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I am trying to grok the fide specification of the swiss pairing algorithm, having a hard time with that. For example rule A.4.b reads:

A4
[...]
b
After two players with different scores have played each other in a round, the higher ranked 
player receives a downfloat, the lower one an upfloat.
[...]

According to definition A.2 the player rank changes dynamically, round after round. My main question here is, at what time the rank of the two players is considered, before or after the round? I personally see arguments for both interpretations.

If I understand correctly, swiss system tries very hard, not to pair moved down players against each other, this is only considered if all other possibilities are exhausted. Hence if a player was moved down because of rule A.4.b or for some other reason he will be likely paired against a player with different score which is lower ranked before the game and subsequently be eligible for another application of rule A.4.b, either unconditionally or if he stays on top of this opponent. Hence because of rule A.4.b I expect downfloats to be positively correlated, which seems to contradict the main pairing construction.

It would be helpful if someone with experience in this matters, could either confirm or correct this interpretation and provide some intuition, for what rule A.4.b was introduced for.

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After two players with different scores have played each other in a round, the higher ranked player receives a downfloat, the lower one an upfloat.

This is just marking the fact that the higher ranked player had to downfloat and the lower ranked player had to upfloat. The algorithm tries to minimize the number of times this happens to the same player, at least for the next couple of rounds.

It uses the ranking from the start of the round, not the one after the game is completed. It would make little sense to consider an upfloated player to have downfloated just because they won the game. I imagine that it's applied after the game just to take into account the possibility of a forfeit; you wouldn't get an upfloat or downfloat marked to you just because you were paired that way if you never actually played the game.

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    Only one of the players is moved down to the next bracket for purposes of the algorithm, but both players are playing someone outside their score group, so both are marked with a float.
    – D M
    Jul 12 at 1:36
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According to definition A.2 the player rank changes dynamically, round after round

No, I don't think you fully understand what is going on. During the pairing process for a particular round in a given score group the ranking within a score group is fixed according to what the original ranking before the first round.

A.2 Order

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

If there are no new players to add and no initial entry errors to correct then the initial rankings apply for the complete tournament.

There are only two situations which can give rise to changes to these rankings. These are:

  1. a new player joins the tournament after the initial ranking. In this case the ranking list is changed by inserting the new player in their slot according to their rating.
  2. an error is discovered in the rating entered for a player in the initial ranking. In this case the rating is corrected and if this changes the place of the player in the ranking then this change in position is confirmed.

An example might make it clearer. Suppose in the first round of an 8 player Swiss, with players called in ranking order, "1", "2", "3", "4", "5", "6", "7", "8", with player 1 having white and where all the white players win. Then for the second round there will be 2 score groups with the 2 groups having the following internal rankings.

Score Group 1 point
1
3
5
7

Score Group 0 points
2
4
6
8

Suppose then that a new player, player 9, joins with a rating between player 2 and 3. Then the new overall ranking would be: 1, 2, 9, 3, 4, 5, 6, 7, 8 and this ranking would remain for the rest of the tournament. However for round 2 this would change the second score group as follows:

Score Group 0 points
2
9
4
6
8

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