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Re below...
Tapir asks why a tiebreaker should measure the same thing as the scoring system. Generally, except in handicap tournaments, the scoring system measures the playing strength during the event. This is good! Unless you want a random tiebreaker you would presumably like the tiebreaker to be measuring something similar OR something related. I don't understand why tapir disagrees.
Tapir: Why? Shouldn't we be confident first, that the single SOS point measures anything at all?
ThorAvaTahr: Why do you start to talk about SOS, should we not stick with stick with the topic and define the points of merit by which we can evaluate the quality of any tiebreaker system.
tapir: As Bass who added this criterion wrote somewhere: "SOS measures the same thing as the main scoring system." Everyone making up a criterion for his preferred tie breaker makes this list so very meaningless.
I must say that "It measures the same thing as the main scoring system measures" seems to be a bit of a duplicate. If the purpose of a tournament is to "find the strongest player", then presumably, the main scoring system was chosen to "find the strongest player", and be extension you will want your tie breaker to "find the strongest player".
Might I propose that we call a tournament ranking system "consistent" if all the placement criteria (main scoring system plus tie breakers) have the same purpose (ie: usually to find the strongest player, or best tournament performance). And by extension: "inconsistent" if some of the placement criteria have a different purpose.
More formally: a set of placement criteria is consistent if each of them has likelihood >= lottery of fulfilling the purpose of the tournament. (Note that this makes lottery consistent with all purposes).
Generally, I think it is also safe to assume that the primary scoring system (ie: Swiss Score or MMS or whatever) can be assumed to be consistent the purpose of the tournament (if not, then the tournament organizers have chosen their scoring system rather badly)
If the tie breaker is not intended to only break ties (most of them do that) but to measure something, we need imho some sort of confidence in its "measurement" including establishing a detection limit of the "strength measurement method" etc. If we get rid of the notion of "measurement", it would really be nice. But so far the claim to still measure anything if one operates a measurement system beyond it's (detection) limits or having a measurement system but not even looking for it's limits can't be serious. We can just weigh all participants and use the last digit (in gram) of their weight as tie breaker, surely there is some relation to weight, but the result is still random.
Criteria designed for a certain outcome make this exercise rather meaningless. The criteria added by Herman are way better.
Nobody has established a degree of confidence for McMahon score. So by extension I could say that you argue against using it. That is not a flippant point by the way.