ICM: chips and prize equity

The Independent Chip Model, or ICM, assigns model prize equity to a tournament stack using the remaining stacks and the ordered prizes. Chip share and prize share can differ because finishing second or third may still pay. The result is an average under a specified ranking model, not an amount a player is guaranteed to receive.

This lesson uses exactly three remaining players so every hypothetical finish order can be inspected. You enter the stacks and prizes; the calculator enumerates all six orders and shows each player's finishing probabilities and prize equity. It supplies no card equities, skill estimates, hand recommendations or predictions about an actual event.

Ace: "The tallest stack has half the chips. Does it get half the prizes?"

Celeste: "We need the prize schedule first. Half the chips is only one part of the calculation."

Keep chip share and prize equity distinct

Here, chip share means a player's stack divided by all chips in the three-player snapshot. It is sometimes described as chip equity, but it is not the expected chip gain from a proposed action. With stacks of 50, 30 and 20, players A, B and C have 50%, 30% and 20% of the chips. Those shares depend only on the stack distribution.

Model prize equity is a different quantity. For each finishing place, multiply the player's model probability of that place by its prize, then add the products. A player's share of the prize pool is that equity divided by the total prizes. The prize schedule affects these values even when every stack stays exactly the same.

The calculator uses fictional prize units. It does not calculate profit, subtract an entry fee, account for previous entries, or value travel and other costs. A model expectation of 70 prize units does not mean a net gain of 70. To discuss net return, you would need a clearly defined cost record and a consistent starting point.

State the ranking assumption

At the first selection step, a player's model chance of finishing first equals that player's fraction of the total chips. After selecting a hypothetical winner, the model removes that player's stack from the selection pool. It chooses second from the remaining players in proportion to their remaining stack amounts. The last unselected player takes third.

This is a mathematical procedure for assigning ranks. It is not a literal elimination history or a claim that the winner's chips disappear during real poker play. There are no hands, blinds or wagers inside the procedure. It uses the same stack-based selection rule at each step so the finish-order probabilities have a precise definition.

A player-specific skill advantage receives no adjustment here. Neither does position, the next blind, table dynamics or a future opportunity to win chips. Equal inputs are treated symmetrically because the model has no information that distinguishes the players beyond their stacks. That is a model limitation, not evidence that the players have equal ability.

Follow one of the six orders

Use stacks A = 50, B = 30 and C = 20. The order A first, B second, C third has probability 50/100 multiplied by 30/50, or 0.30. The second denominator is 50 because A's 50-chip stack has already left the hypothetical selection pool. C is the only remaining player after those two selections.

The order A first, C second, B third has probability 50/100 multiplied by 20/50, or 0.20. Those two orders together account for A's 0.50 first-place probability. Their different second-place outcomes matter when the prizes for second and third differ. The model keeps both paths rather than choosing one as the predicted result.

There are also two orders beginning with B and two beginning with C. B then A then C has probability about 0.214286; B then C then A has about 0.085714. C then A then B has probability 0.125; C then B then A has 0.075. Together with the two A-first orders, all six probabilities sum to one before display rounding.

Build each player's finishing distribution

A finishes second in two paths: B wins first and A is selected next, or C wins first and A is selected next. The calculation is 0.30 x 50/70 + 0.20 x 50/80, about 0.339286. A's first-place chance is 0.50, leaving about 0.160714 for third place.

B's first, second and third probabilities are 0.30, 0.375 and 0.325. C's are 0.20, about 0.285714 and about 0.514286. A player's three probabilities sum to one because that player occupies one finishing place in every order. They are not independent events that can all occur in the same finish.

Each place column also sums to one. Every complete order has exactly one winner, one runner-up and one third-place finisher. These row and column totals are useful checks before any prizes are applied. The interface prints rounded values, so adding the displayed percentages can produce a small rounding difference.

Ace: "I picked a hypothetical winner. Do I keep using all 100 chips to choose second?"

Celeste: "Use only the stacks still in this selection pool. Then weight that branch by the chance of reaching it."

Apply prizes to the probabilities

For the default example, first receives 100 units, second receives 60 and third receives zero. The total prize pool is 160. A's model prize equity is 0.50 x 100 + about 0.339286 x 60 + about 0.160714 x 0. The result is about 70.357143 units.

B's model prize equity is 0.30 x 100 + 0.375 x 60 + 0.325 x 0, or 52.5 units. C's is about 37.142857 units. Their equities sum to the 160-unit prize pool because every possible order distributes the same three prizes exactly once. The model does not create an additional prize by averaging the outcomes.

A's 70.357143 units are about 43.973214% of the pool, compared with a 50% chip share. B has a 32.8125% prize-pool share and C about 23.214286%. The larger stack still has the largest model prize equity. Its fraction of the pool is smaller than its chip fraction because the other stacks retain chances at the lower paid place.

Inspect the three-player calculator

Start with the default stacks and prizes. Open the six-order explanation inside the result and reproduce one path by hand. Then compare chip share, place probabilities and prize equity for the same player. The calculator is deterministic: it enumerates every order in this three-player model rather than sampling simulated tournaments.

All three stacks must be positive and finite. The interface accepts plain numbers up to one million, with at most two decimal places, so you can scale a chip distribution for study. Fractional worksheet units are not a claim that a particular event uses fractional physical chips. Blank, negative, exponent and nonnumeric entries are rejected.

