- The tiebreaker SOS is defined as the Sum of Opponents' Scores.
- Before using a tiebreaker, one should make the more fundamental decision whether to either share final places or break ties. If ties shall be broken, then one hasto choose a tiebreaker.
- A tiebreaker is used within a particular tournament system and its aims. Therefore every tiebreaker has to be reevaluated for its specific usage within everyparticular tournament system and aims.
- Tiebreakers can be used for different purposes: seeding, pairing, final results ordering.
- An ordered combination of successive tiebreakers like SOS - SOSOS might be considered to form a new tiebreaker and should be analysed afresh.
- For the sake of simplicity, in the following it is generally assumed that ties shall be broken, SOS is chosen as the tiebreaker, the behaviour of SOS is similar for allparticular tournament systems and its aims if only it can be applied there at all, SOS is considered mainly for the final results ordering, and SOS is used as the first and only tiebreaker. Future research should study also other assumptions in greater detail.

In practice, it suffices to understand intuitively what the meaning of "stronger" might be. Greater strength of opposition also implies greater strength of the player (for a constant score). Precise values of such probabilities are not known so far. However, if the SOS difference is considerable, then the probability could be estimated as being significantly greater than 0.5.

For a single player, greater SOS for him than smaller SOS for him can be interpreted as greater strength of his opponents during the tournament. If a scoring system finds the "better" player, then SOS finds the "tougher" opposition. SOS measures the same thing in opponents that the score measures in players.

- Every pairing program offers SOS. Some set it as default tiebreaker.
- Some associations or federations have it among their recommended tiebreakers.
- Many tournaments use it.

- It is unclear why winning in earlier rounds should be more important than winning in later rounds (with early wins a player collects more SOS since he gets to play players with more wins earlier).
- A priori one wants to consider all rounds equally important - the behaviour of SOS opposes this.
- Since in a Swiss type tournament players with equal numbers of wins are paired against each other, competitors for the top final places are paired against each other the more likely the later the round. So the opposition becomes more and more worthy in later rounds while SOS decreases rewarding of wins in later rounds.

Almost always players with similar SOS have such a small SOS difference that it is smaller than every meaningful numerical significance would be. In other words, SOS pretends to be more accurate than it is.

The exact significance of SOS is unknown (and also depends on the tournament system, round, etc.) but a small SOS point difference is essentially meaningless. It is doubtful to decide winners on small SOS differences.

It is unclear why SOS should distinguish players at the end of a tournament after the pairing program has done its very best to make SOS of every two players with the same number of wins as close as possible. While the relevant pairing strategy is generally considered good, its side-effect on SOS in the final tournament result list is counter-productive.

It is unclear whether winning or whether losing games against opponents with greater numbers of wins during the tournament is better. SOS is indirectly affected by this aspect because SOS rewards winning in early rounds more than losing in early rounds; a player gets the more opponents with previously the more won games the more games the player has won himself in the early rounds.

Contrary to a widely spread myth - due to the Law of Great Numbers, SOS cannot be fair on average over many tournaments: It requires an infinite number of tournaments to allow that conclusion while no player ever can play an infinite number of tournaments. Even worse, specific titles are issued only once per year, tournament conditions and a player's development change.

E.g., SOS does not break ties in a round-robin. (For reference, nor does SOSOS.)

When using SOS, one should be aware of its imprecision. Within one large score group, the player with the greatest SOS may often have performed better than the player with the smallest SOS. However, the exact order of all players in that score group should not be trusted because the typically small SOS differences between two adjacent players in the final results table are often smaller than the significance of SOS can be.

Then after R rounds player Pn has the expected SOS

sum[for_opponents_in_all_rounds] (opponent's points before actual round + opponent's points in actual round + opponent's points in later rounds) =

sum[r=1..R] ( sum[i=1..r-1]Sni + 1-Snr + 0.5*(R-r) ).

The latter summand could be 0*(R-r) if all Onr would lose in all later rounds or 1*(R-r) if all Onr would win in all later rounds. Therefore the maximal imprecision of the expected SOS of Pn is +- sum[r=1..R] 0.5*(R-r).

The maximal imprecision is independent of Sn!

Closest SOS differ by 1 (if we ignore jigos). Contrarily, this imprecision (error) of SOS can become very great in comparison! So one can hardly assign any reasonable grade of significance for SOS. E.g. in a tournament with R=5, why should a SOS difference 4 between two players be significant but not a SOS difference 3? However, the most clearly the often used significance of a SOS difference 1 (even disregarding R(!)) cannot be justified.

Winning earlier is advantageous since the first summand is greater then.

To conclude, it is hard to justify usage of SOS as the first tiebreaker in a Swiss (or a MacMahon) tournament. SOS may depend on opponents' performance during a tournament, however, the dependency is by far too weak for meaningful numbers. Paul Matthews has shown that it is better than drawing a lot but this is about the best that could be said about SOS. It is much more honest not to use any tiebreaker than to use a particular tiebreaker with such an extremely high error per [used] significance ratio.

The expected SOS of player Pn after round 5 is

2+3.5+2+3.5+2 = 13.

Its maximal imprecision for this player is

+- (2+1.5+1+0.5+0) = +- 5.

So after the tournament Pn might have a SOS from 8 to 18.