How much is the biggest stack worth in ICM with 4 players left?

Seat 1 holds 20,000 chips, 40.0% of the chips in play, and is worth 36.5% of the prize pool when 2 places pay 65/35 percent. That is -3.50 points against its chip share.

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

58.0%. In chips alone the call breaks even at 50%, but the bubble factor of 1.38 raises the requirement because losing costs the smallest stack 21.17% 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 36.5% of the prize pool for 40.0% of the chips.

ICM: One chip leader, 4 Players, 2 Paid

Take 4 players, 2 paid places (65/35 percent) and one chip leader: the largest stack converts 40.0% of the chips into 36.5% of the prize pool, and the smallest turns 20.0% into 21.2%.

How much is each stack worth with 4 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 50,000 chips, and two places paid, 65 and 35 percent. Read each seat's prize equity against its chip share in the table.

Chips and prize equity with 4 players left, payouts 65/35 percent
Seat	Chips	Share of chips	ICM equity	ICM minus chips
Seat 1	20,000	40.00%	36.50%	-3.50 points
Seat 2	10,000	20.00%	21.17%	+1.17 points
Seat 3	10,000	20.00%	21.17%	+1.17 points
Seat 4	10,000	20.00%	21.17%	+1.17 points

Share of the chips against share of the prize pool

Seat 1Chips 40.0%Prize 36.5%
Seat 2Chips 20.0%Prize 21.2%
Seat 3Chips 20.0%Prize 21.2%
Seat 4Chips 20.0%Prize 21.2%

ICM pays less than chip share to the stacks at the top and more to the stacks at the bottom: Seat 1 falls 3.50 points short, and Seat 2 comes out 1.17 points ahead.

How often does each seat finish in the money?

Before payouts are applied, each stack has a chance of every finishing place. The biggest stack cashes 70.0% of the time, the smallest 43.3%, with 2 places paid.

Chance of finishing in each place
Seat	First	Second	Third	Fourth	Paid
Seat 1	40.00%	30.00%	20.00%	10.00%	70.00%
Seat 2	20.00%	23.33%	26.67%	30.00%	43.33%
Seat 3	20.00%	23.33%	26.67%	30.00%	43.33%
Seat 4	20.00%	23.33%	26.67%	30.00%	43.33%

Where each seat's prize equity comes from, in points of the prize pool
Seat	From first	From second	Total
Seat 1	26.00	10.50	36.50%
Seat 2	13.00	8.17	21.17%
Seat 3	13.00	8.17	21.17%
Seat 4	13.00	8.17	21.17%

What is the bubble factor for each seat with 4 players left?

Every call has a bubble factor, the equity lost on a loss over the equity gained on a win. The largest at this table is 1.38 with Seat 2 facing Seat 1; the smallest is 1.18 with Seat 1 facing 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	vs Seat 4
Seat 1	none	1.18	1.18	1.18
Seat 2	1.38	none	1.20	1.20
Seat 3	1.38	1.20	none	1.20
Seat 4	1.38	1.20	1.20	none

How much equity does a call need?

When the smallest stack (Seat 4) calls an all-in from the biggest (Seat 1), the bubble factor is 1.38. A call that needs 50% in chips needs 58.0% once prize money counts, which is 8.0 points more. The same all-in the other way, with the biggest stack calling the smallest, has a bubble factor of 1.18 and needs 54.1%, 4.1 points over the chip figure. The smaller stack is the one that pays the larger ICM premium, because losing the all-in costs it 21.17% of the prize pool and winning adds 15.33%.

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	1.3804	50.0%	58.0%	21.17%	36.50%	0.00%
Biggest stack calls smallest	10,000	1.1795	50.0%	54.1%	36.50%	49.50%	21.17%

Where can I compare other stack patterns?

Each payout structure and player count has the same stack patterns, so you can see which of them moves the numbers most. The ICM calculator takes your own stacks and payouts.

How was this ICM table computed?

Each player finishes first with a chance equal to their share of the chips, and every later place is drawn the same way from the players left. Prize equity is the payout of each place times the chance of finishing there. The bubble factor divides the equity lost by losing an all-in by the equity gained by winning it, with the other stacks unchanged, and a player who reaches 0 chips takes the lowest place.

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 4 players left?: Seat 1 holds 20,000 chips, 40.0% of the chips in play, and is worth 36.5% of the prize pool when 2 places pay 65/35 percent. That is -3.50 points against its chip share.
What equity does the smallest stack need to call the biggest stack's all-in?: 58.0%. In chips alone the call breaks even at 50%, but the bubble factor of 1.38 raises the requirement because losing costs the smallest stack 21.17% 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 36.5% of the prize pool for 40.0% of the chips.