# Mistake in FIDE elo calculation?

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...?

• I suspect they use a rounding scheme that results in the two rounded numbers summing to 0 (though I can't find something to cite). Round away from zero would do the trick. See math.stackexchange.com/a/60690 – Slepz Oct 4 '17 at 20:44
• Indeed, the other factors will likely have a bigger effect. If two players have different K-factors, inevitably rating points will be "created" or "destroyed". (Inflation, if any, likely comes from more players being rated, which allows higher players to "feed" on these players more easily, and even higher players to feed on them etc. These other effects I think are ultimately negligible.) – TMM Oct 4 '17 at 21:06
• @Hoacin: Not necessarily. A difference in K-factor can both create and destroy rating points. On average, they should cancel out. While the rounding issue only creates points. – Maxwell86 Oct 5 '17 at 6:46
• I believe USCF actually stores ratings as floating point numbers even if it publishes them as integers. I wonder if FIDE could be doing the same? – itub Oct 5 '17 at 14:33
• @itub They are not. If you are 1600, gain 3.6 points in one rating period and 3.6 in the next, you would be rated 1608 (which would be impossible without rounding). – TMM Oct 5 '17 at 20:54

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.