Deciding on an n-variable for an Elo rating system

Question

I have recently been looking into the mathematics behind the Elo rating system and its application to areas other than chess. I can understand almost all of it but I'm having a problem finding out the meaning of one of the values, and how it is determined. The value I'm talking about is the n value, which is supposed to be some number of rating points that a person n rated points above their opponent is expected to win ten times as often (If my explanation is not clear enough, it is represented by the number 400 - the value used in chess - in the fomulae on the wikipedia page).

This all makes perfect sense so far, but I was wondering how you determine that value when setting up a new system. For example, in chess the value of n is 400 - why was this number chosen and not something more 'round' like 100 or 500? About the only effect I can see that this number has is the increase in precision (though not accuracy) of the rating for higher values of n, and if this is the case, does the value really matter so long as it remains consistent?

Answer 1

It appears to be basically historical. From Arpad Elo's The Rating of Chess Players, Past and Present:

The present range originally took 2000 as the upper level for the strong amateur or club player, and arranged the other categories above and below (1.26)

Category designations and proficiencies among federations have become generally more comparable with the adoption of the Elo system by FIDE and by many of its member federations, but the numbers assigned to any given level of chess proficiency remain entirely arbitrary. Both the class subdivision into 200 points and the choice of 2000 as the reference point were already steeped in tradition when this author arrived on the scene. [...] These features were retained for their general acceptance by the players. (1.27)

In particular, the Elo system was designed to produce ratings on approximately the same scale as the Harkness system, which had already been in use by the USCF since the 1950s. The ~200 points per class difference in the Harkness system falls out of the simple equations used to describe that system.

Answer 2

Since you referred to the wikipedia page, I assume you understand the probability calculation itself. If not, or if anyone else is interested, I could work through it. I assume that you are just interested in whether the value of 400 was chosen.

From a mathematical perspective, it is totally arbitrary. The value could be 1000, or it could be 1. If the number were larger, the rating scale would end up stretched out. If it were smaller, it would be compressed.

The number 400 works because it means that a difference of 100 points is a pretty meaningful difference. People can relate to that when they look at a rating. People tend to think in 100 point intervals, so if I say that my rating is "in the 800s", people have a reasonable grasp of my abilities, and can instantly make a meaningful comparison to someone who is "in the 900s." If we used 100 instead of 400 in the formula, then instead of 1300 vs 1200, we would be talking about 1225 vs. 1200. It would mean exactly the same thing, but it's easier to relate to that 100 point difference.