Forum for Direct Comparison
DC finds difference in identical performances, based on irrelevant games. [#2123]
Suppose a round robin tournament, where the games are played one after another. It begins with three players who play against each other:
- A wins B
- B wins C
- C wins A
The situation is clearly a three-way tie, that is, every player has performed equally well.
Now enter player D. First he loses to A and B.
- A wins D
- B wins D
Now, in this phase too, A and B have performed equally well, so surely they should end up tied for the first place, with C possibly joining the club if he wins the final game.
Should C also win D, then there would indeed be a three way tie even if Direct Comparison was used. But what happens in the other case?
- D wins C
Oh noes. C lost his game. Now B is royally screwed: his win against C no longer counts for the Direct Comparison tie breaker, so A is declared the winner of the tournament.
So Direct Comparison breaks the tie between identically performing players A and B. Furthermore, it does so solely based on a game between C and D.
-Bass, 2010-01-02
Addendum: Other tie breakers, like SODOS, will also flub this one. Notably, SOS will say that the result is a two-way tie.
Let us remove all the fog candles you throw and concentrate on substance: You have declared the tournament system to be Round-Robin. Under this system, the time order in that the games were played are immaterial. The final placements are determined after the last round. So let us do exactly that for your preferred outcome study case:
| Name | A | B | C | D | Wins | DC |
|---|---|---|---|---|---|---|
| A | X | 1 | 0 | 1 | 2 | 1 |
| B | 0 | X | 1 | 1 | 2 | 0 |
| C | 1 | 0 | X | 0 | 1 | 0 |
| D | 0 | 0 | 1 | X | 1 | 1 |
Players A and B have not (as you claim) performed identically - they have got the same Number of Wins Score. But they have played against different sets of players: {B, C, D} versus {A, C, D}. Therefore also your usage of "Furthermore" is misleading and your headline "DC finds difference in identical performances [...]" is wrong. It is correct to say that "DC might distinguish players with identical Numbers of Wins Scores".
You are right that DC distinguishes the placements of A and B only by one game (the game they played against each other). This is an example for the disadvantage of DC that it evaluates for a second time the games of the subset of the games of the players mutually tied by the Number of Wins Score.
Your headline contains another statement: "based on irrelevant games". Here you have to explain why some games were irrelevant and which. Since all games are relevant for the Number of Wins Score and since DC is applied only after application of the former, none of the games is irrelevant. Instead of being entirely irrelevant, some of the games in the examples are not evaluated for a second time. Those games have, for application of the placement criteria 1) Number of Wins Score, then 2) DC, a _smaller_ relevance than the game that is evaluated for a second time. "smaller" is not the same as irrelevant! What we see here once more is the example for the disadvantage of DC that it evaluates for a second time the games of the subset of the games of the players mutually tied by the Number of Wins Score.
You enlighten that one disadvantage by throwing light on it from different angles. It remains the same disadvantage though, which we all know already.
BTW, also SOS has this disadvantage when being used as a tiebreaker to distinguish players in one score group: It evaluates the games of a player's opponents while it does not evaluate many of the other games of the tournament. I.e., different games are evaluated with a different weight. If players A and B have played the same opponent C, then the games A-C and B-C even get the weight 2 (they are evaluated twice) while other games have the weight 1 or 0. In your own words, SOS used as a tiebreaker has a basic statistical measurement error. It does not matter that SOS if not being used as a tiebreaker does not have this error. As soon as SOS is being used as a tiebreaker to distinguish players of the same score group, quite some games are disregarded by SOS.
It bothers you that a different outcome of a game could affect DC? Ok, let us make the same complaint about SOS, SOSOS and other tiebreakers, too.
In ![[ext]](images/extlink.gif)
your evaluation of the DC tie breaker the first advantage is stated as such:
"Only a Player's Own Performance Affects His Direct Comparison Value"
Given that neither A or B is involved in the game between C and D which decides whether A wins or if there is a tie, that statement is outright false.
