# 撲克ICM數學

The Independent Chip Model (ICM) provides a method to estimate the value "in cash" of a stack in a tournament. Its core assumptions is that the chances of any player winning the tournament are proportional to their stack. For example, we are playing a heads-up sit&go for €10, with Player 1 having 6,000 chips and Player 2 having 4,000 chips, then Player 1's stack is worth €6 and Player 2's stack is worth €4

Similarly, if we have 3 players and it's a winner takes all with a prize pool of €10:

Player 1: 4,000 chips = €4 Player 2: 3,500 chips = €3.50 Player 3: 2,500 chips = €2.50

Let's move to a more complex scenario, with three players fighting for a €90 prize pool (€60 for the winner, €30 for second place). Let's assume stacks are:

Player 1: 10,000 chips Player 2: 5,000 chips Player 3: 5,00 chips

We distinguish 3 scenarios:

• In SCENARIO 1, player 1 wins the tournament and gets €60. The remaining player have an equal chance of finishing second, so their stacks are worth €15 each. Scenario 1 happens with a probability of 50%, since Player 1 has half the total amount of chips.

• In SCENARIO 2, player 2 wins the tournament and gets €60. Player 1 has twice the stack of Player 3, so he is twice as likely to finish second, so his stack is worth €20, while Player 3's stack is worth €10. Scenario 2 happens with a probability of 25%

• In SCENARIO 3, player 3 wins the tournament and gets €60. Player 1 has twice the stack of Player 2, so he is twice as likely to finish second, so his stack is worth €20, while Player 2's stack is worth €10. Scenario 3 happens with a probability of 25%

Now, we only have to add up the values of the stack on each scenario:

Player 1: 60*0.5 + 20*0.25 + 20*0.25 = €40

Player 2: 15*0.5 + 60*0.25 + 10*0.25 = €25

Player 3: 15*0.5 + 10*0.25 + 60*0.25 = €25

With four players and three prizes involved, you can see how we have to make more ramifications on each scenario.