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 33.0% of the prize pool when 3 places pay 50/30/20 percent. That is -7.00 points against its chip share.

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

67.7%. In chips alone the call breaks even at 50%, but the bubble factor of 2.09 raises the requirement because losing costs the smallest stack 22.33% 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 33.0% of the prize pool for 40.0% of the chips.

ICM: One chip leader, 4 Players, 3 Paid

4 players are left, 3 get paid (50/30/20 percent), and the stacks are set out as one chip leader. ICM values the biggest stack at 33.0% of the prize pool for 40.0% of the chips, and the smallest at 22.3% for 20.0%.

How much is each stack worth with 4 players left?

50,000 chips are in play in the one chip leader pattern, where one player holds 20,000 chips, twice each of the others at 10,000. 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	20,000	40.00%	33.00%	-7.00 points
Seat 2	10,000	20.00%	22.33%	+2.33 points
Seat 3	10,000	20.00%	22.33%	+2.33 points
Seat 4	10,000	20.00%	22.33%	+2.33 points

Share of the chips against share of the prize pool

Seat 1Chips 40.0%Prize 33.0%
Seat 2Chips 20.0%Prize 22.3%
Seat 3Chips 20.0%Prize 22.3%
Seat 4Chips 20.0%Prize 22.3%

The chips overstate the big stacks and understate the small ones. Seat 1 is worth 7.00 points less than its chip share, while Seat 2 is worth 2.33 points more.

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 90.0% of the time, the smallest 70.0%, with 3 places paid.

Chance of finishing in each place
Seat	First	Second	Third	Fourth	Paid
Seat 1	40.00%	30.00%	20.00%	10.00%	90.00%
Seat 2	20.00%	23.33%	26.67%	30.00%	70.00%
Seat 3	20.00%	23.33%	26.67%	30.00%	70.00%
Seat 4	20.00%	23.33%	26.67%	30.00%	70.00%

Where each seat's prize equity comes from, in points of the prize pool
Seat	From first	From second	From third	Total
Seat 1	20.00	9.00	4.00	33.00%
Seat 2	10.00	7.00	5.33	22.33%
Seat 3	10.00	7.00	5.33	22.33%
Seat 4	10.00	7.00	5.33	22.33%

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 2.09 with Seat 2 facing Seat 1; the smallest is 1.33 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	1.33	1.33	1.33
Seat 2	2.09	none	1.68	1.68
Seat 3	2.09	1.68	none	1.68
Seat 4	2.09	1.68	1.68	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 2.09. A call that needs 50% in chips needs 67.7% once prize money counts, which is 17.7 points more. The same all-in the other way, with the biggest stack calling the smallest, has a bubble factor of 1.33 and needs 57.1%, 7.1 points over the chip figure. The smaller stack is the one that pays the larger ICM premium, because losing the all-in costs it 22.33% of the prize pool and winning adds 10.67%.

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	2.0937	50.0%	67.7%	22.33%	33.00%	0.00%
Biggest stack calls smallest	10,000	1.3333	50.0%	57.1%	33.00%	41.00%	22.33%

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?

A player finishes first with chance equal to their share of the chips; each next place goes to the players left in proportion to their stacks. Equity is the payout of each place times the chance of finishing there. The bubble factor is prize equity lost by losing the all-in divided by prize equity gained by winning it, with the other stacks unchanged; a player whose stack reaches 0 busts and 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 33.0% of the prize pool when 3 places pay 50/30/20 percent. That is -7.00 points against its chip share.
What equity does the smallest stack need to call the biggest stack's all-in?: 67.7%. In chips alone the call breaks even at 50%, but the bubble factor of 2.09 raises the requirement because losing costs the smallest stack 22.33% 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 33.0% of the prize pool for 40.0% of the chips.