-Bass, 2010-01-02
You have overlooked the context in that I have written my statement: On my page, I declare the most relevant context directly: "here Direct Comparison is studied only when being applied to the final tournament results."
At the moment of application of DC, i.e., after the tournament's last round, "Only a Player's Own Performance Affects His Direct Comparison Value" is correct.
(For different purposes, one might apply DC during the progress of a tournament. In that case, my webpage does not (necessarily) apply. Similarly, one might apply SOS during a tournament, e.g., for making pairings. In that case, such an application of SOS is not to the final tournament results but is to the making of pairings. Each tiebreaker must be studied (partly) afresh for each different purpose of application, each tournament system, each (possibly dynamic) number of rounds, and each (possibly dynamic) number of players in the tournament.)
Depends on your definition of affects.
I think a reasonable definition would be:
Suppose A and B are tied (eg on MMS). We can calculate several tie breaker values.
The tie breaker value is affected by a game (including any unrelated game C-D) if changing the result of that game would change the value of the tie breaker.
So which tie breakers are affected by unrelated games, and which are not?
SOS is affected. If A played C but not D, then changing the result C-D will change the SOS value for A (same for B if he played C or D)
DC is also affected (suppose A>B, B>C, C>A, D>C, ABC won all other games, D lost several). In this scenario A has DC 1, B has DC 0. Change the result to C>D, and A has DC 1, B has DC 1 (and C has DC 1, as he has entered the tied group). So B's DC value was affected by the outcome of the unrelated game C-D
An example of tie breakers which are not affected are CUSS, SOL or prior rating.
(Note that MMS is also a value not affected by unrelated games)
Okay, you must be trying to troll me now, but since I am that kind of a person, I'll bite.
Let's have these two tournaments:
| \ | A | B | C | D | Placement |
|---|---|---|---|---|---|
| A | \ | 1 | 0 | 1 | 1. |
| B | 0 | \ | 1 | 1 | 1. |
| C | 1 | 0 | \ | 1 | 1. |
| D | 0 | 0 | 0 | \ | 4. |
| \ | A | B | C | D | Placement |
|---|---|---|---|---|---|
| A | \ | 1 | 0 | 1 | 1. |
| B | 0 | \ | 1 | 1 | 2. |
| C | 1 | 0 | \ | 0 | 4. |
| D | 0 | 0 | 1 | \ | 3. |
Notice how the only thing changed is the result of C vs D, and in one case B wins the tournament and in the other he does not?
If you are trying to say that you based your tie breaker analysis on the assumption that no tournament can ever be compared to another, then please go hug a cactus.
-Bass, 2010-01-02
You also declare very directly, that "opponents' performance against third players do not affect it".
This is completely untrue, as demonstrated above.
You responded by claiming that your research applies only to finished tournament results. Is that supposed to mean "after all games are played, then further opponents' performance against third players will not affect it"?
I do not get your point at all.
-Bass, 2010-01-02
To everybody, currently (2010-01-04) I have no time to reply but more urgent things to do. Please be patient.
Sorry Bass, but you miss another problem.
In a tie between just two players (A and B)the direct comparison will consider the result of one game. Here the DC value will be affected both by A's performance and B's performance. Therefore there is no case for which this statement is true.
"this statement"? Do you mean "Only a Player's Own Performance Affects His Direct Comparison Value"?
Note that that statement speaks of his DC value. It does not speak of the other player's DC value.
Do not confuse step 1) creation of each player's DC value with step 2) consequences for the places.
EDIT:
Ah: Now I am realizing what you are trying to aim at: Do you mean that A's win against B is his performance while B's loss against A is not A's performance? I see. In a sense, one migth say so. In different sense (it is the same information but just expressed differently), one should not say so. I have meant the latter view. IOW, it depends on how one wants to define "a player's performance during the tournament".
Suppose A and B are tied, and A defeated B.
Was A's DC value influenced by B's performance?
So to clarify, instead of using the ambiguous "his performance", we should use the unequivocal "the results of his games", where results refers to the information related to his Number of Wins Score.
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