Many thanks, jann. I think I understand your ideas better.
Many thanks, xela. You have shed light on the questions involved.
And thanks to others, as well. I don't want to leave anybody out who has contributed to this discussion, but I am only responding with jann and xela's latest notes in mind.
OK, nothing really specific. I may say more mañana, but I actually have a life. I think.
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A rating system is, in a way, a fool's errand. Why is that? Because it pretends that we can represent a player's strength with a single number. We can't.
Everybody is familiar with the situation, even if we don't personally know of one, in which Player A can usually beat Player B, who can usually beat Player C, who can usually beat Player A. If we could represent the strength of each of these players by a single number, the player's rating, then Player A's rating would be greater than Player B's rating, which would be greater than Player C's rating, which would be greater than Player A's rating. Tilt! No puedo, señor.
Why are such situations possible? One reason is that there are several skills and other factors that combine to form skill at go. For instance, there is skill at reading, but actually, there are at least three skills which produce that one. There is skill at invading, skill at sabaki, skill at utilizing thickness, etc. And each of these skills are probably composed of other skills, as well. In addition there are factors such as memory, emotional control, discipline, physical fitness, alertness, etc. If we can represent each of these skills and factors with a single number, we still cannot reduce all of those numbers to a single number. That is why I said that skill at go is a vector, with a number for each factor that makes up a person's go skill.
However, as xela alludes to, other players also matter. For instance, if Player A never played Player C in the situation above, we would happily think that Player A's general go skill was better than Player C's. So really, we should probably think of a player's go strength as a matrix.
Now, the situation with Players A, B, and C, is not so unusual if they are all amateur shodans. But it would be quite unusual, perhaps almost impossible, if Player A were a shodan, Player B were a one kyu, and Player C were a two kyu. There may be a certain two kyu who can regularly beat a certain shodan, playing even, but I have never heard of such a case. In any event, I am willing to say that there is some difference in go skills such that the weaker player will never beat the stronger player on a regular basis. What this means is that we can order some players on the basis of general go strength, but not all players. We say that go strength is
partially ordered.
Ratings, OC, are numbers, and are this completely ordered. Since they are taken to represent go strength, which is only partially ordered, nobody has a precise rating. It is not that there is some uncertainty about a player's rating, which could be reduced with more games played. There is an irreducible uncertainty, such that nobody has a precise rating. Any attempt to assign precise ratings is doomed to failure. As xela points out, "better" is ambiguous.
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What about ranks? Player's ranks are still not completely ordered. It is not unusual for two players of different ranks to play even with each other. It is unusual for an amateur of a lower rank to regularly beat a player of a higher rank, however. So ranks are pretty good indicators of general go skill. They are not nearly as precise as ratings, but that is a good thing, IMO. Ranks better reflects reality than ratings.
The earliest numerical ranking system I am aware of goes back centuries. A one rank difference meant that the higher ranked player could normally take White to make the game even. A two rank difference meant that the higher ranked player alternated between taking White and giving two stones. A three rank difference meant that the higher ranked player gave two stones. Etc. These were pro ranks, OC. Over time pro ranks got closer together. Today the idea that a pro 8 dan would give 4 stones to a pro shodan is absurd. It was absurd 100 years ago, as well. 3 stones was not. Today there is no guarantee that a pro 8 dan can give a pro shodan 2 stones. Times change.
But going back to ancient times, two pros might agree to a lengthy match of several games, often 10 or more. It was normal in these matches for the handicap to change after four wins in a row, regardless of the official handicap. For instance, a player might go from playing Black all the time to playing Black in two games out of three and White in the other.
When I was learning go it was usual for amateurs to change handicaps after a three game winning streak. For instance, the players might be playing even by alternating between taking White and Black, and then, after one player won three games straight, to that player playing White. And then if White won three games straight, she would give two stones. That change was a bigger leap, OC, but we were not worried about niceties.
Note that we usually played even by alternating between taking White and Black, not by komi. Some players used komi, but most of us did not bother with it.
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Against that background, the idea of one player playing Black and the other playing White for 10 games without changing the handicap after Black won 3 or 4 games in a row is bizarre. Komi or no komi. It is a made up scenario to produce a problem that makes no practical sense. I won't say that the problem does not exist, because jann sees one, but I don't.
Now, with some people playing many games a day online with a rating system that, for some reason or other, does not recalculate ratings immediately, something like that scenario could occur. In fact, with thousands of games played on different servers every day, such a scenario may not be all that unusual. But first, it is ephemeral, as modern rating systems are self-correcting. And second, since precise ratings are a pipe dream, anyway, who cares?