The basic choices the give a bye (at the first round) are: the lowest player, the middle player and the top player. The basic choices to score a bye are 0, ½ or 1 point (win). Suppose we prefer top players to play more games compared to bottom players and we like to have fair pairings. The first criteria excludes the top player from having a bye.
The ideal result at the end of the tournament would be that the players with byes are all at the bottom of the result list. One way to achieve this is to give bye always to the weakest player, who yet haven't had a bye. However, if the weakest player gets a point after the first round from a bye, the pairings for the next round would be slightly unfair or suboptimal. In this case giving zero points works better. However getting a bye and no points might feel like a double punishment to players.
A better way could be to choose the middle player for a bye at the first round. You did not specify how many rounds are played in the tournament. If the number of players is no more than double compared to the number of rounds, this works well. The middle player and all below him would have received byes by the end of the tournament. If we had 11 players and 4 or 5 rounds, I might select the player at the 3rd quartile, the 8th for a bye. He would be low enought to deserve a bye, but high enough to get a fair pairing after the second round.
On subsequent rounds I would give the bye to a player near the middle of the lowest score group, who still hasn't got a bye.
The common and simple way the count the SOS is the use the players own score for bye round(s). Of course one could develop a statistical model to make a better estimate, butt maybe it is an overkill.
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