Answers.Body: <p>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.</p>
  
  <p>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.</p>
  
  <p>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 <a href="https://chess.stackexchange.com/questions/18721/why-cant-fide-use-a-non-deterministic-gender-neutral-swiss-pairing-system">perceived unfair pairings</a>; 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.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-09-08T11:38:57.303Z
Answers.LastActivityDate: 2018-09-08T20:16:46.750Z
Answers.LastEditDate: 2018-09-08T20:16:46.750Z
Answers.UserDisplayName: D M
Answers.Body: <p>Thanks for all the comments!</p>
  
  <p>As Slepz noted, rounding away from (or towards) zero would fix the zero-sum property of the elo rating system.</p>
  
  <p>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.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-08-08T11:21:00.863Z
Answers.Text: There are also rounding schemes that differ in how they treat .5 based on whether the integer part of the number is even or odd; so e.g. 2003.5 would be rounded to 2003 and 2004.5 would be rounded to 2005. That would also prevent bias from rounding. But I don't know if they use such a method.
Answers.UserDisplayName: RemcoGerlich
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2018-08-07T18:09:04.263Z
Answers.LastActivityDate: 2018-08-07T18:09:04.263Z
Answers.UserDisplayName: Maxwell86
Body: <p>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 <em>create</em> rating points?</p>
  
  <p>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 <a href="https://ratings.fide.com/calculator_rtd.phtml" rel="nofollow noreferrer">FIDE elo calculator</a>). 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.</p>
  
  <p>An example of such a case can be found here: the elo calculations of <a href="https://ratings.fide.com/individual_calculations.phtml?idnumber=1003623&amp;rating_period=2015-12-01&amp;t=0" rel="nofollow noreferrer">player 1</a> and <a href="https://ratings.fide.com/individual_calculations.phtml?idnumber=1004395&amp;rating_period=2015-12-01&amp;t=0" rel="nofollow noreferrer">player 2</a>.</p>
  
  <p>Am I overlooking something? If not, has this been noticed and pointed out before?</p>
  
  <p>Wouldn't it be more correct if the elo changes are rounded <em>first</em>, before adding them to the old rating? (In the example, -3.5 would be rounded to -4.)</p>
  
  <p>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...?</p>
  
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-04T20:44:16.660Z
Comments.Text: 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 https://math.stackexchange.com/a/60690
Comments.UserDisplayName: Slepz
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-04T21:06:27.133Z
Comments.Text: 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.)
Comments.UserDisplayName: TMM
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-05T06:16:42.220Z
Comments.Text: K-factor can create 30 points in one game if I'm not wrong. This looks like much bigger problem.
Comments.UserDisplayName: hoacin
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-05T06:46:48.140Z
Comments.Text: @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.
Comments.UserDisplayName: Maxwell86
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-05T14:33:15.320Z
Comments.Text: 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?
Comments.UserDisplayName: itub
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-05T20:54:01.237Z
Comments.Text: @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).
Comments.UserDisplayName: TMM
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-05T20:56:10.720Z
Comments.Text: @Maxwell86 Actually it does not necessarily cancel out: different K-factors are specifically linked to (non-uniform) subsets of all players. For instance young players, who are generally underrated and will score more points than average, have larger K-factors than older players. So likely the K-factors "create" many rating points as well.
Comments.UserDisplayName: TMM
Comments.ContentLicense: CC BY-SA 3.0
Comments.CreationDate: 2017-10-06T07:03:24.070Z
Comments.Text: @TMM Well, this might be true (or not), and it's also a part of my question :-D. I wonder if there is any statistical data that shows this.
Comments.UserDisplayName: Maxwell86
Tags: <rating><elo><fide>
Title: Mistake in FIDE elo calculation?
ThreadText: # Mistake in FIDE elo calculation?
  
  (Asked by: username_0 on 2017-10-04)
  
  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](https://ratings.fide.com/calculator_rtd.phtml)). 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](https://ratings.fide.com/individual_calculations.phtml?idnumber=1003623&rating_period=2015-12-01&t=0) and [player 2](https://ratings.fide.com/individual_calculations.phtml?idnumber=1004395&rating_period=2015-12-01&t=0).
  
  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...?
  
  **Tags:** rating, elo, fide
  
  > **Comments:**
  > **username_1**: 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 https://math.stackexchange.com/a/60690  
  > **username_2**: 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.)  
  > **username_3**: K-factor can create 30 points in one game if I'm not wrong. This looks like much bigger problem.  
  > **username_0**: @username_3: 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.  
  > **username_4**: 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?  
  > **username_2**: @username_4 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).  
  > **username_2**: @username_0 Actually it does not necessarily cancel out: different K-factors are specifically linked to (non-uniform) subsets of all players. For instance young players, who are generally underrated and will score more points than average, have larger K-factors than older players. So likely the K-factors "create" many rating points as well.  
  > **username_0**: @username_2 Well, this might be true (or not), and it's also a part of my question :-D. I wonder if there is any statistical data that shows this.  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_5 on 2018-09-08. Score: 2
  
  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](https://chess.stackexchange.com/questions/18721/why-cant-fide-use-a-non-deterministic-gender-neutral-swiss-pairing-system); 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 by username_0 on 2018-08-07. Score: 0
  
  Thanks for all the comments!
  
  As username_1 noted, rounding away from (or towards) zero would fix the zero-sum property of the elo rating system.
  
  Moreover, as mentioned by username_4, 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.
  
  > **Comments:**
  > **username_6**: There are also rounding schemes that differ in how they treat .5 based on whether the integer part of the number is even or odd; so e.g. 2003.5 would be rounded to 2003 and 2004.5 would be rounded to 2005. That would also prevent bias from rounding. But I don't know if they use such a method.  
  
  ---
  
  