Mistake in FIDE elo calculation?

Question

One of the properties of an elo rating system is that the sum of all elo changes is zero. When calculating the new FIDE elo ratings, the elo changes are added to the old ratings and the new ratings are rounded. However, doesn't the last step create rating points?

Say, player A has 110 ratings points more than player B and both players have a K-factor of 10. When player A wins against player B, he gains 3.5 points, while player B loses 3.5 points (see FIDE elo calculator). But after adding the elo changes and rounding the new ratings, player A will gain 4 points and player B will lose only 3. As a result, the zero-sum property is violated.

An example of such a case can be found here: the elo calculations of player 1 and player 2.

Am I overlooking something? If not, has this been noticed and pointed out before?

Wouldn't it be more correct if the elo changes are rounded first, before adding them to the old rating? (In the example, -3.5 would be rounded to -4.)

Or is this effect so small that it can be neglected, compared to other effects like the inflow of new players, different K-factors, etc...?

Answer 1

The effect is small. On average, once out of every 20 rating periods a person plays rated games, a person's rating will be rounded up instead of down. A rating period is one month. Assuming everyone plays every rating period (which of course is not the case), this will create 30 points every 50 years.

There are known deflationary effects from new players - they often enter near the bottom, then gain points, and usually never lose all the points they gain. It's possible the rounding is intentional as one way to insert an inflationary pressure to combat that.

I don't think they will institute decimal points on ratings, regardless. The USCF uses a rating system with a decimal point and a complicated formula. The FIDE rating system, in contrast, was designed so anyone could calculate their own rating change based on their results and the published ratings of their opponents. Adding decimals would make the math slightly more complicated and would make it necessary to publish those decimals for people to calculate their ratings. We've already seen a GM throw a game in protest at perceived unfair pairings; even though pairings are deterministic and were not manipulated in that case, the pairing rules are complicated enough that the GM couldn't easily determine this at the time. Perhaps FIDE doesn't want them questioning their ratings as well.

Answer 2

Thanks for all the comments!

As Slepz noted, rounding away from (or towards) zero would fix the zero-sum property of the elo rating system.

Moreover, as mentioned by itub, by using decimal numbers for both elo change and elo storage, one could solve any rounding issues and make the elo calculations more precise. To me, this seems the most elegant solution, as an elo rating is a continuous variable rather than a discrete one.