ICM From First Principles: How Malmuth-Harville Turns Chips Into Dollars

Chips at a final table are not money — they are lottery tickets with diminishing returns. Here is the exact algorithm that converts a stack into a dollar figure, derived from scratch and worked through with real numbers.

Every tournament player has heard the slogan: "your chips aren't worth their face value." It gets repeated at final tables like scripture, usually right before someone makes a bad call and blames variance. But the slogan is not folklore — it is a theorem. There is an exact, computable algorithm that takes a vector of stacks and a payout structure and returns a dollar figure for each seat. That algorithm is the Independent Chip Model, and the most widely used version of it is Malmuth-Harville.

If you understand ICM as a black box — "the calculator says fold" — you will get the easy spots right and the hard spots wrong. If you understand it as an algorithm, you can reconstruct it at the table, anticipate when it breaks, and know exactly why the chip leader at a final table is poorer than their stack suggests while the short stack is richer than theirs. This article derives the model cleanly, walks a concrete four-handed example with explicit numbers, contrasts Malmuth-Harville with its lesser-known cousin Malmuth-Weitzman, and then shows where static ICM ends and Future Game Simulation begins.

The Core Problem: Chips Are Not Linear in Money

In a cash game, a chip is a dollar. Stack EV and money EV are the same object, which is why cash strategy reduces to maximizing chips won per hand. Tournaments sever that link. You cannot cash out chips; you can only convert your finishing position into a payout. And the payout structure is concave — first pays much less than the proportional value of all the chips in play.

Concretely: in a typical structure, first place might be 50% of the prize pool while holding 100% of the chips at the end. Those 50 percentage points of "missing" equity didn't vanish — they were distributed to everyone else along the way, in proportion to their chances of out-lasting other players. The whole job of ICM is to figure out how to split a fixed prize pool among the remaining players based only on their current chip counts.

The word "Independent" in the name flags the central simplifying assumption: ICM treats finishing probabilities as if they depend only on chip counts, ignoring position, skill, blind level, and future play. That assumption is wrong in detail and useful in aggregate — more on that when we reach FGS.

The Malmuth-Harville Model

Malmuth-Harville rests on a single, clean axiom:

The probability that a given player finishes first is equal to their share of the total chips in play.

If player i holds stack \(s_i\) and the total chips in play is \(T\), then:

\[P(i \text{ finishes 1st}) = \dfrac{s_i}{T}\]

This is the entire engine. Everything else is bookkeeping.

The elegant part is how it handles lower finishing positions. Once you've fixed who finishes first, that player and their chips are removed from consideration, and you ask the same question of the remaining field with the remaining chips. The probability that player j finishes second is the probability that someone else finishes first, times the probability that j "wins" the sub-tournament among the survivors.

Formally, the probability that j finishes second is a sum over every possible first-place finisher k (with k ≠ j):

\[P(j \text{ is 2nd}) = \sum_{k \neq j} P(k \text{ is 1st}) \cdot \dfrac{s_j}{T - s_k}\]

Read that carefully. Conditional on k finishing first, k's chips leave the pool, the new total is \(T - s_k\), and within that reduced field player j finishes "first" (i.e. second overall) with probability \(s_j / (T - s_k)\). Sum over every way the first-place slot could be filled and you have j's exact second-place probability.

Third place recurses one level deeper: sum over all ordered pairs of (1st, 2nd) finishers, remove both stacks, and compute j's share of the remaining-remaining chips. In general, you enumerate the finish orders, weight each order by its Malmuth-Harville probability, and accumulate. For n players left there are n! orderings — trivial for a final table of nine (362,880 orders, milliseconds of compute), which is why ICM at a real final table is exact, not approximate.

Your Dollar Equity

Once you have the full finish distribution — for each player, the probability of finishing 1st, 2nd, 3rd, … — converting to money is a dot product. Let \(\text{pay}[r]\) be the payout for finishing in position r. Then:

\[\text{EV}(i) = \sum_r P(i \text{ finishes in position } r) \cdot \text{pay}[r]\]

That single line is the headline result of the model: your tournament equity in dollars is the sum, over every finishing position, of the probability you land there times what that position pays. ICM is nothing more than a principled way to compute those probabilities.

