Satellites: when folding aces is right

Five players are left and four win a seat. The stacks are:

The chances come from ICM, which estimates how often each stack finishes in each place from the chips alone, then adds up the prizes. With equal prizes it gives each player's chance of a seat.

You are almost certain of a seat. Two players, with 600 and 400 chips, are fighting not to finish fifth. Then the second stack moves all-in for 4,000 and you look down at pocket aces.

Pocket aces win about 85.1% of the time against a random hand. Their equity against a shoving range depends on the hands and suits in that range; it can be higher or lower than against a random hand.

For this simplified calculation, treat the equity as a win probability and leave out ties. If you call and win, the second stack is out, four players remain and everyone has a seat: 100%. If you call and lose, you are left with 1,000 chips and the second stack has 8,000. ICM puts your chance of a seat at 85.1% from there.

Folding is worth about one and a half points more than calling. Under this random-hand example, calling gives away part of a seat.

Set the call equal to the fold and solve for your chance of winning the hand:

For comparison, aces against seven-deuce offsuit have 88.2% equity in the trainer's table, below the threshold. That is an illustration, not a proof about every possible known hand or range; compare the actual matchup with the 94.9% threshold. In chips the call needs a little under 50%, since you call 4,000 to play for a pot of 8,000 plus the blinds. Calling wins chips on average in this example, but folding gives you a better chance of a seat.

Once a seat is secured, extra chips have no prize value. Before then, they can still raise the chance of surviving, but the gain can be tiny. Your 5,000 chips already make a seat almost certain. Winning 4,000 more raises your chance by 0.8 points, to 100%. Losing 4,000 lowers it by 14.1 points, to 85.1%. The possible gain in seat probability is much smaller than the possible loss.

That gap between what a win adds and what a loss costs is the risk premium, and satellites are where it is largest. In a normal tournament it is smaller, because first place pays more than fourth and extra chips still count. With the same stacks and a payout table of 40, 30, 20 and 10, ICM says the same call needs 67.6%, and aces call easily.

The two short stacks have the opposite problem. Folding does not keep their chance of a seat steady, because the blinds come round and shrink them each orbit. They need chips, and they get them from each other or from the middle stacks that cannot afford to call. There is no blanket rule that big stacks fold and short stacks shove: a covering stack can apply pressure, and even a short stack may need to wait when another is closer to busting.

The trainer's satellite spots ask you to make these folds and shoves and grade them under ICM.