6 Nations: what is the score with 1 team playing?

Finally got around to a bit of analysis I thought of at school.

In a round robin contest where everyone plays everyone (like the 6 Nations rugby contest) it seems to me that there is a lot of information redundancy in the network e.g. in a contest of 3 teams A, B and C then after 2 games we ought to have a good idea how the third game will go. If A easily beats team B and team B easily beats team C then there is a fair chance that A will very easily beat team C. We might create a concept of "goodness" to determine that A is better than B which is better than C so A is better than C.

For notation let teams be A,B etc and games be AB, BC where the value of AC = (score of A)/(score of C) in the same game. If AB = 2 (i.e. team A scored twice as much as B) then BA = 1/AB.

On the assumption that a team has some fixed quality of "goodness" then a loop such as above of AB * BC * CA ought to give a value of 1 since if team played itself one would expect it to score as much as itself. Now this is a huge assumption that I will show is false!

In the 6 Nations there are 6 teams and so (n^2 - n)/2 possible games: that is 15 for this contest. So the results space is 15 dimensional with points representing the 15 possible (AB, AC, AD,...) network arcs or strengths. Such a network is "inconsistent" because AB * BC * CA != 1, abbreviated to ABCA = 1

A "consistent network" where all loops multiply to 1 has only n-1 dimensions, in the case of the 6 Nations that is 5. It is the reduction from 15 to 5 dimensions that creates the redundancy that I hope to exploit in predicting games.

The source of this redundancy was that concept of "goodness". The idea is that a team has a goodness that we can determine by considering that were it to play itself it would be as "good" as itself. Now the SRH claims this is an impossibility and a meaningless conception. This is all to say that a team has a self, or a substance which is invariant and determines qualities about that team. So by appealing to the concept of "goodness" of "self" we reduce the world from (n^2-n)/2 complexity to simply n-1. It is no wonder that the human mind appeals to ideas of fixed self, upon which it bases its understandings and which is only updates periodically, in preference to seeing everything as entirely new! It is also interesting that the concept of "loop" is actually the foundation of the concept of "self" here. Previously a link was seen between loops (non halting programs) and Godel statements both of which put limits of systems and raise the question of incompleteness and imply the a meta level. It is this insufficiency of "self" and impossibility of "self" founding it-self and so inherently depending and being built upon what is "non self" that the SRH seeks to prove.

The method used to solve the 6 Nations problem was to construct an expression that determined the distance from the 15 dimensional point to the 5 dimensional point in the consistent network. In a 4 node completely connected system the experimental data consists of 6 arcs and the consistent network needs to obey the loops where ABCDA=1,ABCA=1,BCDB=1,DAB/DB=1,DBCD=1. So if AB=x1,BC=x2 then CA=1/(x1 * x2) . If CD=x3 then DA = 1/(x1 * x2 * x3), and DB = 1/(x2 * x3) so the consistent network has points on

This squared distance between these is thus:

d = (AB - x1)^2 + (BC - x2)^2 + ... + (DB - 1/(x2 * x3) )^2

It occurred to me that this should all be done in log space. I've not thought it through properly yet. Update: NO because the reason for "nearest" was to change the values as little as possible from the ratios established by the data. In log() space this would mean change the scores as little as possible which is another interpretation but since I'm looking for the "goodness" of a team it is sufficient to change this value as little as possible.

This is then minimised either numerically or by partial differentiation in the 3 dimensions, setting to zero, and then solving to give the values of and the full 6 dimensions can then we calculated. These are then the values of arcs in a network where each node is a particular number. This number can then discovered and the relative "goodness" of each team can be found.

After two weekends of play in the 6 Nations it is clear that the predictions this system gives are useless. At best it is 200% out. Maybe for larger systems that 6 teams there is more interaction and so more meaning to "self" but I suspect there is something else wrong.

Teams do not play as well as themselves on each occasion. This is the flaw. Ireland played an amazing game against England on Saturday. Had they done this on all the other occasions they would have won a grandslam. They didn't. So there is no predicting team if they play like different teams on each occasion. It is all down to conditions and just because two teams seem to be the "same", and have the same name, and be the same "selves" on each occasion we should not be fooled into believing that they are the same as before. What is the sound of one hand clapping? What is the score with 1 team is playing? So we can't meaningfully separate a team from its opponents and so we can't meaningfully speak of how "good" a team is. Seems very much to support Buddha!