Direct Comparison - EGF Definition

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Robert Jasiek

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Jun 6, 2009, 1:02:27 PM6/6/09
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Here I explain the meaning of the definition of Direct Comparison in
the EGF Tournament System Rules at
http://home.snafu.de/jasiek/egftsr.html
The definition text is as follows:

* A player's Direct Comparison is the Number of Wins Score of only the
games played against each other.
* If it results in a finer tiebreaking, then the definition is applied
iteratively: If an application still ties some players, then for them
the tiebreaker is applied again, not overwriting but fine-tuning its
previous application. This is sometimes possible if four or more
players are tied before the first application of Direct Comparison.
* However, in a McMahon or Swiss tournament or same stage of a
tournament, the above definition of Direct Comparison is overridden by
giving each player of mutually tied players the value 0 if they all
have not played the same number of games against each other.

The definition presumes an understanding of the Number of Wins Score,
the McMahon tournament system, and the Swiss tournament system. Also
here I assume that the two latter terms are known to the reader. It
should be pointed out though that Direct Comparision is a tiebreaker
that can be applied also in some other tournament systems, in
particular it is pretty common in a round-robin aka league. Although
many readers may know the meaning of Number of Wins Score, to be sure
I explain also this:

The Number of Wins Score of a player is the sum of his game result
values over all rounds of a tournament: 1 per win, 1/2 per jigo, 0 per
loss, etc. You find the complete list in the rules. Example: If, in a
5 rounds tournament, a player has won 3 of his games, made one jigo,
and lost one of his games, then - after the last round - his Number of
Wins Score is 3.5. In a team tournament, substitute "player" by
"team".

Now let us go to the first paragraph of the definition:

* A player's Direct Comparison is the Number of Wins Score of only the
games played against each other.

A particular player is considered in the definition. (Since "a player"
is generic, the definition applies to every player of the tournament
but only one player's Direct Comparison is calculated at a time.)

"Direct Comparison" is the name of the tiebreaker that is being
defined. Quite like one might define SOS in a text like "A player's
SOS is...", here we define Direct Comparison in the text "A player's
Direct Comparison is".

The verb "is" indicates that a definition is being given. At the same
time, the other two paragraphs of the definition should not be
overlooked, of course.

The most difficult part of the first paragraph may be the phrase
"played against each other". Only the "player" is mentioned in the
text, so who the hell is supposed to be playing against each other
here? We know that the thing to be defined is a tiebreaker. The
purpose of a tiebreaker is to make an attempt of breaking a tie among
players that are mutually tied just before application of the
tiebreaker. From this context (another equally likely, possibly
competitive context is unavailable), we imply that "played against
each other" refers to exactly all those players that include the
particular, currently considered player and the other players with
whom he is tied. Since in a tournament a player never plays a game
against himself, he cannot score any wins or jigos from such games.
Points in the Number of Wins Score he can collect only from games
against opponents. Hence, although the particular player is tied with
the other players, it suffices to consider the games he has played
against them.

"the Number of Wins Score of only the games played against each other"
should be clear now: It refers to the particular player's Number of
Wins Score of the games played against those opponents with that he is
still tied just before application of the Direct Comparison
tiebreaker. The "only" expresses that games against other players
(players not being involved in the tie (to be broken and including the
particular player) are ignored.

Example: Round-robin tournament after the last round, Placement
criteria are 1) Number of Wins Score ("Wins") and then 2) Direct
Comparison ("DC"):


Player P1 P2 P3 P4 P5 P6 Wins DC Place
P1 - 1 1 1 1 1 5 - 1
P2 0 - 1 1 1 0 3 1 2
P3 0 0 - 1 1 1 3 0 3
P4 0 0 0 - 1 1 2 - 4
P5 0 0 0 0 - 1 1 1 5
P6 0 1 0 0 0 - 1 0 6


For P1, Direct Comparison is not applied because P1 is not tied on the
Number of Wins Score with any other player. Same for P4.

P2 and P3 are tied on the Number of Wins Score. Therefore, for each of
them, his Direct Comparison value is calculated. The games to be
considered are the games played against each other - this is exactly
the only one game of the pairing P2-P3. So let us consider the
particular player P2. In the game to be considered, he had a win
against P3. Therefore we add 1 for that win to P2's Direct Comparison
value. No more values need to be added since only one game was to be
considered. Thus P2's Direct Comparison sum value is 1. Let us now
consider P3 as the particular player. The same game is considered but,
for P3, it was a loss, i.e., 0 is added to his Direct Comparison
value. Since this is the only game to be considered, we have 0 as P3's
Direct Comparison value.

