Week 10: Tournament equity and the Independent Chip Model

Why a chip isn't worth a dollar once elimination is on the table, made precise with the standard recursive ICM calculation and checked against the naive chip-proportional model it's often confused with

26 April 2027

A different kind of pot#

Every EV calculation so far has priced a single hand in dollars won or lost at that showdown. A tournament changes what’s being priced: chips have no cash value on their own — only finishing position pays, according to a fixed payout structure. Two players with equal stacks and different payout structures ahead of them can have very different dollar equity, even though their chip counts say they’re identical.

The naive model, and why it’s tempting#

A natural first guess: a player’s share of the total prize pool should equal their share of the total chips in play. It’s simple, and it’s what a lot of players do in their heads at a final table. This week checks it against a real model rather than assuming it.

The Independent Chip Model, defined#

The standard model (Malmuth–Harville) makes one assumption: the probability a specific player finishes best among whoever’s left equals their share of the chips remaining in play. Formally, for a set of players still in the tournament:

P(player i finishes 1st) = stack_i / (sum of all remaining stacks)

Places beyond 1st are computed the same way, recursively: P(player i finishes in the next-best remaining place) is the probability someone else takes the place ahead of them, times the probability i finishes best among whoever’s left after that. This is a real, well-defined recursive calculation — not a heuristic — and it’s what the rest of this page computes directly rather than quoting a solver’s output.

A three-player final table#

Three players remain, 10,000 total chips, a $1,000 prize pool paying $600 / $300 / $100 for 1st / 2nd / 3rd:

Running the recursive ICM calculation gives each player’s probability of finishing in each place:

(Every row sums to 100%, and so does every column — each place is assigned to exactly one player’s probability mass, in expectation.)

The dollar figures#

Multiplying each row by the payouts gives each player’s $EV:

(Both columns sum to exactly $1,000 — the whole prize pool, split two different ways.)

Reading the gap#

The naive model overvalues the chip leader by $82.14 and undervalues the short stack by $57.14. The reason isn’t a rounding quirk — it’s the shape of the payout structure. A’s extra chips over B and C are only worth their proportional share if finishing 1st and finishing 3rd paid proportionally too, but here 1st pays 6x what 3rd pays while A holds only 2.5x C’s chips. Once C busts, a large share of the remaining prize pool is still locked into 1st and 2nd — money A can’t guarantee just by holding more chips now. Symmetrically, C’s stack under-represents C’s dollar equity because C is already guaranteed some placement money simply by being one of three players left, a floor the linear chip-share model doesn’t see.

What survives from earlier weeks, and what doesn’t#

Pot odds and Bayesian updating (week 9’s audit) still price a single hand in chips won or lost. ICM is the layer on top: it’s what converts those chip swings into dollar value once elimination is possible. A chip-EV positive call can be a dollar-EV negative call near the bubble, if it risks busting out of guaranteed money — the two are no longer the same calculation, and conflating them is the single most common tournament mistake this model is built to catch.

Before next week#

ICM prices tournament chips against a payout structure. Week 11 goes back to a cash-game setting and asks a related but different question: given a bankroll instead of a fixed prize pool, how much of it should a positive-EV bet actually risk?