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

Seat 1 holds 15,000 chips, 42.9% of the chips in play, and is worth 36.8% of the prize pool when 3 places pay 50/30/20 percent. That is -6.07 points against its chip share.

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

54.2%. In chips alone the call breaks even at 50%, but the bubble factor of 1.18 raises the requirement because losing costs the smallest stack 6.43% 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 36.8% of the prize pool for 42.9% of the chips.

ICM: One short stack, 3 Players, 3 Paid

3 players are left, 3 get paid (50/30/20 percent), and the stacks are set out as one short stack. ICM values the biggest stack at 36.8% of the prize pool for 42.9% of the chips, and the smallest at 26.4% for 14.3%.

How much is each stack worth with 3 players left?

The stacks are one short stack: one player holds 5,000 chips, a third of each of the others at 15,000. That makes 35,000 chips in play, and three places paid, 50, 30 and 20 percent. The table gives each seat's share of the prize pool next to its share of the chips.

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	15,000	42.86%	36.79%	-6.07 points
Seat 2	15,000	42.86%	36.79%	-6.07 points
Seat 3	5,000	14.29%	26.43%	+12.14 points

Share of the chips against share of the prize pool

Seat 1Chips 42.9%Prize 36.8%
Seat 2Chips 42.9%Prize 36.8%
Seat 3Chips 14.3%Prize 26.4%

The chips overstate the big stacks and understate the small ones. Seat 2 is worth 6.07 points less than its chip share, while Seat 3 is worth 12.14 points more.

How often does each seat finish in the money?

Equity comes from the places a stack can finish in. With 3 paid, Seat 1 (15,000 chips) is in the money 100.0% of the time and Seat 3 (5,000 chips) 100.0%.

Chance of finishing in each place
Seat	First	Second	Third	Paid
Seat 1	42.86%	39.29%	17.86%	100.00%
Seat 2	42.86%	39.29%	17.86%	100.00%
Seat 3	14.29%	21.43%	64.29%	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	21.43	11.79	3.57	36.79%
Seat 2	21.43	11.79	3.57	36.79%
Seat 3	7.14	6.43	12.86	26.43%

What is the bubble factor for each seat with 3 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 1.62 with Seat 1 facing Seat 2; the smallest is 1.06 with Seat 1 facing Seat 3.

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	1.62	1.06
Seat 2	1.62	none	1.06
Seat 3	1.18	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 1.18. A call that needs 50% in chips needs 54.2% once prize money counts, which is 4.2 points more. The same all-in the other way, with the biggest stack calling the smallest, has a bubble factor of 1.06 and needs 51.5%, 1.5 points over the chip figure. The smaller stack is the one that pays the larger ICM premium, because losing the all-in costs it 6.43% of the prize pool and winning adds 5.43%.

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	1.1842	50.0%	54.2%	26.43%	31.86%	20.00%
Biggest stack calls smallest	5,000	1.0615	50.0%	51.5%	36.79%	41.43%	31.86%

Where can I compare other stack patterns?

Compare this pattern with its siblings, with the other payout structure, and with one player more or fewer. The ICM calculator takes your own stacks, and the push or fold calculator prices a shove with the same model.

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 15,000 chips, 42.9% of the chips in play, and is worth 36.8% of the prize pool when 3 places pay 50/30/20 percent. That is -6.07 points against its chip share.
What equity does the smallest stack need to call the biggest stack's all-in?: 54.2%. In chips alone the call breaks even at 50%, but the bubble factor of 1.18 raises the requirement because losing costs the smallest stack 6.43% 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 36.8% of the prize pool for 42.9% of the chips.