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

Seat 1 holds 10,000 chips, 33.3% of the chips in play, and is worth 33.3% of the prize pool when 3 places pay 50/30/20 percent. That is +0.00 points against its chip share.

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

57.1%. In chips alone the call breaks even at 50%, but the bubble factor of 1.33 raises the requirement because losing costs the smallest stack 13.33% 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 33.3% of the prize pool for 33.3% of the chips.

ICM: Even stacks, 3 Players, 3 Paid

With 3 players left and 3 paid (50/30/20 percent), the biggest stack of 10,000 chips holds 33.3% of the chips but is worth 33.3% of the prize pool. The smallest, 10,000 chips, holds 33.3% and is worth 33.3%.

How much is each stack worth with 3 players left?

The stacks are even stacks: every player holds 10,000 chips. That makes 30,000 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	10,000	33.33%	33.33%	+0.00 points
Seat 2	10,000	33.33%	33.33%	+0.00 points
Seat 3	10,000	33.33%	33.33%	+0.00 points

Share of the chips against share of the prize pool

Seat 1Chips 33.3%Prize 33.3%
Seat 2Chips 33.3%Prize 33.3%
Seat 3Chips 33.3%Prize 33.3%

Chips and prize money do not trade one for one. The biggest loss against chip share is 0.00 points for Seat 3, and the biggest gain is 0.00 points for Seat 1.

How often does each seat finish in the money?

Equity comes from the places a stack can finish in. With 3 paid, Seat 1 (10,000 chips) is in the money 100.0% of the time and Seat 3 (10,000 chips) 100.0%.

Chance of finishing in each place
Seat	First	Second	Third	Paid
Seat 1	33.33%	33.33%	33.33%	100.00%
Seat 2	33.33%	33.33%	33.33%	100.00%
Seat 3	33.33%	33.33%	33.33%	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	16.67	10.00	6.67	33.33%
Seat 2	16.67	10.00	6.67	33.33%
Seat 3	16.67	10.00	6.67	33.33%

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

Bubble factor compares what a lost all-in costs with what a won one gains, both in prize equity. In this spot it runs from 1.33 (Seat 1 calling Seat 2) up to 1.33 (Seat 1 calling Seat 2). A figure of 1 would mean chips and prize money agree.

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	1.33	1.33
Seat 2	1.33	none	1.33
Seat 3	1.33	1.33	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 1.33. A call that needs 50% in chips needs 57.1% once prize money counts, which is 7.1 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. Here the bigger stack pays the larger premium: losing costs it 13.33% of the prize pool and winning adds 10.00%.

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	1.3333	50.0%	57.1%	33.33%	43.33%	20.00%
Biggest stack calls smallest	10,000	1.3333	50.0%	57.1%	33.33%	43.33%	20.00%

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?

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 10,000 chips, 33.3% of the chips in play, and is worth 33.3% of the prize pool when 3 places pay 50/30/20 percent. That is +0.00 points against its chip share.
What equity does the smallest stack need to call the biggest stack's all-in?: 57.1%. In chips alone the call breaks even at 50%, but the bubble factor of 1.33 raises the requirement because losing costs the smallest stack 13.33% 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 33.3% of the prize pool for 33.3% of the chips.