From the perspective of fairness and objective impartiality there are some issues with the league system, specifically the match making system, that I would like to point out. Essentially, since the league can be seen as a sorting algorithm, there is the question whether the algorithm is valid (i.e., arranging a verifiably orderable sequence in the proper order) or even deterministic (i.e., for a given sequence of quantities, yields the same sorted order regardless of the initial order of the quantities). I will go on to demonstrate that neither condition is satisfied by the current implementation.
Axiom 1: A league consists of concurrent matches of three servers facing each other.
Corollary: The number of servers in a league is N_s = 3i, i = 1, ….
Axiom 2: After a match, the winning server is awarded 5 points, the second one 3 points and the final one 1 point.
Corollary: An alternative but equivalent way to implement the ranking system is to postulate that after a match a server gets -1 rank for every server it bested in the match, and +1 rank for every server that bested it. The axiomatic reward R can be expressed as a function of that rank as R = 3 – 2r. However, this alternative counting method makes it easier to calculate pairwise effects.
Theorem: A series of matches where a server is matched the same number of times with every other server in the league is both valid and deterministic.
Side note 1: For a server to be matched the same number of times with every other server in the league, the number of servers in the league must be N_s = 2j + 1, j = 1, …. Combined with the corollary of axiom 1, it directly follows that N_s = 6n + 3, n = 0, ….
Side note 2: The number of rounds required to satisfy the condition is N_r = k(3n + 1), k = 1, ….
Proof: If w_1 < w_2 are the inherent, objective rankings of two servers, then it directly follows that their league ranks are W_1 = (w_1 – 1) – (N_s – w_1) = 2w_1 – N_s – 1 < W_2 = 2w_2 – N_s – 1. Q.E.D.
Theorem: A series of matches where a server is not matched the same number of times with every other server in the league is not necessarily either valid or deterministic. The proof by using simple counterexamples can be left as an exercise for the reader.
I would like to draw attention to the schedule of the league I am most familiar with myself. In the NA silver league it is a fact that two of the nine servers, Fort Aspenwood and Stormbluff Isle are so much stronger than the rest that there isn’t any realistic chance of them losing ranks to any of the other server. In such a situation, the respective order of the two strong servers will be determined by their mutual matches, and that satisfies the validity criteria. However, in order to the remaining ranks to have any relevance, the other servers must face these two servers the same number of times during the season.
This simple requirement is not satisfied. While most servers will lose 3 ranks by the “luck” of being matched against FA or SBI, there are three exceptions. Northern Shiverpeaks will face each strong server once, thus losing only 2 ranks. Ehmry Bay will face the strong servers twice, losing 4 ranks. As an egregious anomaly, Dragonbrand will face the strong servers three times, thus by default losing no fewer than 6 ranks! As a side note, this is not a whine since I’m a Yak myself. However, the chosen schedule very effectively denies DB the possibility to finish anywhere around its true strength, unless FA and/or SBI should decide to game the system just for git and shiggles.
This is an unfortunate side effect of having a single season duration with leagues of different sizes. Had all leagues consisted of nine servers, it would have been possible to satisfy the validity criteria with seasons that last multiples of 4 weeks, since 4 is the smallest N_r for N_s = 9. As an example, here is one such minimal schedule that nevertheless satisfies the validity theorem:
Round 1: (1 2 3) (4 5 6) (7 8 9)
Round 2: (1 4 7) (2 5 8) (3 6 9)
Round 3: (1 5 9) (2 6 7) (3 4 8)
Round 4: (1 6 8) (2 4 9) (3 5 7)
Also, before anyone start to speak about tactics and servers rising up to the occasion, those considerations are completely irrelevant. This discussion is about the validity of the match schedule and ranking system itself, and how the current algorithm fails to produce a correct result even if used to sort a sequence of numbers in ascending order.
TL;DR version: The league schedule is bad. Please fix by the next season.