撲克ICM數學


1

讓我自己的ICMizer咯咯地笑,但是卻找不到如何找到以下內容的方法:

示例:買進:$ 10

玩家人數:10

支出:1-$ 50、2-$ 30、3-$ 20

初始堆棧:1,000個芯片

現在讓我們假設一段時間後只剩下4個玩家,這是他們的籌碼量:

玩家1:5,000個籌碼

玩家2:2,000籌碼

玩家3:2,000籌碼

玩家4:1,000籌碼

現在這些籌碼的價值是什麼?只需將堆棧大小和支出輸入ICM計算器,您將獲得以下結果:

玩家1:5,000籌碼≅ $ 37.18

玩家2:2,000籌碼≅ 24.33美元

玩家3:2,000籌碼≅ 24.33美元

玩家4:1,000籌碼≅ $ 14.17

如何計算粗體數字?我知道它必須處理yourChips / totalChipsInPlay以及與通過進入第一,第二,第三和第四來進行排列有關的事情,但是我不能在數學上將它們放在一起。

謝謝!

1

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.