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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.stack.imgur.com/jMSn3.png

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

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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.

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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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