Chess tie break.

Now day, can considered all chess tournament are using the Swiss Pairing System.
Swiss Pairing System is designed to maximize the fair treatment of the players. This means that players having the same score should have met as equal opposition as possible during a tournament. If the Sonnenborn-Berger and/or Buchholz and/or Median, of all players in the same score-group, is nearly equal, the goal is reached. As a Swiss System is a more or less statistical system, this goal can only be reached approximately.
The approach is the attempt to equalize the strengh of the opponents of all players in a given score group. Therefore the pairing of each round will tend to pair players who have high Sonnenborn-Berger (or Buchholz or Median) with players having low Sonnenborn Berger (or Bucchholz or Median) in the same score-group.
The ratings of the players should be taken into consideration only when the Sonnenborn-Berger (or Buchholz or Median) is equal (e.g. in the first two rounds); otherwise, only current data of the tournament itself should be the basis for measuring the "strength" of the players and making the pairings thereafter -Fide handbook.

Basic pairings principles:

• Two players who have played each other shall not be paired again. 

• Before making the pairings in each round, players in every SG (including "floaters" from another SG) shall be arranged in the order of their (1) Sonneborn-Berger (SB); (2) Buchholz; (3) Median (4) rating. The player with the highest SB shall be No. 1 in the SG. Players with the same SB shall be arranged in the order of their Buchholz and so on.

• For accelerating pairing, in the first two rounds, an ‘imaginary’ point shall be added to the score of each of the players in the top half of the initial list of participants (arranged in the order of their R). This imaginary point shall then be deducted before making the pairings of the third round.

Berger or Sonneborn-Berger
This is calculated by adding scores of the opponets who were beaten by a given player and half the scores of the opponents who she drew with. This has been adopted from round-robin tournaments and is usually used as a secondary method.


Buchholz (FIDE) or Solkoff 
This is the sum of opponents' scores. The idea is that the same score is more valuable if achieved against players with better performances in a given tournament. Looks like an ideal tie-breaking method and has been used since the Swiss system was invented. However it has some weaknesses which are addressed by other methods (see Median-Buchholz, Progress, Berger).

Median-Buchholz (FIDE) or Median
Same as above but discarding the highest and the lowest opposition's scores.
Its idea is to eliminate distortions in Buchholz values caused by taking into account games against run-away winners and bottom placed players.

example:
Suppose a player participated in a Swiss tournament and he scored as shown in the table below.

Example: Player #57 faced the players mentioned in the first line. In the second line the score of #57 against these opponents is listed, and in the third line you find the total scores of his opponents.
The Buchholz score of player #57 is the sum of the scores: 3½ + 4½ + 2 + 5 + 4 + 6 + 5½ = 36.
The Median Buchholz is the Buchholz score reduced by the highest and the lowest score of the opponents.
 In Table 1 we would have to reduce the scores of #8 (2 points) and #28 (6 points).
The Median Buchholz is in this case: 3½ + 4½ + 5 + 4 + 5½ = 22½.

The following is very important and also a little bit complicated:

Progress (FIDE) or Cumulative 
Calculated by adding points from a progress table eg if your scores were: Win, Loss, Win, Draw then your progressive scores are 1, 1, 2, 2.5 and your Progress tie-break value is 6.5
This is an attempt to put a higher value on scores which were achieved by scoring better in the initial rounds than by finishing from behind. It is common knowledge that the latter is usually much easier to achieve.
The problem is that the order of the Progress tie-breaks is known before the last round (last round scores will change the actual value but not the order within a point group). This may encourage some undesirable tournament "tactics" in the last round.