Tournaments 
Highways and Byways of Byes

From Backgammon Times, Volume 1, Number 1, September 1981. 
Backgammon players who enter tournaments often complain about byes. They think they don't get enough byes or that somebody else gets too many. If byes are given randomly, then it is certain that at least some of the complainers are right. If more players understood the value of byes, they would complain even more vehemently.
The effect of a bye is to double the recipient's chances and equity. This assumes that each player has a 50–50 chance to win a match against an opponent chosen at random. Players with better than 50% chances benefit slightly less from byes, players with poorer than 50% chances benefit slightly more in proportion to their chances of winning the match cancelled by the bye.
Let's suppose that in a tournament with an entry fee of $100, the promoters rake in 20% for themselves and return 80% to the players in the prize pool. An entrant's equity, before his firstround match, may be $80. But if byes are given in the first round, the recipient of a bye may have a $150 equity while another player who is forced to play his firstround match has only $75 equity. Similarly, if byes are given in the second round, then the firstround winner who must play a secondround match has only $150.
It is reasonable to assign byes randomly, but only among players who enter the tournament promptly, excluding latecomers. From one point of view, this looks like a reward for promptness. But it is also a requirement for fairness. Otherwise one entrant may inadvertently receive two byes while other entrants receive none, as happened in a recent regional tournament.
An outoftown player registered in advance, sending in a check for his entry fee. His plane had not yet arrived when the firstround pairings were drawn. He was given a bye in a random draw. Bracket sheets were then drawn, and the player was paired in the second round against another player who had a firstround bye. Then all the players with firstround byes started their secondround matches ahead of time.
The outoftowner never arrived. Perhaps he changed his mind about coming to the tournament. Perhaps his phone message never got to the tournament director when he cancelled his registration. Perhaps, tragically, due to no fault of his own, his plane crashed. No matter. His secondround opponent received a second bye. Since this opponent was a player with considerable power and influence within local backgammon circles, the two byes raised suspicions of favoritism and reflected poorly on the tournament director.
For this reason, any player not yet present must be excluded from the drawing for byes.
There is an alternative method of dealing with byes with avoids inequities. Again, let's suppose there is a 20% rake from a $100 fee, with $80 going into the prize pool. The promoters announce a prize pool of $2560 if the field reaches 32, $5120 if the field reaches 64 entrants. Any intermediate field size will result in the same prize pool as the next lower power of 2.
Now let's suppose that 48 players enter—32 of them will play firstround matches, 16 will receive byes. Does this give to those 16 twice the equity of the other 32? Very well. The director now rebates to the unfortunate 32 half of the entry fee; each pays $50 instead of $100, and 80%—$40—goes into the prize pool, which totals 16 × $80 + 32 × $40 = the $2560 promised for a field of 32. The two groups of entrants have different equities, but each gets equity in proportion to the entry fee paid.
If this system is adopted, byes need no longer be assigned randomly. "Antibyes" can be offered to those players desiring to play the extra match, or to players who are willing to accept a smaller chance of winning a prize in exchange for a lower entry fee. Of course, within each group, those receiving byes and those receiving antibyes, the pairings will be random, both in the first round and in subsequent rounds.
Not only is this system unquestionably equitable, but it also permits the tournament organizers to cater to the preferences of their patrons.
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