so i get the whole way how to calculate the ICM. However, i'm not sure completely how to use the Malmuth-Harville written down in the book "mathematics of poker". It's important for me to understand the formula.

Page from the book https://i.sstatic.net/jMSn3.png

So could anyone please calculate an example with this formula? Let's say three people. Payout 65%-35%. Thanks in advance :)

2 Answers 2


As I understand it, this equation predicts the probability of a player coming in second (or any place other than first) by creating a ratio between the average chance of winning when all 3 players are at the table and the average chance of winning when j players have finished the tournament.

We can use ICM and replace the average chance of winning with a win percentage for a specific player and find out their chances.

let's make up some numbers and say the win probabilities of the players are as follows:

Player 1: 80%
Player 2: 17%
Player 3: 3%

Calculating the probability of Player 2 getting second place can be done this way:

(.17) / (1 - .80) = .85

this splits up the equity that is not obtained by Player1 between the remaining two players.

.85 can then be multiplied by the payout to determine equity in second prize.


For 3 players, let's set the chances of winning at 50%, 30%, and 20% for players A, B, and C respectively.

When A does win, what the first formula says is that B's chance of then getting 2nd place is (.30) / (1 - .50) = 60%. This makes sense because of the two remaining players, B and C, their "skill" still relates comparatively to what their chances of winning had been-- 30:20.

But to get a complete picture of your chances of second place, you must do the same thing with all other potential 1st place winners. So given that player C wins (20% of the time), you have (.30) / (1 - .20) = 37.5%.

Now your overall odds of getting second place is the sum of [ (odds of 2nd place given that player X wins) * (odds of player X winning) ]. In this example then, you end up with ((.60)(.50)) + ((.375)(.20)) = 37.5%.

Your overall equity then would be (.30)(1st place money) + (.375)(2nd place money) + (1 - (.30) - (.375))(3rd place money).

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