1. Treatment of unplayed games for Buchholz tie break: the FIDE rule
by Luigi Forlano

2. Why we need adjustments
A Swiss tournament is a statistical system in which an equal, limited number of games is played by each participant
Tiebreak systems based on the sum of opponents’ scores are not fair to the players who did not play one or more of the games – particularly when those players were not responsible for the unplayed game!
Each unplayed game has consequence on the player himself and on all of his opponents
Hence, it is fair to assign to each unplayed game a reasonable result, to produce a standing not distorted by unfair side effects caused by unplayed games
The new result should turn out to be neither an advantage nor a punishment for those players whose standing have been affected by the unplayed game, but just an adjustment to let the tiebreak system work normally.

3. The FIDE rule for Buchholz
For tiebreak purposes, if a Player A does not play at a given round it is necessary to determine:
(a) the result of the unplayed game, and (b) the tiebreak points got by his opponents for the unplayed game.

4. (a) Result of unplayed games
The purpose is achieved by means of a virtual opponent V.
1. at the beginning of the round, V has as many points as player A.
2. The result by default of player A is treated as a normal result:
If player A is absent then V wins: A – V = 0 – 1
If player A wins (because of a BYE or due to the absence of his opponent) then V loses: A – V = 1 – 0
If player A draws (1/2 point BYE) then V too draws: A – V = 1/2 – 1/2;
3. For each subsequent round, V gains half a point.

5. (b) Points got by A’s opponents
In order to reducing the consequences for the A’s opponents, each unplayed game is counted as a draw.
This means that the total score of player A to be used by his opponents for Buchholz calculation needs to be adjusted: all the wins and losses by default should be considered as draws.

6. Example 1
In a 9 round Swiss, player A achieves 6 points, including a default win in round 3.
After round 2, A had a 2 points score.
His virtual opponent gets: x 0.5 = 5 points Buchholz
The contribution of A for his opponents’ Buchholz is 5.5 = 5 + ½ instead of 6
Draw for the next 6 rounds

7. Example 2
In a 9 round Swiss, player A was absent in round 7 and scored 6 points after round 9.
After round 6, A had 4 points.
His virtual opponent get : x 0.5 = 6 points Buchholz
Draw for the next 2 rounds
The contribution of A for his opponents’ Buchholz is 6.5 = 6 + ½ instead of 6

8. A complete case study We must adjust all these results
ID NAME Pts |
1 ANDREA | +W5 =B4 -W2 -B9 +W8
2 BRUNO | +B6 =W9 +B1 -W3 +B7
3 CARLO | +W7 -B5 +BYE +B2 +W4
4 DARIO | +B8 =W1 -B9 =W5 -B3
5 FRANCESCO | -B1 +W3 -BYE =B4 -W6
6 GIORGIO | -W2 -B7 +BYE -BYE +B5
7 LUIGI | -B3 +W6 -BYE +BYE -W2
8 MARIO | -W4 +BYE -BYE +BYE -B1
9 ROBERTO (withdrawn) 3.5 | +BYE =B2 +W4 +W
We must adjust all these results
First let’s consider all of them DRAW and calculate the new scores…

9. A complete case study
ID NAME Pts |
1 ANDREA | +W5 =B4 -W2 -B9 +W8
2 BRUNO | +B6 =W9 +B1 -W3 +B7
3 CARLO | +W7 -B5 = +B2 +W4
4 DARIO | +B8 =W1 -B9 =W5 -B3
5 FRANCESCO | -B1 +W3 = =B4 -W6
6 GIORGIO | -W2 -B7 = = +B5
7 LUIGI | -B3 +W6 = = -W2
8 MARIO | -W4 = = = -B1
9 ROBERTO (withdrawn) 3.5 | = =B2 +W4 +W1 =
New Pts
-----
2.5
3.5
2.0
1.5
Now let’s calculate the score of the virtual opponent…

10. A complete case study Now we can calculate the Buchholz as usual
ID NAME Pts |
1 ANDREA | +W5 =B4 -W2 -B9 +W8
2 BRUNO | +B6 =W9 +B1 -W3 +B7
3 CARLO | +W7 -B B2 +W4
4 DARIO | +B8 =W1 -B9 =W5 -B3
5 FRANCESCO | -B1 +W =B4 -W6
6 GIORGIO | -W2 -B B5
7 LUIGI | -B3 +W W2
8 MARIO | -W B1
9 ROBERTO (withdrawn) 3.5 | =B2 +W4 +W
New Pts
2.5
3.5
2.0
1.5
Buc Tot
12.5
13.5
11.5
13.0
11.0
10.5
14.5
Now we can calculate the Buchholz as usual

11. A complete case study 3.5 + 2 + 3 + 1.5 + 3.5 = 13.5
ID NAME Pts |
1 ANDREA | +W5 =B4 -W2 -B9 +W8
2 BRUNO | +B6 =W9 +B1 -W3 +B7
3 CARLO | +W7 -B B2 +W4
4 DARIO | +B8 =W1 -B9 =W5 -B3
5 FRANCESCO | -B1 +W =B4 -W6
6 GIORGIO | -W2 -B B5
7 LUIGI | -B3 +W W2
8 MARIO | -W B1
9 ROBERTO (withdrawn) 3.5 | =B2 +W4 +W
New Pts
2.5
3.5
2.0
1.5
Buc Tot
12.5
13.5
11.5
13.0
11.0
10.5
14.5
For example for player 7 that had two virtual opponents we have
= 13.5