Why not just use the μ (average skill) for the rankings?
For the most part, a ranking that used just the μ value would be the same as subtracting 3 * σ. The only differences appear for new players and cases where one player has played more than another. The σ value will eventually flatten out at around 2 – 3 in this system, so people that play a lot will all be ranked according to their average skill, but when a player is just starting out, their rank will jump up and down a lot until it starts to stabalize on their real skill level. It’s better to put an uncertain player at the bottom of their point range, than to put them up higher where more established players will feel their rank is cheapened by being ranked the same. This method also is more heartening for a new player as when playing more games, their rank will keep going up until their rank is more certain, instead of hitting a point and then hopping around a lot.