Enter exactly three prizes in finishing order. Equal prizes and unpaid places are allowed, but a later place cannot pay more than an earlier one. Use zero deliberately for an unpaid place. The stack labels A, B and C identify players; the prize labels identify places. First-place prize does not belong to player A merely because it is entered first.

Use winner-take-all as a check

Select Winner take all. The stacks remain 50, 30 and 20 while the prizes become 100, 0 and 0. Only the first-place probability has prize value, so the model equities are 50, 30 and 20 units. Each player's prize-pool share now equals that player's chip share under the stated first-place assumption.

This equality follows from the payout vector. It does not show that chip share and prize share are interchangeable under every schedule. Restoring a paid second place changes the valuation while preserving the same stack-based finish-order probabilities. A calculation that kept the original prize shares after that change would be using the wrong schedule.

Equal prizes expose another boundary

Select Equal prizes to pay 40 units for every place. Each remaining player then has model prize equity of 40, regardless of stack size. The total pool is 120, and each player has one third of its value. Every path pays that player the same 40, so the rank probabilities no longer change the player's expected prize.

The all-zero vector is also accepted as a mathematical boundary. It gives zero prize equity to everyone, while the chip-based finishing probabilities remain defined. A normalized share of a zero prize pool is undefined, so the interface says so. This is a worksheet consistency check, not a proposed payout structure for a real event.

Scale the chips without changing the model

Multiplying every stack by the same positive factor leaves every selection ratio unchanged. Stacks 50, 30, 20 and stacks 500, 300, 200 therefore produce the same finishing probabilities and prize equities when prizes stay fixed. Doubling only one player's stack is different because it changes the distribution and the total chips.

Multiplying every prize by two doubles each model prize equity. It does not change the finishing probabilities, and it does not change normalized prize-pool shares when the pool is positive. These tests separate the stack-based ranking calculation from the payout calculation. They also help catch accidental mixing of chip and prize units.

Compare snapshots without inventing a hand

Keep prizes at 100, 60, 0 and compare A's stack in three snapshots. In the baseline 50, 30, 20 distribution, A has about 70.357143 prize units. If 10 chips move from B to A, the distribution becomes 60, 20, 20 and A's model equity becomes 78. If 10 move from A to B, the distribution becomes 40, 40, 20 and A's model equity becomes 62.

The gain is about 7.642857 prize units, while the loss is about 8.357143. Both stack changes involve 10 chips, but the prize-equity changes have different magnitudes under this payout schedule. Those are valuations of specified snapshots. No probability has been assigned to reaching either snapshot, and no poker action has been priced.

To evaluate a hypothetical action, a separate model would have to supply its possible resulting stack distributions, their probabilities and all relevant alternatives. This calculator supplies none of those inputs. The difference between two ICM values cannot by itself tell you whether to call, fold, raise or accept a deal.

Ace: "I changed the prize envelopes, but left all three stacks alone. Why did the values move?"

Celeste: "The finish probabilities stayed the same. You changed what each finish pays."

Keep the remaining roster complete

The model requires exactly three remaining players. Leaving out a short stack changes the selection pool and every player's finishing distribution. This page does not approximate a larger field by ignoring everyone except three selected players. If more or fewer players remain, use a calculator that explicitly supports that roster.

A zero-stack player is outside this positive-stack snapshot model. Do not replace zero with a tiny invented stack just to pass input validation. Update the remaining roster and its applicable prizes in a tool that supports the resulting player count. Simultaneous eliminations, tied places and unusual payout rules require the event's actual rules and additional handling.

All prizes must use the same unit and describe the remaining places consistently. If third place is paid, entering zero changes the question being answered. If you omit a guaranteed payout from every place, the equities describe only the remaining prize differences, not the entire prize amount. Name that adjustment in the study note.

Understand what the model leaves out

The calculation has no blind schedule, seat positions, player skills, future betting, card holdings or opponent ranges. It does not model re-entry, bounty payments, progressive knockout accounts or special noncash awards. Those features need their own explicit treatment. A precise stack-only calculation cannot supply information that was never entered.

Floating-point arithmetic and six-decimal display rounding remain even though no tournament simulation is used. Very small positive probabilities can display as 0%. Internal values are not replaced by those rounded labels. Reproduce calculations from the original inputs rather than feeding rounded output percentages into another calculation and expecting identical totals.

An ICM number is not an accepted deal or an offer to exchange chips. This worksheet performs no transaction. Its output is the value assigned by the stated mathematical model.

Read the result as a model statement

A useful study record names all three stacks, all three ordered prizes and the stack-proportional ranking assumption. Then it reports a model average, such as about 70.357143 prize units for A under the default inputs. Keep the word model attached to the number so a computed expectation does not become a claim about an actual finish.

Does the chip leader's first-place probability predict the winner?

It is the model's stack-proportional first-place assignment. It is not a verified forecast for the players in a particular tournament. The calculation has no skill or future-game inputs with which to adjust it. A player can finish in any of the three places represented by the model, including when that player's current stack is the largest.

Why show six orders if I only need prize equity?

The six orders make the calculation inspectable. Each order has a probability and assigns every prize once. Adding those contributions produces the place probabilities and expected prizes. Seeing the paths helps distinguish a complete deterministic enumeration from an unexplained number, while leaving the model's real-world limits visible.

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