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

Seat 1 holds 15,000 chips, 30.0% of the chips in play, and is worth 29.1% of the prize pool when 3 places pay 50/30/20 percent. That is -0.90 points against its chip share.

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

58.4%. In chips alone the call breaks even at 50%, but the bubble factor of 1.40 raises the requirement because losing costs the smallest stack 12.71% 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 29.1% of the prize pool for 30.0% of the chips.

ICM: One short stack, 4 Players, 3 Paid

With 4 players left and 3 paid (50/30/20 percent), the biggest stack of 15,000 chips holds 30.0% of the chips but is worth 29.1% of the prize pool. The smallest, 5,000 chips, holds 10.0% and is worth 12.7%.

How much is each stack worth with 4 players left?

50,000 chips are in play in the one short stack pattern, where one player holds 5,000 chips, a third of each of the others at 15,000. The prize pool pays three places paid, 50, 30 and 20 percent, and the table converts each seat's chips into prize equity.

Chips and prize equity with 4 players left, payouts 50/30/20 percent
Seat	Chips	Share of chips	ICM equity	ICM minus chips
Seat 1	15,000	30.00%	29.10%	-0.90 points
Seat 2	15,000	30.00%	29.10%	-0.90 points
Seat 3	15,000	30.00%	29.10%	-0.90 points
Seat 4	5,000	10.00%	12.71%	+2.71 points

Share of the chips against share of the prize pool

Seat 1Chips 30.0%Prize 29.1%
Seat 2Chips 30.0%Prize 29.1%
Seat 3Chips 30.0%Prize 29.1%
Seat 4Chips 10.0%Prize 12.7%

Chips and prize money do not trade one for one. The biggest loss against chip share is 0.90 points for Seat 3, and the biggest gain is 2.71 points for Seat 4.

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 86.0% of the time, the smallest 42.1%, with 3 places paid.

Chance of finishing in each place
Seat	First	Second	Third	Fourth	Paid
Seat 1	30.00%	29.05%	26.90%	14.05%	85.95%
Seat 2	30.00%	29.05%	26.90%	14.05%	85.95%
Seat 3	30.00%	29.05%	26.90%	14.05%	85.95%
Seat 4	10.00%	12.86%	19.29%	57.86%	42.14%

Where each seat's prize equity comes from, in points of the prize pool
Seat	From first	From second	From third	Total
Seat 1	15.00	8.71	5.38	29.10%
Seat 2	15.00	8.71	5.38	29.10%
Seat 3	15.00	8.71	5.38	29.10%
Seat 4	5.00	3.86	3.86	12.71%

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 2.40 with Seat 1 facing Seat 2; the smallest is 1.15 with Seat 1 facing Seat 4.

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	2.40	2.40	1.15
Seat 2	2.40	none	2.40	1.15
Seat 3	2.40	2.40	none	1.15
Seat 4	1.40	1.40	1.40	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 1.40. A call that needs 50% in chips needs 58.4% once prize money counts, which is 8.4 points more. The same all-in the other way, with the biggest stack calling the smallest, has a bubble factor of 1.15 and needs 53.6%, 3.6 points over the chip figure. The smaller stack is the one that pays the larger ICM premium, because losing the all-in costs it 12.71% of the prize pool and winning adds 9.07%.

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	5,000	1.4016	50.0%	58.4%	12.71%	21.79%	0.00%
Biggest stack calls smallest	5,000	1.1541	50.0%	53.6%	29.10%	35.43%	21.79%

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 15,000 chips, 30.0% of the chips in play, and is worth 29.1% of the prize pool when 3 places pay 50/30/20 percent. That is -0.90 points against its chip share.
What equity does the smallest stack need to call the biggest stack's all-in?: 58.4%. In chips alone the call breaks even at 50%, but the bubble factor of 1.40 raises the requirement because losing costs the smallest stack 12.71% 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 29.1% of the prize pool for 30.0% of the chips.