progressBubble with unequal stacks: who shoves
The big stack shoves almost anything. The short stack can be the tightest of all.
Equal stacks are the clean case. Real bubbles are not equal. Here is a four-handed bubble, three paid (50 / 30 / 20), with these stacks: cutoff 8 BB, button 15 BB, small blind 25 BB, big blind 12 BB. First in, folded to him:
| Seat | Stack | Chip EV shove | ICM shove | Callers behind (ICM) |
|---|---|---|---|---|
| Cutoff | 8 BB | 38% | 26% | 7% / 12% / 14% |
| Button | 15 BB | 36% | 38% | 9% / 5% |
| Small blind | 25 BB | 68% | 100% | 5% |
The small blind has 25 BB, the biggest stack, and only the 12 BB big blind behind. In our model he shoves 100% of hands (any two cards): the big blind, who cannot afford to lose his stack on the bubble, calls only 5%. In chip EV the small blind shoves 68%.
The cutoff is the shortest stack at 8 BB, with three bigger stacks behind him. In our model he shoves only 26% of hands under ICM, less than the 38% of chip EV. The three players behind him continue against his shove with 7% / 12% / 14% of hands.
What to take away
- The biggest stack is the one that profits most from the bubble: opponents fold far more than chip EV says.
- A short stack in front of several bigger stacks should not copy the equal-stack bubble: here it shoves less than chip EV says.
- The 15 BB button is close to chip EV: it shoves 38% under ICM against 36% in chip EV.
- Always look at who is behind you and how much they have.
How these numbers are made
They come from our own push/fold model of the whole hand. Every player, in turn, can fold, go all-in or call an all-in. The payouts use the Malmuth-Harville ICM, and the ranges are found by repeated best responses until they settle (they move by about a percentage point between runs; a second, independent solver agrees with the chip EV numbers to within about a point and a half). The blinds are 0.5 and 1 BB and the antes total 1 BB. Stacks are before posting. There is no postflop play, because an all-in and a call run to showdown. It is an approximation of the real game, and ICM itself ignores skill and position after the flop. It is a map for study, not an exact solution.
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Questions
Who benefits most from the bubble?
The biggest stack. In our example the 25 BB small blind shoves 100% of hands because the 12 BB big blind calls only 5%.
Should a short stack shove wider on the bubble?
Not automatically. In our example the 8 BB cutoff shoves 26% under ICM against 38% in chip EV, with three bigger stacks behind him.
Why do the numbers differ from the equal-stack case?
Because the cost of being called depends on who calls and what they have behind them.
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progressGTO guides use our own solution book (v13, 8-max tournaments with ante, chip EV): an approximation of equilibrium with a modelled postflop, not an exact solution. Numbers are computed from the book and from the exact ICM model.
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