabilities equal to those shares. P(A first) = 0.50, P(B first) = 0.30, P(C first) = 0.20.
3. For each possible winner, re-share the remaining chips. If A wins, B and C hold 5,000 chips between them: B's share is 3,000 / 5,000 = 0.60 and C's is 2,000 / 5,000 = 0.40. If B wins, A and C hold 7,000: A 5/7, C 2/7. If C wins, A and B hold 8,000: A 5/8, B 3/8.
4. Compute each player's chance of finishing second by summing over who wins:

• P(A second) = P(B first) × 5/7 + P(C first) × 5/8 = 0.30 × 0.7143 + 0.20 × 0.625 = 0.2143 + 0.1250 = 0.3393
• P(B second) = P(A first) × 0.60 + P(C first) × 3/8 = 0.30 + 0.075 = 0.3750
• P(C second) = P(A first) × 0.40 + P(B first) × 2/7 = 0.20 + 0.0857 = 0.2857

1. Get third place by subtraction. With three players, each one finishes somewhere, so P(third) = 1 − P(first) − P(second). A: 1 − 0.50 − 0.3393 = 0.1607. B: 1 − 0.30 − 0.375 = 0.3250. C: 1 − 0.20 − 0.2857 = 0.5143.
2. Check the columns. Each place must be taken by exactly one player, so each column must sum to 1: second place 0.3393 + 0.3750 + 0.2857 = 1.0000, third place 0.1607 + 0.3250 + 0.5143 = 1.0000.
3. Multiply by the prizes. Equity = P(first) × $500 + P(second) × $300 + P(third) × $200.
4. Check the total. The equities must add up to the prize pool still to be paid, $1,000.

For four or more players the same steps apply, but the number of finishing orders grows quickly, which is why calculators exist. The calculator is doing step 4 for every possible order.

The worked result

The equity lines in full:

• A: 0.50 × $500 + 0.3393 × $300 + 0.1607 × $200 = $250.00 + $101.79 + $32.14 = $383.93
• B: 0.30 × $500 + 0.375 × $300 + 0.325 × $200 = $150.00 + $112.50 + $65.00 = $327.50
• C: 0.20 × $500 + 0.2857 × $300 + 0.5143 × $200 = $100.00 + $85.71 + $102.86 = $288.57

The last column is what each player would get if chips were worth a fixed amount each, $0.10 per chip here. The model says A's 5,000 chips are worth $383.93, not $500, and C's 2,000 chips are worth $288.57, not $200. Every player is guaranteed $200 for third place, so the $1,000 prize pool is really $600 of guaranteed money plus $400 that the chips are competing for. A big stack cannot win more than first prize, and a short stack keeps a floor no matter how small it gets. That squeeze at both ends is what makes chip value non-linear.

Calling an all-in when places pay differently

The non-linearity changes decisions. Take the same table and ignore blinds and antes, so only the all-in itself moves chips. Player B moves all in for 3,000 and Player A, who covers, decides whether to call.

If A folds, the stacks stay as they were and A's equity is $383.93.

If A calls and wins, B finishes third and collects $200. A has 8,000 and C has 2,000. Heads-up, A's chance of first is 0.80 and of second 0.20, so A's equity is 0.80 × $500 + 0.20 × $300 = $460.00.

If A calls and loses, the stacks are A 2,000, B 6,000, C 2,000. Running the same procedure: P(A first) = 0.20; P(A second) = 0.60 × 2/4 + 0.20 × 2/8 = 0.30 + 0.05 = 0.35; P(A third) = 0.45. A's equity is 0.20 × $500 + 0.35 × $300 + 0.45 × $200 = $100 + $105 + $90 = $295.00.

A's call breaks even in dollars when p × $460.00 + (1 − p) × $295.00 = $383.93. Solving: p = ($383.93 − $295.00) / ($460.00 − $295.00) = $88.93 / $165.00 = 0.539. A needs to win the hand about 53.9 percent of the time to call. In chips alone, risking 3,000 to win 3,000 would need exactly 50 percent.

The gap grows when the payout steps are steeper. Change the prizes to $650 for first, $350 for second and nothing for third, a true bubble with three left and two paid. A's fold equity becomes 0.50 × $650 + 0.3393 × $350 = $443.75. Winning the call leaves A heads-up with 80 percent of the chips: 0.80 × $650 + 0.20 × $350 = $590.00. Losing leaves A with 2,000 chips: 0.20 × $650 + 0.35 × $350 = $252.50. The break-even becomes ($443.75 − $252.50) / ($590.00 − $252.50) = $191.25 / $337.50 = 0.567, or 56.7 percent. The same chips and the same opponent now demand a hand that is clearly ahead, because losing costs more dollars than winning gains.

The asymmetry runs the other way for the shover. B risks finishing third but gains chips that are worth less per chip once B is the leader. The model does not tell anyone whether to shove or call; it tells them what price in hand equity a call or a shove has to beat.

What the model leaves out

Skill. The first-place assumption treats every player as equally likely to win each chip. A strong player's chips are worth more than the model says, a weak player's less. George T. Gilbert's paper "The Independent Chip Model and Risk Aversion" (arXiv, 2009) describes ICM as a model for tournaments between players of roughly equal abilities, and notes that it did not arise from any model of how chips actually move from player to player.

Blinds and position. A player who is about to pay the big blind with a short stack is in a worse position than the same stack on the button. The model gives both the same number.

Future play. The model is a snapshot of the current stacks. It does not account for how the next few orbits are likely to go, how stacks will change as blinds rise, or how aggressive each opponent is.

Large stacks. The same paper reports a test of the model against World Poker Tour final-table data in which it modestly underestimated how often large stacks went on to win, a result it says could come from flaws in the model or from differences in skill.

Even with those limits, the arithmetic is the reason the model is used. It turns "how much is this stack worth" into a calculation anyone can check with the eight steps above and a column total.