The Buchholz system is a ranking system in chess developed by Bruno Buchholz in 1932 in order to determine ranks in a Swiss system tournament where players have the same score. It sums up the score of the players' opponents and thus favors those who have confronted better opponents.
The major criticism of this system is that tie-break scores can be distorted by the set of opponents that each player plays (especially in early rounds). To avoid this problem a version of Buchholz, the Median-Buchholz System is sometimes used. In the Median-Buchholz System the best and worst scores of a player's opponents are discarded, and the remaining scores summed.
As an example of the system in action, here is the cross-table of the eighth Correspondence Chess World Championship Final (here cs indicates conventional score, ns Neustadtl score, bu Buchholz score, mb Median-Buchholz score):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cs ns bu mb
1. Sloth X ½ ½ 1 ½ ½ 1 1 ½ 1 ½ 1 1 1 1 11 69.5 94 82
2. Zagorovsky ½ X 0 ½ 1 ½ 1 1 1 ½ 1 1 1 1 1 11 66.75 83.5 72.5
3. Kosenkov ½ 1 X ½ ½ ½ ½ ½ 1 1 ½ 1 1 1 1 10½ 67.5 94.5 82.5
4. Khasin 0 ½ ½ X ½ 1 ½ 0 1 1 ½ 1 ½ 1 ½ 8½ 54.75 78.5 67.5
5. Kletsel ½ 0 ½ ½ X ½ ½ ½ ½ 0 1 1 ½ 1 1 8 47.75 79 68
6. De Carbonnel ½ ½ ½ 0 ½ X ½ ½ 0 1 ½ ½ 0 1 1 7 45.25 78 67
7. Arnlind 0 0 ½ ½ ½ ½ X ½ 1 0 ½ ½ 1 1 ½ 7 42.5 69 58.5
8. Dunhaupt 0 0 ½ 1 ½ ½ ½ X 0 ½ 1 0 1 ½ 1 7 41.5 63.5 53
9. Maedler ½ 0 0 0 ½ 1 0 1 X 1 ½ ½ ½ ½ 1 7 41.5 61 50
10. Estrin 0 ½ 0 0 1 0 1 ½ 0 X 1 1 1 0 1 7 40.5 49.5 38.5
11. Walther ½ 0 ½ ½ 0 ½ ½ 0 ½ 0 X 0 1 ½ 1 5½ 33.25 61 50
12. Boey 0 0 0 0 0 ½ ½ 1 ½ 0 1 X ½ ½ 1 5½ 28.5 43.5 36.5
13. Abramov 0 0 0 ½ ½ 1 0 0 ½ 0 0 ½ X ½ 1 4½ 24.75 41.5 33
14. Siklos 0 0 0 0 0 0 0 ½ ½ 1 ½ ½ ½ X 1 4½ 22.75 37.5 30.5
15. Nun 0 0 0 ½ 0 0 ½ 0 0 0 0 0 0 0 X 1 7.75 15.5 7
Although the Neustadtl score settles most of the ties created by the conventional scoring of points, there was still a tie between Dunhaupt and Maedler. Buchholz and Median-Buchholz scores gave Dunhaupt a higher ranking, thus breaking the tie.