08-30-2005, 10:01 PM
I have recently begun reading various poker forums and I have been bothered by how people use ROI as a way of comparing results in Single Table Tournaments(STT). Although ROI provides general information about your earnings, it must be adjusted before it can be used as a basis for comparing results across different types of STTs.
As an extreme example, let's suppose you play 100 10-person sit-and-gos and 100 2-person sit-and-gos and you win every one of them. Your ROI for the 10-person games is 354.55% while your ROI for the heads-up games is 90.48%. The huge discrepancy between these ROI values is due to key differences in the two types of games.
In some sense these two values must be equivalent because they represent the same level of results in both types of STTs. Since each type of STT has its own scale for ROI it is necessary to convert the ROI from one scale to the other in order to compare ROIs across game types. To do this we need to use a formula, just like converting Fahrenheit temperatures to Celsius.
The formula to covert ROI1 to ROI2 is: ROI2 = ROI1 * (MaxROI2/MaxROI1).
Before I explain this formula and some of its implications I must first issue a couple of disclaimers. I have only been playing poker for a year and have just started reading the forums, so I apologize if any of this information has been presented elsewhere. Also I am not a mathematician or statistician so I also apologize if I have made any mistakes in math or in terminology. Finally, I also apologize that this post is rather lengthy.
OK, the conversion formula is based on three key concepts. The first is that, when it comes to ROI, any specific STT game belongs to a family of games that share three common properties - the number of players(n), the fee percent(r), and the payout structure(f/s/t). These three properties form the signature for a game type, which I write as n-r-f/s/t. You may have noticed that the buy-in amount isn't included. That's because when you have identical results for games of the same type the ROIs for those games will be identical, regardless of the buy-in.
For example, I typically play in STTs on UltimateBet with a signature of 10-10-50/30/20, a 10 person game with a 10% fee and a 50%/30%/20% payout for first through third place. If I play 100 games of this type with a $5 buy-in and place first 10 times, second 20 times, and third 10 times, my ROI would be 18.18%. I would have an identical ROI if I had the same results playing $1, $10, or $20 games.
The reason that buy-in is irrelevant for games of the same type is because they all operate on the same scale. All games of all types have a minimum ROI of -1. Since ROI = (AmountWon - Investment)/Investment, when your results are the absolute worst and your AmountWon = 0 the ROI is -1. On the other end of the scale is the maximum ROI (MROI). This occurs when the amount won is the highest possible. MROI was illustrated in the opening example where the 10-person game (10-10-50/30/20) has an MROI of 354.55% while the heads-up game (2-5-100/0) has an MROI of 90.48%.
The use of MROI in converting ROI from one game type to another is the second key concept. Each game type has its own MROI value. Given a game type's signature its MROI can be calculated using the formula MROI = (nf/(1+r)) - 1, where f and r need to be expressed as decimals instead of percentages. For example, the MROI for a 9-person game (9-10-50/30/20) would be calculated as ((9 * .5)/(1+.1)) - 1 = 3.0909, or 309.09%. As you can see from this formula neither the number of games played nor the buy-in is involved in calculating MROI. In fact, the MROI value is a constant for each game type. Below I have included a chart of MROI values for different types of games. Because all games of the same type have the same signature and MROI, it is not necessary to convert ROI when you are comparing results of games within the same game type. The conversion formula reflects this because when MROI2 = MROI1 the formula reduces to ROI2 = ROI1 * 1.
However it is necessary to convert ROI when comparing results across game types. The derivation of the formula to do this conversion is based on the third key concept - the ROI of one game type is equivalent to the ROI of another game type when the profit ratio of the first game is equal to the profit ratio of the second game. Profit ratio is simply your profit divided by your maximum possible profit, which is equivalent to ROI divided by MROI. Therefore, the third concept may be expressed mathematically as ROI2 = ROI1 when ROI2/MROI2 = ROI1/MROI2. This expression yields the conversion formula.
Suppose you have an ROI of 18.18% in a 10-person game (10-10-50/30/20), as in the example above. The formula tells us that this would equate to an ROI of 4.64% playing a heads-up game (2-5-100/0). So, if you have an ROI of 3.64% at the heads-up game (57 wins out of 100), we can conclude that your 10-person results are slightly better than your heads-up results. Some other equivalent ROIs are 15.85% in a 9-person game (9-10-50/30/20) and 14.45% in a 6-person game (6-10-70/30).
By using the concepts of Game Types and MROI it was possible to determine the conversion formula needed to compare ROI results. I have compiled the MROI values for various game types I have located on UltimateBet and Full Tilt.
Game Type.........MROI%....Site (buy-in)
10-7.5-50/30/20...365.12....UltimateBet ($200)
10-9-50/30/20......358.72....UB ($100)
10-10-50/30/20....354.54....UB (under $100)
9-8-50/30/20.......316.67....Full Tilt ($200)
9-9-50/30/20.......312.84....FT ($100)
9-10-50/30/20.....309.09....FT (under $100)
6-7.5-70/30.........290.70....UB ($200)
6-9-70/30............285.32....UB ($100)
6-10-70/30..........281.82....UB (under $100)
2-3-100/0...........94.17.....UB ($1000)
2-4-100/0...........92.31.....UB ($500)
2-5-100/0...........90.48.....UB & FT (under $500)
As an extreme example, let's suppose you play 100 10-person sit-and-gos and 100 2-person sit-and-gos and you win every one of them. Your ROI for the 10-person games is 354.55% while your ROI for the heads-up games is 90.48%. The huge discrepancy between these ROI values is due to key differences in the two types of games.
