# Where do the default values in the Elo ratings formulas come from?

After doing some reading about the Elo ratings system, I am trying to implement one. I have some questions on the default values in the formulas.

If player A has rating $r_a$ and player b has rating $r_b$. Then with

$x = \frac{r_a}{400}$ and $y = \frac{r_b}{400}$

The two main formulas are:

$E = \frac{10^x}{10^x + 10^y}$,

$r_{post} = r_{pre} + K * (S - S_{exp})$

where $E$ is expected score, also the probability of player A beating player B, and $r_{post}$ is a player's new ranking based on their expected score and actual score, 1 if they won 0 if they lost.

My question is where do the numbers 400 and 10 come from? Are they arbitrary? I know that in Elo with a pair of players that are 200 points apart, the better player is predicted to win ~75% of the time, is this why 400 was chosen, to satisfy this condition? My guess is this stems from the fact that Elo assumes players ratings are gaussian, and they all have the same standard deviation. What about 10, and why is it raised to an exponent? All I know is the formula comes from the Bradley–Terry model. Lastly, I've read the default average rating in the Elo system is 1500, is this connected to 400 as well?

• They are essentially arbitrary, fixing the scale of the elo system? – kjetil b halvorsen Apr 9 '16 at 20:45

I found a slightly different equation for the calculation of the estimate: E = 1/(1+10^((ra-rb)/400)). And the number 400 contained in the formula was chosen by Arpad Elo so, that the Elo numbers are as compatible as possible with the numbers of the formerly used rating system by Kenneth Harkness. In fact one can see the Harkness-Model as a stepwise linear approximation towards the Elo-Model. If the rating difference between player A and B, i.e. |R_B - R_A| is more than 400 points, the number 400 (or -400) is used instead of the real difference.

Regarding last question (Lastly, I've read the default average rating in the Elo system is 1500, is this connected to 400 as well?), the answer is: no.

The starting rating is arbitrary (in Elo, as in Glicko, etc. E.g. in rankade, our ranking system, players starts at 2000), it doesn't matter what number you pick.

Starting Elo at 1500 leads to

• serious difficulties to go under 1000 (so nearly all scores are in 4-digits format),
• a significant quality benchmark at 2000 and
• an unsurpassed limit at 3000 (Magnus Carlsen peaked at 2882),

but none of these seems to be the main reason for choosing it.

Even if it's out of your question, probably most debate is on $K$ constant value, indeed.

• I am surprised to see the claim about 3000 being a limit. I can imagine a group of players attaining, say, the 2800 level and then playing exclusively amongst themselves. In effect, they will have restarted the process at a new average value of 2800, so why would 3000 (or any fixed value) be a barrier? – whuber Apr 14 '16 at 17:07
• You're right, it's not a barrier, so I wrote unsurpassed, not unsurpassable. :) – Tomaso Neri Apr 14 '16 at 18:32