Math ko

[Bill Spight]

Intransitivity is easily explained by the fact that there are several different skills at go. I.e., a player's "correct" rating is a vector, not a single number. Reducing the rating to a single number loses information. The single number rating calculated by a rating system is an average. In addition, if the player's opponents in different systems have significantly different skills, then the player's single number ratings in the different systems will almost surely differ.

[Bill Spight]

With B's rating as 1, from B's perspective, A's perfect score over B means A's exp is an idiotic-sounding 'infinite' a pretend number. Likewise with B's rating of 1 B's perfect score over C means C's exp has to be 0. Now, ignoring temporarily the silliness of the next phase as those handling imaginary numbers did, if C's exp is 0, and C scores perfectly over A, it implies A has an exp that makes C's 0 look infinite. What could this mystery value be?

Statisticians are way ahead with regard to this question. Going back to Laplace, if A and B play N games and A wins N games, then Laplace estimated the probability that A would beat B on their next game as (N+1)/(N+2), not 1. There are problems with Laplace's estimate, which were well known by the 20th century. But an estimate of 1 is, without other evidence, ridiculous. That is obviously so with cases of intransitivity.