In order to give a more accurate representation of the skill ranking of players in the JBC, I have attempted to adjust the PPMs of the players to account for the fact that the average opponent PPMs are not the same (i.e., you would obviously have a high PPM if you played unskilled players and conversely nearly zero if you always played Climo or Feldberg). From statistics theory we can represent the probability that player A of skill level SA beats player B of skill level SB by the expression,
Here skill levels SA and SB are known or approximated. The exponent K determines how much skill affects the probabilities. Here we approximate skill by PPM and use K=2 (I found values of 2 to 3 in the literature for sports matches). Next I adjust the PPMs by using the probability that the average opponent for a player beats a fictitious player with PPM=0.5 (a 50/50 player). This is done by,
PPM_Adjusted = PPM + P(average opponent beats PPM=0.5) - 0.5
This formula adjusts the PPM up or down depending on whether or not the average opponent PPM for a player is above or below 0.5. This is by no means the only way to do it. I just chose this way because it gave reasonable results. I still need to factor in the players partner in the PPM adjustment for doubles - for now the adjusted PPMs for doubles do not account for the doubles partner.