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There have been some threads lately where people want an estimate of their rating based on their performance. Using the average between the best win and worst loss is kind of a cop out when computers are now used to run tournaments.
I have a better algorithm for estimating ratings. It would take into account all the matches played and the ratings of all the players. Given any outcome I can find the estimated rating that will maximize the probability of that outcome occurring. It takes a little number crunching because the process is iterative.
For instance if a player beats a 1800 and loses to 2 2000 players his rating would be 2057. If the he beats a 1800 player and loses to a 2000 player and a 3000 player his rating is 1899.999993 or basically 1900. This is the same that you would expect if the player only played 2 matches and beat the 1800 player and lost to the 2000 player. You can see the loss to the 3000 player is not weighted very heavily. In the same way, wins against weak players will be lightly weighted. The algorithm takes care of the weighting process. The first example would look like this
pwl(r-1800)^2+pwl(2000-r)^2+pwl(2000-r)^2=???
The idea is to find the rating r that would maximize ???
Since the probabilities are between 0 and 1 then 3 is the maximum value of ???
There can be multiple players with unknown ratings and my system will still work but there must be some "anchor" ratings to keep all the ratings from drifting high or low.
One of the functions used is a probability of win loss function
pwl(?R)=1/(1+exp(-?*?R))
the probability of winning if rated 100 points higher is
pwl(100)=0.837879
obviously the probability of winning if rated 100 points lower is
pwl(-100)=0.162121
The two probabilities should add up to 1.
The first trick is finding the value of ? so the the pwl() function matches the USATT tables as close a possible. My pwl function looks a lot like the ratings central function. The USATT tables are a joke because they are just that, tables with discrete jumps in rating changes rather a smooth function like my pwl() function and the ratings central function.
So why hasn't anybody found a better solution better than best win and worst loss? You get what you put up with if you are lucky.
I have a better algorithm for estimating ratings. It would take into account all the matches played and the ratings of all the players. Given any outcome I can find the estimated rating that will maximize the probability of that outcome occurring. It takes a little number crunching because the process is iterative.
For instance if a player beats a 1800 and loses to 2 2000 players his rating would be 2057. If the he beats a 1800 player and loses to a 2000 player and a 3000 player his rating is 1899.999993 or basically 1900. This is the same that you would expect if the player only played 2 matches and beat the 1800 player and lost to the 2000 player. You can see the loss to the 3000 player is not weighted very heavily. In the same way, wins against weak players will be lightly weighted. The algorithm takes care of the weighting process. The first example would look like this
pwl(r-1800)^2+pwl(2000-r)^2+pwl(2000-r)^2=???
The idea is to find the rating r that would maximize ???
Since the probabilities are between 0 and 1 then 3 is the maximum value of ???
There can be multiple players with unknown ratings and my system will still work but there must be some "anchor" ratings to keep all the ratings from drifting high or low.
One of the functions used is a probability of win loss function
pwl(?R)=1/(1+exp(-?*?R))
the probability of winning if rated 100 points higher is
pwl(100)=0.837879
obviously the probability of winning if rated 100 points lower is
pwl(-100)=0.162121
The two probabilities should add up to 1.
The first trick is finding the value of ? so the the pwl() function matches the USATT tables as close a possible. My pwl function looks a lot like the ratings central function. The USATT tables are a joke because they are just that, tables with discrete jumps in rating changes rather a smooth function like my pwl() function and the ratings central function.
So why hasn't anybody found a better solution better than best win and worst loss? You get what you put up with if you are lucky.