P5 and P6 are tied on the Number of Wins Score. The only interesting
game is from the pairing P5-P6. We proceed with the calculations quite
like we proceeded for P2 and P3.

Next we consider the second paragraph of the definition:

* If it results in a finer tiebreaking, then the definition is applied
iteratively: If an application still ties some players, then for them
the tiebreaker is applied again, not overwriting but fine-tuning its
previous application. This is sometimes possible if four or more
players are tied before the first application of Direct Comparison.

A typical application is in EGC side tournaments, which start with a
first stage in that groups of players each play a league. It has been
a tradition to apply Direct Comparison iteratively there. Therefore
this practice has found its way also into the definition.

It would not help much to try explaining the definition text with a
lot of words. An example is probably easier:


Player P1 P2 P3 P4 P5 P6 P7 P8 Wins DC1 DC2 Place
P1 - 1 1 0 0 1 1 0 4 2 1 3 (shared)
P2 0 - 1 0 0 1 1 1 4 1 - 6
P3 0 0 - 1 1 1 1 0 4 2 1 3 (shared)
P4 1 1 0 - 1 1 0 0 4 3 - 2
P5 1 1 0 0 - 1 1 0 4 2 1 3 (shared)
P6 0 0 0 0 0 - 0 0 0 - - 8
P7 0 0 0 1 0 1 - 0 2 - - 7
P8 1 0 1 1 1 1 1 - 6 - - 1


Players P1 to P5 are tied on Wins. Their games played against each
other are considered for Direct Comparison in its first iteration
step: P1-P2, P1-P3, P1-P4, P1-P5, P2-P3, P2-P4, P2-P5, P3-P4, P3-P5,
P4-P5. For each particular player of these five players, the sum of
their number of wins made IN THOSE GAMES is formed. This is given in
the column DC1.

Then still three of the players P1 to P5 are tied: The players P1, P3,
P5. So a next iteration step of Direct Comparion Application is
performed to them. It considers only these games: P1-P3, P1-P5, P3-P5.
The result is shown in the column DC2.

The Place is determined by first looking at Wins, then at DC1, then at
DC2. Unfortunately, even iterative application of Direct Comparison
has not broken all the ties since during the second iteration step a
so called three-way-tie has occurred between the players P1, P3, P5.

Consider a slightly different example:


Player P1 P2 P3 P4 P5 P6 P7 P8 Wins DC1 DC2 Place
P1 - 1 1 0 1 1 0 0 4 3 0 3
P2 0 - 1 0 0 1 1 1 4 1 0 6
P3 0 0 - 1 1 1 1 0 4 2 - 4
P4 1 1 0 - 1 1 0 0 4 3 1 2
P5 0 1 0 0 - 1 1 1 4 1 1 5
P6 0 0 0 0 0 - 0 0 0 - - 8
P7 1 0 0 1 0 1 - 0 3 - - 7
P8 1 0 1 1 0 1 1 - 5 - - 1


During the second iteration step of Direct Comparison Application, we
have the still tied players P1 and P4 and consider only their game
P1-P4, which P4 won. Therefore P4 gets a 1 and P1 gets a 0 in the DC2
column.

Furthermore we have the still tied players P2 and P5 and consider only
their game P2-P5, which P5 won. Therefore P5 gets a 1 and P2 gets a 0
in the DC2 column.

This time, iteratively applied Direct Comparison dissolves all ties.

Finally let us move to the third definition paragraph:

* However, in a McMahon or Swiss tournament or same stage of a
tournament, the above definition of Direct Comparison is overridden by
giving each player of mutually tied players the value 0 if they all
have not played the same number of games against each other.

While a round-robin or multiple round-robin tournament lets all
players play the same number of games against each other (even if we
consider only a subset of the players), in a tournament with a Swiss
or McMahon system, a set of mutually tied players might not have
played the same number (in almost all cases: 1) of games against each
other. If exactly two players are tied, then they have played the same
of games against each other: the one game in that they were paired
against each other. With 3 or more tied players, a complete implied
league among them is scarce but possible in principle.

Obviously it would be unfair to apply Direct Comparison to an
incomplete league. Therefore the tiebreaker is applied only trivially
then: each player involved in such a tie is given the default Direct
Comparison value of 0.

Example: Pairings of some players that are tied on Wins in a Swiss
tournament (X means "pairing did occur", % means "pairing did not
occur):


P1 P2 P3 P4
P1 - X X X
P2 X - X X
P3 X X - %
P4 X X % -


Since the pairing P3-P4 did not occur, all the players P1 to P4 get 0
as their Direct Comparison value.

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