In some sense these two values must be equivalent because they represent the same level of results in both types of STTs. Since each type of STT has its own scale for ROI it is necessary to convert the ROI from one scale to the other in order to compare ROIs across game types. To do this we need to use a formula, just like converting Fahrenheit temperatures to Celsius.
The formula to covert ROI1 to ROI2 is: ROI2 = ROI1 * (MaxROI2/MaxROI1).
Before I explain this formula and some of its implications I must first issue a couple of disclaimers. I have only been playing poker for a year and have just started reading the forums, so I apologize if any of this information has been presented elsewhere. Also I am not a mathematician or statistician so I also apologize if I have made any mistakes in math or in terminology. Finally, I also apologize that this post is rather lengthy.
OK, the conversion formula is based on three key concepts. The first is that, when it comes to ROI, any specific STT game belongs to a family of games that share three common properties - the number of players(n), the fee percent(r), and the payout structure(f/s/t). These three properties form the signature for a game type, which I write as n-r-f/s/t. You may have noticed that the buy-in amount isn't included. That's because when you have identical results for games of the same type the ROIs for those games will be identical, regardless of the buy-in.
For example, I typically play in STTs on UltimateBet with a signature of 10-10-50/30/20, a 10 person game with a 10% fee and a 50%/30%/20% payout for first through third place. If I play 100 games of this type with a $5 buy-in and place first 10 times, second 20 times, and third 10 times, my ROI would be 18.18%. I would have an identical ROI if I had the same results playing $1, $10, or $20 games.
The reason that buy-in is irrelevant for games of the same type is because they all operate on the same scale. All games of all types have a minimum ROI of -1. Since ROI = (AmountWon - Investment)/Investment, when your results are the absolute worst and your AmountWon = 0 the ROI is -1. On the other end of the scale is the maximum ROI (MROI). This occurs when the amount won is the highest possible. MROI was illustrated in the opening example where the 10-person game (10-10-50/30/20) has an MROI of 354.55% while the heads-up game (2-5-100/0) has an MROI of 90.48%.
The use of MROI in converting ROI from one game type to another is the second key concept. Each game type has its own MROI value. Given a game type's signature its MROI can be calculated using the formula MROI = (nf/(1+r)) - 1, where f and r need to be expressed as decimals instead of percentages. For example, the MROI for a 9-person game (9-10-50/30/20) would be calculated as ((9 * .5)/(1+.1)) - 1 = 3.0909, or 309.09%. As you can see from this formula neither the number of games played nor the buy-in is involved in calculating MROI. In fact, the MROI value is a constant for each game type. Below I have included a chart of MROI values for different types of games. Because all games of the same type have the same signature and MROI, it is not necessary to convert ROI when you are comparing results of games within the same game type. The conversion formula reflects this because when MROI2 = MROI1 the formula reduces to ROI2 = ROI1 * 1.
However it is necessary to convert ROI when comparing results across game types. The derivation of the formula to do this conversion is based on the third key concept - the ROI of one game type is equivalent to the ROI of another game type when the profit ratio of the first game is equal to the profit ratio of the second game. Profit ratio is simply your profit divided by your maximum possible profit, which is equivalent to ROI divided by MROI. Therefore, the third concept may be expressed mathematically as ROI2 = ROI1 when ROI2/MROI2 = ROI1/MROI2. This expression yields the conversion formula.
Suppose you have an ROI of 18.18% in a 10-person game (10-10-50/30/20), as in the example above. The formula tells us that this would equate to an ROI of 4.64% playing a heads-up game (2-5-100/0). So, if you have an ROI of 3.64% at the heads-up game (57 wins out of 100), we can conclude that your 10-person results are slightly better than your heads-up results. Some other equivalent ROIs are 15.85% in a 9-person game (9-10-50/30/20) and 14.45% in a 6-person game (6-10-70/30).
By using the concepts of Game Types and MROI it was possible to determine the conversion formula needed to compare ROI results. I have compiled the MROI values for various game types I have located on UltimateBet and Full Tilt.
Game Type.........MROI%....Site (buy-in)
10-7.5-50/30/20...365.12....UltimateBet ($200)
10-9-50/30/20......358.72....UB ($100)
10-10-50/30/20....354.54....UB (under $100)
9-8-50/30/20.......316.67....Full Tilt ($200)
9-9-50/30/20.......312.84....FT ($100)
9-10-50/30/20.....309.09....FT (under $100)
6-7.5-70/30.........290.70....UB ($200)
6-9-70/30............285.32....UB ($100)
6-10-70/30..........281.82....UB (under $100)
2-3-100/0...........94.17.....UB ($1000)
2-4-100/0...........92.31.....UB ($500)
2-5-100/0...........90.48.....UB & FT (under $500)