Elo Rating Math Help (Different from stickied thread)


Before I get flamed, yes, I've read the stickied thread in this forum, and it doesn't answer my 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?


Also, does anyone know of any other system that uses the Elo rating system, but does not use n=400?


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.


Excellent, that's exactly what I wanted to know. Thanks.


I would also hazard the guess, that having a larger value of n means that when Elo scores are calculated from one game to the next, that the rounding error of keeping the scores as integer values is mostly negligible. If the range were compressed, it would become more important to keep the scores with fractional precision (e.g 1200.23 vs 1224.71) in order that rounding errors are reduced. This might be seen as unwieldy, or unpleasant. I gathter that among GM's they might worry about fractional points but for the average player it's handy to just round to nearest integer.