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- Ein Deal stellt beim Poker die Möglichkeit dar, das Preisgeld eines Pokerturniers unabhängig vom späteren Spielverlauf unter den verbliebenen Spielern zu verteilen. (de)
- In poker, the Independent Chip Model (ICM) is a mathematical model used to approximately calculate a player's overall equity in a tournament. The model uses stack sizes alone to determine how often a player will finish in each position (1st, 2nd, etc.). A player's probability of finishing in each position is then multiplied by the prize amount for that position and those numbers are added together to determine the player's overall equity. The ICM is also known as the Malmuth–Harville method. In 1973 David Harville published a method to calculate the probability for a horse to finish 1st, 2nd, etc. in a horse race. In 1987 Mason Malmuth, independent of Harville, used this method to calculate the probability for a tournament player to finish 1st, 2nd, etc. The term ICM is often misunderstood to mean a simulator that helps a player make decisions in a tournament. Such simulators often make use of the Independent Chip Model but are not strictly speaking ICM calculators. A true ICM calculator will have the chip counts of all players, as well as the payout structure of the tournament, as input and each player's equity as output. The ICM can be applied to answer specific questions, such as:
* The range of hands that a player can move all in with, considering the action so far and the stack sizes of the other players still in the hand
* The range of hands that a player can call another player's all in with, and recommends either calling or moving all in over the top, considering all the stacks still in the hand
* When discussing a deal, how much money each player should get The calculation using the ICM can be elaborated as below: 1.
* Every player's chance of finishing 1st is proportional to its chip count 2.
* If player i did not finish 1st, given player k finished 1st, player i chance of finishing 2nd is P(Xi,2|Xk,1) = xi/(1-xk） 3.
* Following this logic, given m1 finish 1st, m2 finished 2nd, mj-1 finish j-1th, the chance of player i finish jth place is P(Xi,j|Xm1,1, Xm2,x.....Xmj-1,j-1) = xi/(1-xm1-xm2-....-xmj-1) 4.
* Sum of the value in each permutation (using enumeration, computation complexity = O(n!)) For example: 3 players A,B,C have 50%, 30%, 20% chips, the payout is 1st place 70, 2nd place 30 P(A=1,B=2,C=3) = 0.5*(0.3/(1-0.5))=0.3 P(A=1,C=2,B=3) = 0.5*(0.2/(1-0.5))=0.2 P(B=1,A=2,C=3) = 0.3*(0.5/(1-0.3))=0.2143 P(B=1,A=3,C=2) = 0.3*(0.2/(1-0.3))=0.0857 P(C=1,A=2,B=3) = 0.2*(0.5/(1-0.2))=0.125 P(C=1,A=3,B=2) = 0.2*(0.3/(1-0.2))=0.075 ICM(A) = 70*(0.3+0.2)+30*(0.2143+0.125)=45.18, unit chip value = 45.18/50 = 0.9036 ICM(B) = 70*(0.2143+0.0857)+30*(0.3+0.075)=32.25, unit chip value = 32.25/30 = 1.075 ICM(C) = 70*(0.125+0.075)+30*(0.2+0.0857)=22.57, unit chip value = 22.57/20 = 1.1285 ICM precision 2-players case For any one of the 2 players the probability to finish 1st is exactly equal to its share of the tournament chips. The ICM gives perfect results. 3-players case The Finite element method (FEM) can be used to compute for any player its exact probabilities to finish 1st, 2nd etc. and its exact tournament equity. Those exact values allows for an evaluation of the precision of the ICM. The FEM is used to compute the exact values for all the repartitions of 200 big blinds between 3 players. The table hereafter summarizes the comparison of the approximate ICM values versus the exact FEM values. The big blind repartition 25-87-88 gives the largest difference between an ICM and a FEM probability (0.0360) and the largest tournament equity difference ($0.36 for tournament payouts $50/$30/$20). The relative difference between an ICM and a FEM tournament equity [(ICM- FEM)/FEM)] is 1.42% The big blind repartition 25-87-88 gives the largest relative difference between an ICM and a FEM tournament equity (1.43%). The big blind repartition 198-1-1 gives the largest relative difference between an ICM and a FEM probability (4900%). However that large relative difference has no impact on the tournament equity. Although the ICM does a poor job when computing the exact probability of a player to finish 1st, 2nd , etc., it gives fairly good tournament equities for the 3-players case. 4-players case The FEM is used to compute the exact values for all the repartitions of 100 big blinds between 4 players. The table hereafter summarizes the comparison of the approximate ICM values versus the exact FEM values. The ICM gives fairly good tournament equities for the 4-players case. (en)
- Au poker, l'Independent Chip Model (ICM) est un modèle mathématique utilisé pour calculer approximativement l'équité (c'est-à-dire l'espérance) globale d'un joueur dans un tournoi. Le modèle utilise uniquement les profondeurs de tapis (c'est-à-dire le nombre de jetons détenus par chaque joueur) pour déterminer la fréquence à laquelle un joueur finira à chaque position d'un tournoi (qu'il soit à une seule table, alors dit sit-n-go, ou multi-tables, alors appelé MTT). La probabilité qu'un joueur termine à chaque position est ensuite multipliée par le montant du prix pour cette position et ces nombres sont additionnés pour déterminer l'équité globale du joueur. (fr)
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- Ein Deal stellt beim Poker die Möglichkeit dar, das Preisgeld eines Pokerturniers unabhängig vom späteren Spielverlauf unter den verbliebenen Spielern zu verteilen. (de)
- Au poker, l'Independent Chip Model (ICM) est un modèle mathématique utilisé pour calculer approximativement l'équité (c'est-à-dire l'espérance) globale d'un joueur dans un tournoi. Le modèle utilise uniquement les profondeurs de tapis (c'est-à-dire le nombre de jetons détenus par chaque joueur) pour déterminer la fréquence à laquelle un joueur finira à chaque position d'un tournoi (qu'il soit à une seule table, alors dit sit-n-go, ou multi-tables, alors appelé MTT). La probabilité qu'un joueur termine à chaque position est ensuite multipliée par le montant du prix pour cette position et ces nombres sont additionnés pour déterminer l'équité globale du joueur. (fr)
- In poker, the Independent Chip Model (ICM) is a mathematical model used to approximately calculate a player's overall equity in a tournament. The model uses stack sizes alone to determine how often a player will finish in each position (1st, 2nd, etc.). A player's probability of finishing in each position is then multiplied by the prize amount for that position and those numbers are added together to determine the player's overall equity. The ICM can be applied to answer specific questions, such as: The calculation using the ICM can be elaborated as below: For example: ICM precision 2-players case (en)
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