Tournament poker: what changes when chips are not money

Take four players left in a sit-and-go that pays three places: 50, 30, 20 percent of the prize pool. The stacks add up to 10,000 chips.

If chips were money, the leader with 40% of the chips would own 40% of the prize pool. But first place pays only 50%, however many chips the winner holds, and the three players who avoid finishing fourth collect a prize.

ICM, also called the Malmuth–Harville method, comes from a 1973 paper by the statistician David Harville on betting on horse races. It assumes each player's chance of finishing first is their share of the chips. Take that winner out, and each remaining player's chance of finishing second is their share of the chips left, and so on down the places. Weight each finishing place by what it pays and you have each stack's share of the prize pool, its ICM equity.

Ana's 40% of the chips is worth 33.6% of the prizes. Dev, with 10% of the chips, is worth 13.3%, because a short stack still often lasts long enough to collect third place.

Four players are left and three are paid, so this is the bubble. Ana moves all in, and Ben has to decide whether to call with 3,000 chips. Ana covers Ben: if Ben calls and loses, Ben is out in fourth with nothing. To keep the calculation simple, leave out the blinds and tied pots.

In chips the answer is easy. Ben risks 3,000 to win 3,000, so the call needs 50% equity, the chipEV price. ICM prices three outcomes in shares of the prize pool instead:

Fold

29.5%

Stacks stay as they are

Call and win

40.5%

Ben doubles to 6,000

Call and lose

0%

Out in fourth

The call breaks even where winning and losing balance against folding:

Ben needs 73% equity to call, not 50%. The difference, 23 percentage points, is the risk premium: the extra equity ICM asks for because busting costs more than doubling up gains. Even pocket aces win only about 85% of the time against a random hand, so against a real shoving range only the strongest starting hands can call, and most hands that would call in a cash game fold here.

The cost of busting also explains why different stacks play differently near the money. A player who covers another risks only part of their stack in an all-in, while the other risks everything. Ana can lose this all-in and still be in the tournament with 1,000 chips. Ben would be out.

• The big stack can shove wide, because the stacks it covers have to call tight.
• Middle stacks face the biggest risk premiums, especially against a stack that covers them, and fold the most.
• The short stack faces pressure from the blinds, but still has a chance of a payout to protect. Its shoving and calling ranges depend on who covers it and whether another player is close to busting.

The gaps between payouts drive all of this. The bigger the pay jump for outlasting someone, the more a medium stack gives up by busting first, and the tighter it plays. In a satellite, where every seat pays the same, the effect is extreme: a stack that is already safe can fold almost anything.

At short stacks, push/fold is a useful model: the opener may only shove or fold, and players behind may only call or fold. Small raises and limps can still be part of a fuller strategy, even below 15 big blinds. Restricting the choices makes the model easier to solve and study. Add payouts to see how ICM changes a particular shoving or calling range; the direction depends on the stacks and seats.

ICM leaves out several things that can affect your decision:

• It treats a player's chance of winning as their share of the chips, so it ignores skill. A skill advantage can change a stack's value, but the model does not estimate it.
• It ignores the blinds, position and who is about to post, all of which change what a stack is worth over the next few hands.
• It prices a single decision. It does not see that folding now may leave you shoving into the big blind next orbit.

Even with those limits it gets the direction right: near the money, calls need more equity than the chips alone suggest. The same arithmetic prices a final-table deal, as an ICM chop.