# Theory behind SOS

This page tries to describe the "meaning" and properties of the SOS tie breaker. See, however, SOS/Discussion for further or contrary aspects of SOS.

## SOS in McMahon tournaments

1. When a tournament organiser chooses to use the McMahon system, he makes a decision to reward the player with the higher final McMahon Score (MMS).
2. By tautology, MMS is an indicator of "best ability to collect MMS". Just what this ability includes is, interestingly, irrelevant.
3. MMS has the property[1] that in a given game, a fixed amount of MMS is divided to the players. The more MMS one player gets, the less the other gets. Therefore..
4. Collecting MMS in a game against a player with the worse ability to collect MMS is easier than collecting MMS in a game against a player with a better ability to collect MMS. Substituting definition from 2 and shuffling a bit:
5. Opponent's MMS is an indicator of how difficult it was to collect MMS in a given game.
6. Summing is one way to combine the overall difficulty from many games, so the sum of all opponent's scores indicates how difficult it was for the player to collect MMS in the whole tournament.
7. Although seemingly obvious from a statistical estimation point of view, this is the leap of faith: If the same amount of MMS was collected, it is likely that better ability was required when the collecting was more difficult.
8. Combining the last two points, among players with the same amount of MMS, Sum of opponents' MMSs is an indicator of a better ability to collect MMS. Which becomes
9. In case of a tie, SOS breaks the tie using better ability to collect MMS as a tie breaker, or combining with 2:
10. SOS provides more resolution to MMS.

## SOS in other tournament systems

You might have noticed that only point 3 above states anything about the properties of MMS.

Therefore, you can substitute any kind of score as long as 3 holds for that system.

There are no widely known tournament systems for go that use scores that do not have this property[1]. Therefore it generally holds that

SOS can be used as a tie breaker to provide a finer resolution to any score based go tournament system[2].

## Improving on SOS

The general opinion about SOS is somewhere along the lines of "If you absolutely cannot tolerate ties, SOS is among the least bad of tie breakers." See SOS/Discussion to find out about individuals' opinions.

When measuring someone's ability to gain points in games, nothing beats actually playing those games. So a rematch is usually considered the definitive tie breaker.

Also, it should be possible to construct a statistically better tie breaker by using a method more sophisticated than just summing the difficulty indicators.

Moreover, if extra details are known about the nature of the scores, then those details can be used to improve the tie breaker. (For example, in a team tournament there are table scores in addition to the primary score of team points.)

The most important thing about improving on SOS is though: If your primary scoring system rewards something you do not want it to reward, then SOS breaks ties in a manner you do not want it to. Choosing a better tournament system will cure this symptom.

## Resolution concerns

Having no tie breaker at all is always a possibility. Allowing ties is not a bad idea at all.

One SOS point is often considered a ridiculously small margin to decide the winner of the tournament by. This is, however, a very common practise.

On the other hand, it might be considered unfair to ignore a notable SOS difference, on the account of one of the tied players having had to work much harder on those points.

In theory, one could reach a middle ground by breaking ties when the SOS difference is, for example, 3 SOS points or more. When more than 2 people are tied, this will result in very bizarre looking results though. (A possible result might be that Adam is tied for places 1 and 2, Bengt is tied for places 1 through 3 and Cthulhu is tied for the second and third places only..) A method for sensibly dividing any prize money according to this kind of results exists, but finding it is left as an exercise to the adventurous reader.

If you need to break ties even in the case where MMS and SOS both agree that the players are equally matched, you might as well toss a coin. Should you suffer from a lack of coins, you can also try using SOSOS as your source of randomness.

[1] Strictly speaking, this is a lie: the players get MMS at the beginning of the McMahon tournament without regard to other players' MMS. The initial MMS should be considered to be the tournament organiser's best a priori guess about the players' go playing abilities, presented in the form of points earned in simulated rounds. Go ability happens to be the part of "ability to collect MMS" that affects MMS collection during actual play, so the two luckily happen to be closely linked. Because it is just a guess though, it is not guaranteed that the initial amount correctly reflects the player's MMS collecting abilities. However, any non-blatant initial errors will even out during the actual tournament rounds, so even though point 3 does not strictly speaking hold for MMS, in the final results the effect is approximated well enough that point 5 holds even without it. (If it did not, then MMS would not reflect go skills, and would quickly disappear from use)

[2] Exception: SOS can't be used to break ties in a Round Robin tournament. This is because in a Round Robin you play against all players except yourself. Consequently your SOS will be the total score of all players minus your own score. So players with the same score will have the same SOS, so SOS doesn't break any ties.