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入正题：

Let:

E = the equity of calling and winning

E = the equity of calling and losing

E = the equity of folding

P = the probability of winning

For simplicity, we will ignore the possibility of a split pot. For most hands this minor factor can be safely ignored.

Now:

E x P E x (1 – P) = E

Since E in this hand is zero, we can ignore that term and end up with:

0.1844 x P = 0.0991

P = 53.74%

According to the ICM, we need to win 53.74% of the time to make this a call. Since 2h2d only beats AcKd 52.34% of the time and ties 0.31%, this would be a fold.

Furthermore, we can calculate exactly how much making this call would cost you. The equity of calling is:

P x E P x E

= 0.5234 x 0.1844 0.0031 x 0.1000

= 0.0994

Subtracting this from E gives 0.18% of prize pool. In a $100 buyin SNG, calling here costs $1.80.

Since the player with the AcKd is a dog to us, he fairs much worse:

P x E P x E

= 0.4730 x 0.1844 0.0031 x 0.1000

= 0.0875

This is a loss of 1.16%. This hand would cost him $11.60 in a $100 buyin SNG.

Alert readers might recall that the total amount of equity in a tournament is constant. But in this hand, both players involved in the hand have lost equity. Where has this equity gone?

The answer is that it is distributed evenly amongst all the other players in the tournament. In an SNG, every player has a stake in every hand. If you have ever been short stacked and heaved a sigh of relief at someone else busting on the bubble, you will be intuitively familiar with this idea. Even though you weren’t involved in the hand, it’s obvious that your equity in the tournament just took a big jump. Similar effects are taking place on every hand of the tournament, albeit usually in a much more minor way.

This net equity loss isn’t limited to hands where the participants are allin. Suppose two players have a confrontation on the first hand of a 2000-chip SNG where they are both exactly 50% to either win or lose 500 chips. Their possible equities after the hand are:

2500 chips: 0.1223

1500 chips: 0.0767

Averaging these gives 0.0995, so each player has lost (on average) 0.05% of the prize pool, or about 50 cents at a $100 buyin tournament. That money has been redistributed amongst all the other players in the SNG. Having another identical confrontation just makes the problem worse. There’s a 50% chance that they’ll both end up with 2000 chips again, returning everyone’s equity to 0.1. But the other 50% of the time, the stacks will become even more unbalanced:

3000 chips: 0.1438

1000 chips: 0.0524

For the player who started with 2500 chips, his average equity after this hand is 12.19%, a further loss of 0.04%. The player with 1500 chips gets an average equity after the hand of 7.62%, a further loss of 0.05%. Once again, the equity is distributed amongst players not involved in the hand.

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