How do I determine the bye player in a Swiss tournament?

Question

The FIDE basic rules for Swiss systems mention:

(c) Should the number of players to be paired be odd, one player is unpaired. This player receives a pairing-allocated bye: no opponent, no colour and as many points as are rewarded for a win, unless the rules of the tournament state otherwise.

(d) A player who has already received a pairing-allocated bye, or has already scored a (forfeit) win due to an opponent not appearing in time, shall not receive the pairing-allocated bye.

However, who gets the bye is never spoken of. So my questions are, how do I determine the bye player:

• in the first round, and
• in other rounds?

I am looking for how FIDE deals with it, and not any other Swiss system.

Answer 1

In the first round, it should go to the player with the lowest initial ranking. After that, it depends, but it would ordinarily go to a low ranked player in the lowest score group.

According to rule A.2:

For pairings purposes only, the players are ranked in order of, respectively
(a) score
(b) pairing numbers assigned to the players accordingly to the initial ranking list and subsequent modifications depending on possible late entries or rating adjustments

Rule B.2 says:

To make the pairing, each bracket will be usually divided into two subgroups, called S1 and S2.
S1 initially contains the highest N1 players (sorted according to A.2), where N1 is either M1 (in a heterogeneous bracket) or MaxPairs (otherwise).
S2 initially contains all the remaining resident players.

And rule B.3 says:

S1 players are tentatively paired with S2 players, the first one from S1 with the first one from S2, the second one from S1 with the second one from S2 and so on.

There is ordinarily no reason to change the tentative pairings in the first round, and everyone's score is 0. If there are an odd number of players, then S2 will contain one more player than S1. When you pair S1 with S2 as the rule says, there will be one player left over, and this player receives the bye. So, the bye in the first round would ordinarily go to the last player in S2, which in this case is simply the player with the lowest initial ranking.

In later rounds, however, those tentative pairings could change. As you noted, the bye cannot go to the same player twice. Other pairing rules also come into play; for example, if switching who gets the bye will improve the number of players who get the color they are due, then it will be switched. You may have to pair everyone before you can confidently determine who gets the bye in later rounds.

The bye would ordinarily go to someone in the lowest score group - however, it is also possible that score groups will collapse into each other in certain circumstances (see rule A.9), and this could result in someone with a higher score getting the bye instead of someone with a lower score, even if neither has yet received a bye.

Answer 2

Note:   Independently from the route followed, the assignment of the pairing-allocated bye (see C.2) is part of the pairing of the last bracket.

https://www.fide.com/component/handbook/?id=170&view=article

In simple terms, this means that the bye normally goes to one of the players currently tied for last place, but there may be complications if this conflicts with other rules (for example, if every player in the last bracket already got a bye).

It is not stated, but I think that it makes sense to give the bye to someone in the last bracket because that way the bye is least likely to influence who wins the tournament.