Satellites
The Satellite Bubble Formula: Equity Needed to Call for Every Table Size and Seat Count
In a satellite every seat is worth the same. With N players left, S seats and every stack the same, you need to win (N − 1) ÷ (2N − 1 − S) of the time to call an all-in. That is 80.0% with 5 left for 4 seats and 90.0% with 10 left for 9. With 101 left for 100 it is 99.0%. On the bubble the bubble factor is simply N − 1: 4 at 5 for 4, 9 at 10 for 9.
Pocket aces have 85.25% equity against a random hand. Need more than that and no hand can call a random shove. On the bubble that starts at 7 players: 7 left for 6 seats needs 85.7%, and aces fold. For a table with uneven stacks, the satellite bubble calculator does the exact sum.
Equity needed to call on a satellite bubble, by players left and seats
The cells in bold need more than aces have against a random hand.
| Players left | Seats | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| 3 | 50.0% | 66.7% | |||||||||
| 4 | 50.0% | 60.0% | 75.0% | ||||||||
| 5 | 50.0% | 57.1% | 66.7% | 80.0% | |||||||
| 6 | 50.0% | 55.6% | 62.5% | 71.4% | 83.3% | ||||||
| 7 | 50.0% | 54.5% | 60.0% | 66.7% | 75.0% | 85.7% (more than aces have) | |||||
| 8 | 50.0% | 53.8% | 58.3% | 63.6% | 70.0% | 77.8% | 87.5% (more than aces have) | ||||
| 9 | 50.0% | 53.3% | 57.1% | 61.5% | 66.7% | 72.7% | 80.0% | 88.9% (more than aces have) | |||
| 10 | 50.0% | 52.9% | 56.3% | 60.0% | 64.3% | 69.2% | 75.0% | 81.8% | 90.0% (more than aces have) | ||
| 11 | 50.0% | 52.6% | 55.6% | 58.8% | 62.5% | 66.7% | 71.4% | 76.9% | 83.3% | 90.9% (more than aces have) | |
| 12 | 50.0% | 52.4% | 55.0% | 57.9% | 61.1% | 64.7% | 68.8% | 73.3% | 78.6% | 84.6% | 91.7% (more than aces have) |
The first column is winner-take-all, and it is 50% at every table size: with one prize, a chip is worth the same whether you win it or lose it. In every other column the number is highest on the diagonal, where one player misses out, and drops with each extra player left. Open 5 left for 4 seats, 7 for 6 or 10 for 9 in the calculator and it shows the same numbers.
Satellite bubble factor by players left and seats
The bubble factor, (N − 1) ÷ (N − S), is what a lost all-in costs you for every unit a won one gains. The bubble factor calculator works it out for cash payouts too.
| Players left | Seats | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| 3 | 1.00 | 2.00 | |||||||||
| 4 | 1.00 | 1.50 | 3.00 | ||||||||
| 5 | 1.00 | 1.33 | 2.00 | 4.00 | |||||||
| 6 | 1.00 | 1.25 | 1.67 | 2.50 | 5.00 | ||||||
| 7 | 1.00 | 1.20 | 1.50 | 2.00 | 3.00 | 6.00 | |||||
| 8 | 1.00 | 1.17 | 1.40 | 1.75 | 2.33 | 3.50 | 7.00 | ||||
| 9 | 1.00 | 1.14 | 1.33 | 1.60 | 2.00 | 2.67 | 4.00 | 8.00 | |||
| 10 | 1.00 | 1.13 | 1.29 | 1.50 | 1.80 | 2.25 | 3.00 | 4.50 | 9.00 | ||
| 11 | 1.00 | 1.11 | 1.25 | 1.43 | 1.67 | 2.00 | 2.50 | 3.33 | 5.00 | 10.00 | |
| 12 | 1.00 | 1.10 | 1.22 | 1.38 | 1.57 | 1.83 | 2.20 | 2.75 | 3.67 | 5.50 | 11.00 |
The same formula, counted by who misses out
Write k for the players who will miss out, N − S. The equity you need is (N − 1) ÷ (N − 1 + k), and the bubble factor is (N − 1) ÷ k. One player missing out makes the call (N − 1) ÷ N, so with 100 seats it is 99.0%. The calculator stops at 12 players, but the formula doesn't care how big the field is.
| Players left | Seats | Bubble factor | Equity needed |
|---|---|---|---|
| 101 | 100 | 100.00 | 99.0% |
| 110 | 100 | 10.90 | 91.6% |
| 120 | 100 | 5.95 | 85.6% |
| 301 | 300 | 300.00 | 99.7% |
With 120 players left for 100 seats, 20 still have to go out, and an equal-stack call needs 85.6%. Aces have 85.25%.
Paul Harris wrote about a Heartland Poker Tour mega satellite with 16 players left for 15 seats. At his table the short stack moved all in and the chip leader called. Harris, in the big blind, showed his aces and folded: the short stack at the other table was about to bust, and it did. He didn't need to win another hand. An equal-stack call at 16 for 15 would need 93.8%, and he wasn't even facing that. A PokerStrategy hand from a $25 satellite had 9 players left for 5 tickets, with $50 back for 6th and 7th. The stacks were uneven, and their own working put the bubble factor at 6.6 and the equity needed at about 87%. Aces fold there too. With equal stacks and nothing for 6th and 7th, 9 for 5 would need 66.7%.
Where the formula comes from
Give everyone one chip's worth of stack, so there are N in play. ICM gives each player a chance of finishing first equal to their share of the chips, takes the winner out, and repeats for each place. With equal stacks, everyone's chance of a seat is S ÷ N.
- Lose the all-in and you are the first one out, in Nth place, which pays nothing.
- Win it and you have 2 of the N chips, with N − 1 players left. You miss a seat only if none of the first S places picks you. When j others have already been placed, the chance the next place goes to someone else is (N − 2 − j) ÷ (N − j). Multiply that from j = 0 to S − 1 and nearly everything cancels, leaving (N − S)(N − S − 1) ÷ (N(N − 1)). Your seat chance is 1 minus that, which is S(2N − 1 − S) ÷ (N(N − 1)).
- The call breaks even when your chance of winning times your seat chance if you win equals your seat chance if you fold, S ÷ N. Divide and the S and the N cancel: (N − 1) ÷ (2N − 1 − S).
Winning adds S(N − S) ÷ (N(N − 1)) to your seat chance, and losing costs S ÷ N, so the bubble factor is the ratio, (N − 1) ÷ (N − S). I checked both against the exact ICM engine for all 65 cells in the grids. The page won't build if any cell is off by more than a ten-millionth of a point, and none is.
What the formula leaves out
Every stack here is the same. Real stacks aren't, so put yours in the satellite bubble calculator. The pot holds only the two all-ins. Away from the bubble, blinds and antes in the middle usually lower the equity you need, and the bubble factor chart adjusts for them. On the bubble they barely help, because winning already gets you a seat, and only a blind or ante you posted yourself makes the call cheaper.
The shover also holds a random hand here. A real shover has a range, and most hands do worse against it. Aces do about the same, a little better against a wide range and worse against a tight one. The satellite safe stack tables cover the other side: the stack where you never have to face this call. The lesson Folding aces in a satellite plays the hand out.