old version of the "direct encounter is not a rematch" argument:
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.
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.
- 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?
- 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.
- I am talking about likely differences in skill, which are what an MLES algorithm tries to find. Actual differences in skill are another matter.
- When you are trying to estimate an unknown distribution (the players' relative strengths) from samples (the 6 games played), but your results suggest that there was not enough data to decide, do you a) pick one of the samples and multiply it, while not multiplying the other samples (direct comparison) or b) take another sample (rematch). One of these is a valid statistical method and the other is an invalid one. This is a big enough difference to claim these methods are not similar.
- (Do we really need endless indents in the text layout? They make reading more difficult.)
- Maximum likelihood? I see. Another endless topic, which I do not want to reenter now.
- Now you have stated a reason why direct comparison and rematch are different. Earlier I stated why they are related. They have both relations and differences.
A rematch between Adam and Bengt, on the other hand, will give a whole game worth of extra data to make decisions by, so the better player can easily be chosen to be the winner.