# Sonneborn-Berger

__Keywords__: Tournament

Sonneborn-Berger is a method to determine a tournament result if two players have an equal number of wins. It takes the opponents' wins into account.

Example: Each player has played two games against each other player. 1 indicates a win, 0 a loss.

+---------+---------+------------------+--------------+ | 1st | 2nd | won games | Sonneborn- | | round | round | | Berger | |---------+---------| | | | A B C D | A B C D | wins | opponent | wins | +---+---------+---------+-------+----------+--------------| | A | - 1 1 1 | - 0 1 0 | 4 | B C C D | 4+3+3+1 = 11 | | B | 0 - 1 1 | 1 - 0 1 | 4 | A C D D | 4+3+1+1 = 9 | | C | 0 0 - 1 | 0 1 - 1 | 3 | B D D | 4+1+1 = 6 | | D | 0 0 0 - | 1 0 0 - | 1 | A | 4 = 4 | +---+---------+---------+-------+----------+--------------+

Player A and B both have 4 wins, thus the Sonneborn-Berger method is used to determine the tournament winner: For each won game the opponent's final primary score is added to the winner's Sonneborn-Berger score. For example, A won against B, C, C and D, his Sonneborn-Berger score is 4+3+3+1 = 11. B has a Sonneborn-Berger score of 9, thus A is the tournament winner.

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### Discussion

Dieter: Is this what we call SOS (Sum of Opponent's Scores) in MacMahon - The Program ?

SAS: No, it's what is sometimes called SODOS (Sum Of Defeated Opponents' Scores), at least in the absence of draws. SOS makes more sense than Sonneborn-Berger for McMahon Pairing tournaments. Sonneborn-Berger is mainly used for all-play-all tournaments, where SOS doesn't work at all.

Tim Brent: There is also a modified version at Little Golem where the score is double the opponent's points (i.e. if second has 2 points and first won, the Sonneborn-Berger score is 4).

MRFvR: The true definition of Sonneborn-Berger is the sum (for all opponents) of the product of the points you achieved against an opponent for the total points achieved by that opponent. Thus, in Little Golem, where victory is valued 2 points, that gives the double of opponent's points