velobici: Direct Encounter compared with Additional Round(s) of Play (2008-02-22 13:34) [#4367]

From the Rematch page:

A direct encounter is not a rematch

A direct encounter is not a rematch, and should not be treated as such. Here's a simple example why:

Consider 4 players (named Adam, Bengt, Cthulhu and Donald) who play a round-robin of 3 rounds. Adam and Bengt get 2 wins each, and Cthulhu and Donald get one win each, like this:

        A  B  C  D  Total
Adam    X  1  1  0   2
Bengt   0  X  1  1   2
Cthulhu 0  0  X  1   1
Donald  1  0  0  X   1

Now one might think that since Adam has won against Bengt, Adam must be the stronger player. However, this thinking is flawed: the fact that Bengt has won against Donald, who in turn has beaten Adam, actually gives a (marginally) better likelihood to Bengt being the stronger player.

Disucssion

RobertJasiek: The reasoning of direct comparison is being a shortcut to an otherwise necessary rematch between the tied players. This thinking is not flawed; what you call "flawed" here is the construction of another view, one that puts greater weight on a certain type of likelihood (to be defined) than on the shortcut to a rematch view. With a different type of likelihood definition (A has beaten B more often than B has beaten A, so empirically it is more likely that A will beat B again), also a likelihood view supports direct comparison. Show your definition of likelihood and explain why indirect relations between the players like B>D>A should be more relevant than direct relations like A>B. In my opinion, the more direct relations should have the greater weight in all reasonable definitions of likelihood.

Bass: You show your definition of likelihood and explain why the likely skill gap between A and B, measured from "A won against B" should be bigger than the likely skill gap between B and A when "B won against D who won against A". After all, in the latter gap there was enough room for a whole another player. Or better yet, ask Matti, as I suggested in the SOS/Discussion where this is slightly more relevant. The purpose of having this example on this page is to show that direct comparison is not even remotely similar to a rematch. The remarks about flawed logic are only there to make that point clearer.

RobertJasiek:

1. I have not argued to construct skill gaps but to count the number of players in a chain: A>B is a chain of 2 players; B>D>A is a chain of 3 players. If you speak of skill gaps, then we should notice that the gap between A and B is the same as the gap between B and A. So which relevance should skill gaps have here at all?
2. Why do you think that the example shows that direct comparison is not similar to a rematch? It is not the same for sure, but "not similar" is another category.