As with the first poker quiz, Poker Quiz: Pick Your Poison (published on February 1, 2008), the answers I received to questions in Poker Quiz II: Final Table Deals (published February 13, 2008) were very well-reasoned, well articulated, and all over the place. For those of you who do not remember, the quiz gave three final table scenarios and asked readers to determine an equitable deal (chop the prize money) for all players based on their chip stacks. The scenario questions were:
Assume the payout structure for all questions is:
- 1st Place - $400
- 2nd Place - $250
- 3rd Place - $150
- 4th Place - $100
The scenarios were:
- Scenario 1 - Player A: 40,000 chips; Player B: 20,000 chips; Player C: 10,000 chips
- Scenario 2 - Player A: 40,000 chips; Player B: 20,000 chips; Player C: 10,000 chips; Player D 5,000 chips
- Scenario 3 - Player A: 40,000 chips; Player B: 40,000 chips; Player C: 20,000 chips; Player D 10,000 chips
There were no right or wrong answers, but I do believe some deals were far more fair than others. Overall, people seemed to use one of three methods to compute a fair deal. I will discuss each of those methods below.
Each method is based on the principle that a player’s chances of winning a tournament is directly proportional to the percentage of chips he/she possesses. For example, a player with 25,000 of the 100,000 chips in play has a 25% chance of winning the tournament. Unfortunately, some of the methods used to divide payouts during a deal do not correctly apply this principle.
Chip Percentage Method (CPM):
The Chip Percentage Method (CPM) is the simplest to compute, but also the least likely to result in a fair and equitable deal for all parties. Simply put, the CPM awards prize money based on the percentage of chips each person holds and nothing more. It appeared about 1/3 of people who took the quiz used this method to reach their answers.
To use the CPM to compute a person’s portion of the prize pool, simply add the prize money yet to be awarded and multiply it by the percentage of chips you have in relation to all chips in the tournament. For example, to compute the amount of prize money that the CPM would award to Player A in the first scenario, simply add the remaining prize money ($800 = $400 for 1st place + $250 for 2nd Place + $150 for 3rd Place) and multiply it by the percentage of chips held by Player A (57.1% = 40,000 chips held by Player A/ $70,000 chips in play). Therefore, according to CPM, Player A should be awarded approximately $457.
Payout = $ * PnC / TC
where PnC = Player Chips and C = Total Chips
While the CPM is definitely the easiest method to use when computing equitable divisions of the prize money, it is clearly not the most fair. Compare the CPM prize money awarded to Player A ($457) to the 1st Place prize money if the tournament is completed ($400). The CPM vastly over-estimates the amount of equity the chip leader has in a tournament. So much so, that in many situations, it awards a player more than he/she could earn by finishing the tournament.
In contrast, it vastly under-estimates the amount of equity a player with a short stack has in the tournament. Again, looking to Scenario 1, the worst Player C can do is earn $150 by finishing 3rd, yet the CPM would only award Player C $114. Does this seem fair.
Of course, there are scenarios in which the vastly exagerated estimations are not as outlandish, but the illustration shows how the CPM does not provide a fair representation of each player’s chances of winning the tournament. The reason for this is simply the misapplication of mathematical principles. A player who has a 60% chance of winning a tournament cannot have a 60% chance of finishing 2nd and a 60% chance of winning 3rd; however, that’s the portion of each places prize money that the CPM awards the player.
Chip Chop Method
To make up for the mathematical deficiencies in the CPM, many players use the chip chop method. In fact, this method is used by some online poker hosts to provide players with a starting point for negotiating deals. The method is very similar to the CPM; however, it awards all players the least amount of money they can win before distributing the remainder of the prize money based on chip counts. About 1/2 of the people who took the quiz used this method to determine their answers.
To use the Chip Chop Method to compute a person’s portion of the prize pool, begin by awarding everybody the smallest amount of prize money they can win. For example, in Scenario 2, the lowest amount the four remaining players can win is 4th place prize money - $100. From there, simply add the remaining amount of prize money yet to be awarded and multiply it by the percentage of chips each player has in relation to all chips in the tournament. For example, to compute the amount of prize money that the Chip Chop Method would award to Player A in the second scenario, simply add the remaining prize money after awarding each player 4th place money ($500 = $400 for 1st place + $250 for 2nd Place + $150 for 3rd Place + $100 for 4th Place - $400 already awarded) and multiply it by the percentage of chips held by Player A (57.1% = 40,000 chips held by Player A/ $70,000 chips in play). After getting that total, add on the $100 already awarded and you have the amount. Therefore, according to CPM, Player A should be awarded approximately $366.67 ($100 + ($500 remaining in prize pool * 40,000 chips held by Player A / 75,000 total chips in play).
Payout = G + ($ - (G * N) * (PnC * C)
where G = Minimum Guarantee, N = Number of Players, PnC = Player Chips, and C = Total Chips in Play
While this method is definitely superior to the CPM and still relatively easy to use when computing equitable divisions of the prize money, it is still is not the most fair. Again, the reason for this is simply the misapplication of mathematical principles. While the chip chop method takes a certain portion off the top and splits it equally, it still distributes the remainign prize money based strictly on the amount of chips a player has in his/her stack. A player who has a 60% chance of winning a tournament cannot have a 60% chance of finishing 2nd and a 60% chance of winning 3rd; however, that’s the portion of each places prize money that the chip chop method awards the players after taking the minimum amount that can be won and awarding it to each player.
Independent Chip Model (ICM)
Fortunately, there is a method for making deals that accounts for these inequities - one I believe to be the most fair. The Independent Chip Model (ICM) takes into account the probabilities each player has of finishing in each position. By computing the likelihood of each player finishing in each position, the ICM can compute the amount of equity each player has in each individual prize. Only about 1/6 of the people who took the quiz used the ICM to compute an equitable split.
The math of the ICM is a little tricky, but there are ICM calculators available online to do the math. For example, http://www.chillin411.com/icmcalc.php provides the ability to compute a chop for up to ten players over five positions. If you do not have access to the Internet, you can download the CCT Equity Estimator - Version 1.1 - Trial Edition. This version of the CCT Equity Estimator uses the ICM to compute a chop for up to four players over four positions. It is Microsoft Excel driven and can be downloaded directly to your computer.
The basic premise is to determine the likelihood of each possible order of finish happening based on chip counts and use the probabilities to add up each person’s equity. For example, in a three-player chop, there are 6 different possible orders of finish. In a four-player chip, there are 24 possible orders of finish. If you can determine the likelihood of each finish, you will know how likely each player is to finish in each position and be able to identify an more even split.
The tables below show what I believe are the optimal chops for each scenario, based on the ICM:
Scenario 1 - 3-Way Chop
Scenario 2 - 4-Way Chop
Scenario 3 - 4-Way Chop
Of course, the key is knowing all of these methods and using them as simply a starting point for your deal negotiations. Knowing what deals may be out there for you can be very helpful, but knowing what deal is the most fair and knowing why it is the most fair is invaluable.
Stumble it!