A Concrete Four-Left Example

Theory absorbs better through numbers. Four players remain. Stacks and a prize pool of $10,000 paid 50 / 30 / 15 / 5:

| Player | Stack | Chip share | Payout for that finish | |---|---|---|---| | A | 5,000 | 50% | 1st = $5,000 | | B | 3,000 | 30% | 2nd = $3,000 | | C | 1,500 | 15% | 3rd = $1,500 | | D | 500 | 5% | 4th = $500 | | Total | 10,000 | 100% | Pool = $10,000 |

Note the deliberate trap baked into this setup: the payouts (50/30/15/5) exactly mirror the chip shares (50/30/15/5). If chips were linear in money, every player's $EV would equal their chip-share dollars: A = $5,000, B = $3,000, C = $1,500, D = $500. ICM will show that none of those hold.

Step 1 — P(1st) is just chip share

Straight from the axiom, and it sums to 1 exactly:

(0.5000 + 0.3000 + 0.1500 + 0.0500 = 1.0000.) ✓

Step 2 — One worked branch of P(2nd)

Let's compute P(D finishes 2nd) by hand, summing over who could finish first:

Sum: 0.05000 + 0.02143 + 0.00882 = 0.08025. So D, holding 5% of the chips, finishes second about 8% of the time. The same recursion applied to every player (here computed by full enumeration of all 24 orderings) gives the complete finish distribution below. The 1st-place column is exact by axiom; the 2nd/3rd/4th columns are the correct Malmuth-Harville recursion, rounded:

| Player | P(1st) | P(2nd) | P(3rd) | P(4th) | |---|---|---|---|---| | A | 0.5000 | 0.3288 | 0.1456 | 0.0255 | | B | 0.3000 | 0.3687 | 0.2613 | 0.0699 | | C | 0.1500 | 0.2222 | 0.4197 | 0.2081 | | D | 0.0500 | 0.0803 | 0.1733 | 0.6965 |

Each row sums to 1.0, and each column sums to 1.0 — both are sanity checks the model must pass. Notice how the short stack D is overwhelmingly likely to bust first (69.65%) but is no longer guaranteed to: survival is probabilistic all the way down.

Step 3 — Convert to dollars

Dot each player's finish distribution with the payout vector [5000, 3000, 1500, 500]:

| Player | Chip share | "Chip-equity" $ (linear) | ICM $EV | Delta vs chips | |---|---|---|---|---| | A | 50% | $5,000 | $3,717.74 | −$1,282.26 | | B | 30% | $3,000 | $3,033.17 | +$33.17 | | C | 15% | $1,500 | $2,150.19 | +$650.19 | | D | 5% | $500 | $1,098.89 | +$598.89 | | Total | 100% | $10,000 | $10,000.00 | 0 |

The dollars sum back to the full $10,000 prize pool — equity is conserved, never created or destroyed. That is the crux of the entire model, sitting in plain numbers:

This is not a quirk of these particular stacks; it is structural. The concavity of the payout ladder plus the fact that even a 1-chip stack is guaranteed some finish position means short stacks are systematically over-valued in dollars relative to chips, and big stacks systematically under-valued. Everyone is "pulled toward the middle" of the pay scale.

Why This Changes How You Play

The practical consequence lives in one phrase: risk premium. Because your last chips lost are worth more (in $) than your next chips won, the breakeven equity for a stack-off rises above the naive chip-EV breakeven. A spot that's a clear chip-EV call can be a clear ICM fold.

Run the leader A's situation through the lens above: A risking chips against a medium stack is wagering dollars at an unfavorable exchange rate — A pays the full marginal-chip cost on losses but collects a discounted marginal-chip value on wins, because doubling up doesn't double A's money (A is already near the top of the curve). The short stack D, by contrast, has a low risk premium against the bigger stacks: D's chips are cheap to risk because D's downside is small and well-compensated by laddering equity. This is the mathematical backbone of "big stacks should bully, but not against the other big stack" and "ICM pressure crushes the medium stacks hardest" — the medium stacks have the most ladder equity to lose and the least to gain.

