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I'm much less opposed to league rating if we keep WHR next to it AND keep it visible. If league rating is already reset every 4 month, it is NOT needed to also reset WHR. Just leave WHR as is, then no spare month is needed.
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I'm much less opposed to league rating if we keep WHR next to it AND keep it visible. If league rating is already reset every 4 month, it is NOT needed to also reset WHR. Just leave WHR as is, then no spare month is needed.
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Whether the points are taken from before or after the game does not necessarily matter: For an elo system, the points after the previous game are the same as the points before the next game. For WHR, the points before and after the game are the same according to WHR-internal time after everything has been calculated. But there is a second type of time: real time. After a game happens in real time, the whole WHR-internal timeline changes. It makes sense to calculate everything according to the last real time status of the WHR-internal timeline, but not according to the real timeline of the last WHR-internal time. The former causes unintuitive changes while not playing, though.
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Whether the points are taken from before or after the game does not necessarily matter: For an elo system, the points after the previous game are the same as the points before the next game. For WHR, the points before and after the game are the same according to WHR-internal time after everything has been calculated. But there is a second type of time: real time. After a game happens in real time, the whole WHR-internal timeline changes. It makes sense to calculate everything according to the last real time status of the WHR-internal timeline, but not according to the real timeline of the last WHR-internal time. The former causes unintuitive changes while not playing, though.
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The latest numbers that @GoogleFrog suggested remind me a lot of Fibonacci numbers. So why not directly use those for consistency? According to the Moivre-Binet formula, the x-th Fibonacci number converges to an exponential function a*b^x with {{{a = 1/sqrt(5), b = (1+sqrt(5))/2 = 1.618.}}} @GoogleFrog's number for player at ladder rank x is approximately a*b^(10-x). As the constant scaling factor a*b^10 doesn't matter, we may as well say b^-x. A continous exponential function compared to discrete Fibonacci numbers has the advantage that lower ranked players are not excluded.
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The latest numbers that @GoogleFrog suggested remind me a lot of Fibonacci numbers. So why not directly use those for consistency? According to the Moivre-Binet formula, the x-th Fibonacci number converges to an exponential function a*b^x with {{{a = 1/sqrt(5), b = (1+sqrt(5))/2 = 1.618.}}} @GoogleFrog's number for player at ladder rank x is approximately a*b^(10-x). As the constant scaling factor a*b^10 doesn't matter, we may as well say b^-x. A continous exponential function compared to discrete Fibonacci numbers has the advantage that lower ranked players are not excluded.
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I think @GoogleFrog chose those numbers because he intuitively wanted the exponential behavior. But what is a good value for b? Points proportional to exp(natural WHR) make a lot of sense from the math of WHR, so I think it'd be nice to make it so. {{{natural WHR = (WHR - 1500)/(400/ln(10))}}}But if we want to use ladder positions instead of WHR, we need a way to estimate WHR from ladder positions. By assuming that WHR is distributed equidistantly over ladder positions and looking at the current top1 and top50 values, we get {{{b = 10^((3434-2250)/(50-1)/400) = 1.15}}} which is already more sane than the above 1.618.
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I think @GoogleFrog chose those numbers because he intuitively wanted the exponential behavior. But what is a good value for b? Points proportional to exp(natural WHR) make a lot of sense from the math of WHR, so I think it'd be nice to make it so. {{{natural WHR = (WHR - 1500)/(400/ln(10))}}}But if we want to use ladder positions instead of WHR, we need a way to estimate WHR from ladder positions. By assuming that WHR is distributed equidistantly over ladder positions and looking at the current top1 and top50 values, we get {{{b = 10^((3434-2250)/(50-1)/400) = 1.15}}} which is already more sane than the above 1.618.
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However, WHR distances are not equal. Better ladder positions have higher WHR distances between each other. Therefore, the above assumed equidistant distribution does still exclude all but the top players. By assuming a Gaussian normal distribution, I come to the conclusion that the ideal number of points for a game proportional to exp(natural WHR) is {{{exp(sigma*erfinv(1-(2*x-1)/N))}}}where sigma is the natural WHR standard deviation of all players, erfinv the inversion of the error function erf, x the ladder position and N the number of all players. Every league start, sigma can be set as the WHR standard deviation of all players of the previous league divided by 400/ln(10). Usually, sigma is 1, but for us I guess rather sigma is 4. Similar to b, sigma determines how much better players get more income.
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However, WHR distances are not equal. Better ladder positions have higher WHR distances between each other. Therefore, the above assumed equidistant distribution does still exclude all but the top players. By assuming a Gaussian normal distribution, I come to the conclusion that the ideal number of points for a game proportional to exp(natural WHR) is {{{exp(sigma*erfinv(1-(2*x-1)/N))}}}where sigma is the natural WHR standard deviation of all players, erfinv the inversion of the error function erf, x the ladder position and N the number of all players. Every league start, sigma can be set as the WHR standard deviation of all players of the previous league divided by 400/ln(10). Usually, sigma is 1, but for us I guess rather sigma is 4. Similar to b, sigma determines how much better players get more income.
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I think, @GoogleFrog had mostly the top players in mind because he is a top player himself. For sigma >= 2, the above formula is similarly extreme as his suggestion for top5 players, then gets more to the b=1.15 solution for top50 players and so on for average players.
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League Ideas