2008-03-01, Robert Jasiek

Quality of Direct Comparison



Only a Player's Own Performance Affects His Direct Comparison Value

Only a player's own performance during the tournament affects his Direct Comparison value. Round numbers, pairings, opponents' performance against third players do not affect it. Only the player himself is responsible for it.

Meaningful Design Aim

For players tied at the top of the final result table, the design aim of the tiebreaker Direct Comparison is very meaningful: These tied players are the players still competing for place 1, so they are compared among each other. This comparison is like a subtournament in that only their performance against each other is considered. If a playoff tournament were to be hold, then these players should be its participants. In such a playoff, only their performance against each other would matter. The design aim of Direct Comparison agrees to that nature.

The Numerical Precision is the Significance

Direct Comparison is exactly as accurate as it pretends to be: Its precision and its significance are 1 if there are no jigos (or default jigos). They are 1/2 if there are jigos. The significance errors of SOS do not have any impact on a player's Direct Comparison value. Other significance errors with a numerically relevant impact on the latter are not known. In other words, Direct Comparison values do not contain any noise.


Scarcely Breaks Ties in Some Tournament Systems

In McMahon or Swiss tournaments, Direct Comparison scarcely breaks ties because often not all players of a tie group have played against each other. One should note, however, that breaking ties for the typically small score group at the top is possible more often. E.g., if exactly two players are tied at the top, then their tie is broken unless they had a jigo. If exactly three players are tied, then one player might have beaten both competitors (the tie is dissolved) or there can be a three-way-tie (cycle; the tie remains). Future research should study how frequently Direct Comparion breaks a tie in the final top score group of a McMahon or Swiss tournament. In round-robin tournaments, ties are broken more often than not.

Only Part of a Player's Performance Is Considered

Direct Comparison ignores the games against opponents not in the group of tied players. The games considered by Direct Comparison, those against opponents in the group of tied players, are effectively evaluated for the second time because they have already been considered for the score criterion. Apart from the design aim of a tiebreaker, it is unclear why some games of a player should be evaluated the more often than other games.

Potentially Great Impact of a Significance Error

Should some significance error with a numerical impact on a player's Direct Comparison value exist at all, then it would be relatively great in comparison to the value's size. So far this is an only theoretical danger.


Advantages and disadvantages are in a balance. This can be said of only few tiebreakers and so Direct Comparison is among those being worth considered if ties need to be broken at all.

In round-robin tournaments, Direct Comparison is clearly the first choice. There also SODOS is popular but why should one's won games be more important than one's lost games and why not vice versa? Direct Comparison does not have this disadvantage.

In McMahon or Swiss tournaments and for the purpose of breaking the final top score group's tie, opinions vary greatly whether Direct Comparison or SOS (or one of its derivates) is the better tiebreaker. The author's experience is as follows:

If the top score group was large, then SOS felt like a random lottery. Getting place 1 due to SOS was no joy. Direct Comparison would have been inapplicable, but at least it would not have been associated with an awkward feeling of arbitrary ordering.

In a big 5 rounds McMahon tournament, the author lost in round 1 and then fought his way back. Another player had won his first 4 games. They were paired against each other in the last round. The author won and so both were tied on McMahonScore. The used tiebreaker SOS gave the author 2nd place. Had Direct Comparison been used instead, it would have been 1st place. The author felt and still feels very strongly that usage of SOS had been unfair because it rewarded winning in earlier rounds more than the decisive direct encounter of the top two players. What are round numbers compared to the most important game of the tournament? Nothing.