You don't have to eyeball this. Plug the stacks and payouts into shadepoker's ICM Calculator, read off each seat's $EV, and the risk premium for any given confrontation falls out of comparing the before/after dollar equity of winning versus busting. That before-vs-after dollar swing — not the chip swing — is the number your river decision should be priced against.

Malmuth-Harville vs Malmuth-Weitzman

Malmuth-Harville is the default in essentially every commercial ICM tool, but it is not the only finish-order model. Malmuth-Weitzman answers the same question — given chip counts, what is the finish distribution — with a different conditioning rule.

The distinction is in how lower places are derived. Harville builds the distribution forward: fix who finishes first (probability = chip share), remove them, recurse on the survivors. Weitzman instead reasons from the bottom up — modeling the probability of finishing last as inversely related to chip stack, then recursing upward through the elimination order. The two models agree exactly on P(1st) and on the two-player case, but they assign slightly different probabilities to the middle finishing positions in larger fields, and therefore slightly different dollar equities.

Which is "right"? Neither is empirically perfect — both are simplifications of a real elimination process that depends on blinds, position, and skill. Harville won the popularity contest because its forward recursion is intuitive, fast, and matches observed tournament data acceptably well. Weitzman tends to be marginally more pessimistic for big stacks and marginally kinder to short stacks than Harville. The disagreement is real but usually small relative to the modeling error both share against reality. For practical purposes: know that "ICM" almost always means Malmuth-Harville, know that Weitzman exists as a principled alternative, and don't agonize over the second-decimal differences between them — they are dwarfed by the assumptions both make.

Beyond Static ICM: Future Game Simulation

Here is the honest limitation. Static ICM assumes the tournament resolves with no further play — as if the finish distribution crystallizes the instant you freeze the stacks. It embeds two fictions:

  1. No future hands. Blinds don't go up, antes don't drain stacks, nobody open-shoves into anybody. The chips are treated as a fixed lottery.
  2. No skill. Every player is identical. A world-class reg and a recreational player with the same stack get the same equity. Position relative to the button — a massive short-stack edge — is invisible to the model.

These fictions matter most when there's a lot of future play left: deeper stacks, big blinds relative to stacks, and a button that will sweep around several times before anyone busts. Static ICM systematically misprices those spots because it ignores the positional and initiative advantages that accrue over the next orbit.

Future Game Simulation (FGS) is the refinement. Instead of freezing the stacks, FGS simulates the next k hands of play — typically a small lookahead like one to four hands — using a simplified strategy (often a push/fold or solver-derived model) for how the blinds will be contested, and only then applies static ICM to the resulting distribution of stacks. In effect, FGS lets the chips "play forward" a little before cashing them out into dollars, capturing the value of position and the cost of being in the blinds soon.

The payoff: FGS rewards having position on a short stack, penalizes being about to post big blinds with a marginal holding, and generally narrows some of static ICM's harshest folds in spots where you'll get to use your skill and position before anyone is eliminated. The cost is compute and modeling complexity — the simulation is only as good as the strategy you assume for those future hands, and the state space explodes quickly with the lookahead depth, which is why FGS depths are kept shallow. Think of FGS as static ICM plus a short, principled glimpse into the next orbit. For final-table and bubble decisions where future play is thin, static Malmuth-Harville is already excellent; for deeper mid-stage pay-jump spots, FGS meaningfully corrects it.

The Takeaway

ICM is not a vibe. It is a concrete algorithm: P(1st) equals chip share, lower places follow by removing the higher finishers and recursing, and your dollar equity is the finish distribution dotted with the payout ladder. Run it on any four-handed spot and the same structural truth appears every time — the chip leader is worth less than their stack, the short stack worth more, and the medium stacks are the ones the model squeezes hardest.

The players who internalize this don't just "play tight on the bubble." They price every all-in against the dollar swing rather than the chip swing, they know when their risk premium frees them to gamble and when it shackles them, and they understand exactly which assumptions of the model are about to break — which is the moment to lean on FGS or pure judgment instead. That gap, between treating chips as money and knowing the precise exchange rate, is the gap between min-cashing and final-table scoring. Plug a real spot into shadepoker's ICM calculator, read the dollar equities, and start making the conversion second nature.