How much is the biggest stack worth in ICM with 3 players left?

Seat 1 holds 20,000 chips, 50.0% of the chips in play, and is worth 38.3% of the prize pool when 3 places pay 50/30/20 percent. That is -11.67 points against its chip share.

What equity does the smallest stack need to call the biggest stack's all-in?

59.1%. In chips alone the call breaks even at 50%, but the bubble factor of 1.44 raises the requirement because losing costs the smallest stack 10.83% of the prize pool.

Why is ICM equity different from chip share?

Because the prize pool is paid by place, not by chips. Doubling a stack does not double its payout, so each extra chip is worth less than the one before it. Here the biggest stack is worth 38.3% of the prize pool for 50.0% of the chips.

ICM: One chip leader, 3 Players, 3 Paid

With 3 players left and 3 paid (50/30/20 percent), the biggest stack of 20,000 chips holds 50.0% of the chips but is worth 38.3% of the prize pool. The smallest, 10,000 chips, holds 25.0% and is worth 30.8%.

How much is each stack worth with 3 players left?

Here the stacks follow the one chip leader pattern, where one player holds 20,000 chips, twice each of the others at 10,000. Together they hold 40,000 chips, and three places paid, 50, 30 and 20 percent. Read each seat's prize equity against its chip share in the table.

Chips and prize equity with 3 players left, payouts 50/30/20 percent
Seat	Chips	Share of chips	ICM equity	ICM minus chips
Seat 1	20,000	50.00%	38.33%	-11.67 points
Seat 2	10,000	25.00%	30.83%	+5.83 points
Seat 3	10,000	25.00%	30.83%	+5.83 points

Share of the chips against share of the prize pool

Seat 1Chips 50.0%Prize 38.3%
Seat 2Chips 25.0%Prize 30.8%
Seat 3Chips 25.0%Prize 30.8%

Chips and prize money do not trade one for one. The biggest loss against chip share is 11.67 points for Seat 1, and the biggest gain is 5.83 points for Seat 2.

How often does each seat finish in the money?

ICM gets each equity from the chance of finishing in every place. The Seat 3 stack (10,000 chips) is paid 100.0% of the time, and the Seat 1 stack (20,000 chips) is paid 100.0% of the time.

Chance of finishing in each place
Seat	First	Second	Third	Paid
Seat 1	50.00%	33.33%	16.67%	100.00%
Seat 2	25.00%	33.33%	41.67%	100.00%
Seat 3	25.00%	33.33%	41.67%	100.00%

Where each seat's prize equity comes from, in points of the prize pool
Seat	From first	From second	From third	Total
Seat 1	25.00	10.00	3.33	38.33%
Seat 2	12.50	10.00	8.33	30.83%
Seat 3	12.50	10.00	8.33	30.83%

What is the bubble factor for each seat with 3 players left?

The bubble factor is the prize equity a player loses by losing an all-in divided by the equity gained by winning it. The highest figure here is 1.44, for Seat 2 calling Seat 1, and the lowest is 1.13, for Seat 1 calling Seat 2.

Bubble factor when the row seat calls an all-in from the column seat
Seat	vs Seat 1	vs Seat 2	vs Seat 3
Seat 1	none	1.13	1.13
Seat 2	1.44	none	1.18
Seat 3	1.44	1.18	none

How much equity does a call need?

When the smallest stack (Seat 3) calls an all-in from the biggest (Seat 1), the bubble factor is 1.44. A call that needs 50% in chips needs 59.1% once prize money counts, which is 9.1 points more. The same all-in the other way, with the biggest stack calling the smallest, has a bubble factor of 1.13 and needs 52.9%, 2.9 points over the chip figure. The smaller stack is the one that pays the larger ICM premium, because losing the all-in costs it 10.83% of the prize pool and winning adds 7.50%.

Break-even equity for two all-in calls, risking and winning the same chips
Call	All-in (chips)	Bubble factor	Chip break-even	ICM break-even	Equity now	Equity if win	Equity if lose
Smallest stack calls biggest	10,000	1.4444	50.0%	59.1%	30.83%	38.33%	20.00%
Biggest stack calls smallest	10,000	1.1250	50.0%	52.9%	38.33%	45.00%	30.83%

Where can I compare other stack patterns?

Open a sibling to see how the same payouts value a different split of the chips. For your own stacks and payouts, use the ICM calculator.

How was this ICM table computed?

Under the Malmuth-Harville model a stack wins first place in proportion to its chips, then the next place is awarded among the rest in the same proportion, and so on. Equity adds up the payout of each place weighted by those chances. For the bubble factor, a player whose stack falls to 0 busts into the lowest place and the other stacks stay put.

Source: Malmuth-Harville ICM, computed by GeorgeLiPoker's own library. The model follows D. A. Harville, 'Assigning Probabilities to the Outcomes of Multi-Entry Competitions', Journal of the American Statistical Association, 1973.

Questions

How much is the biggest stack worth in ICM with 3 players left?: Seat 1 holds 20,000 chips, 50.0% of the chips in play, and is worth 38.3% of the prize pool when 3 places pay 50/30/20 percent. That is -11.67 points against its chip share.
What equity does the smallest stack need to call the biggest stack's all-in?: 59.1%. In chips alone the call breaks even at 50%, but the bubble factor of 1.44 raises the requirement because losing costs the smallest stack 10.83% of the prize pool.
Why is ICM equity different from chip share?: Because the prize pool is paid by place, not by chips. Doubling a stack does not double its payout, so each extra chip is worth less than the one before it. Here the biggest stack is worth 38.3% of the prize pool for 50.0% of the chips.