This applies at moment after such rules have been set. During moments before, there is rather the question first whether and possibly why ties shall be brokwn.

I agree, but is there honestly an argument for why the reasoning for one is more appropriate than the reasoning for another? The validity of the advantages against the disadvantages of each are in the eye of the beholder only - fairness with regards to awarding the most appropriate result is entirely subjective, dependent on what people feel individually is "fair".

Let's assume, as done in many rating systems:
rA is a scalar measuring player A's strength
rB is a scalar measuring player B's strength

p(A beats B) = 1/(1 + exp(rA - rB))

NA number of wins of Player A in round robin
NB number of wins of Player B in round robin

I believe that the following conjecture holds:

In a round robin if NA = NB, then rA = rB.

I have not constructed a proof, but any maximumlikelihood computation on Round robin results I have done has supported it.

If it is true, it depends on the functional form of the probability function. The AGA system uses a cumulative normal model and the assertion doesn't hold in that case.

I think D should be 4th. When DC is possible, I think it should be used before SODOS, at least for the round-robin type tournaments we are discussing here.

When I think about SODOS, it seems to me that any argument for it can be reversed into an equally valid argument for SOLOS. So the choice between SODOS and SOLOS seems arbitrary. (For example, the SODOS rewards E for winning against stronger opponents. But why not instead penalize E for losing against weaker opponents?) By comparison, the argument for DC can't be reversed into an equally valid argument for the other player. SODOS can still be useful, and I think it can be used when DC is not possible, when some method of tie breaking must be found. But in the case where DC is possible, I think it makes a better tie breaker.

I think this all becomes a little less clear when dealing with a typical McMahan tournament, because of the issue of one player possibly facing easier opponents overall than another player. But even in this case, any argument for SODOS can be reversed in to an argument for SOLOS, so even in this case I think DC is a better tie breaker.

In addition, I suspect that there may be complications in applying DC when there are cycles (A beats B, B beats C, C beats A) where each player in the cycle has the same number of wins. If we can't use DC due to the cycle, and then we use some other tie breaker to eliminate one player, do we then go back and use DC to decide between the remaining two?
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Eric