# Is Elo conserved?

Does the average elo of a group that plays only amongst themselves stay constant?

For example, say 100 players start with 2000 rating and they play only amongst themselves many times. Will their total Elo remain at 2000*100, decay, or increase over time?

I'm asking this because if the top 10-20 players keep playing super tournaments, will their rating be higher or lower or same as their actual strength?

• This can be easily answered by running a simulation. May 10 at 5:45
• Keep in mind that the number of players does not actually stay constant.
– usul
May 10 at 14:41

After a game, the new rating for a player is updated according to the formula:

r' = r + K · (s - E[S])


...where r is the elo rating, K is a "constant" factor, s is the score (i.e. 1 for win, 0 for loss, 0.5 for draw), and E[S] is the expected score (e.g. Magnus Carlsen has E[S] = 0.9999999 against Average Joe).

Player A and player B have a total score of s_A + s_B = 1, since either exactly one player wins, or both players draw. By the proof given on Wikipedia, player A's expected score E[S_A] and player B's expected score E[S_B] also sum to 1, i.e., E[S_A] + E[S_B] = 1.

Thus, if K = K_A = K_B, the total new rating of the players after the game is:

r_A' + r_B' = r_A + r_B + K · ((s_A + s_B) - (E[S_A] + E[S_B]))
= r_A + r_B + K · ((1) - (1))
= r_A + r_B


That is, the total rating is conserved after a game iff K = K_A = K_B.

In Elo's original work, K = K_A = K_B = 10 is symmetric, but more modern rules may sometimes use asymmetric factors for K_A, K_B depending on the situation.

• Unrelated, but now that Carlsen has taken up poker, I want to know what his NLH ELO is ;-) May 12 at 17:11

It depends.

If they play games at the same rate (i.e nobody plays 2 more games than anyone else) and nobody's rating goes up to 2400 (or 2300 if they are all juniors) and nobody dies or moves away (unless they do so when their rating is exactly 2000) and nobody new joins and all of the players are either juniors (U18) and remain juniors or adults, then yes, total rating will remain constant.

The reason for all these carve outs is the way the ratings are calculated uses a fudge factor "K" which depends on these things and can introduce asymmetry into the calculations whereby when two players play the result can see one player gain/lose more or fewer ratings points than the other player loses/gains.

Here is the relevant section of the rules:

8.3.3 K is the development coefficient.

K = 40 for a player new to the rating list until they have completed events with at least 30 games.

K = 20 as long as a player's rating remains under 2400.

K = 10 once a player's published rating has reached 2400 and remains at that level subsequently, even if the rating drops below 2400.

K = 40 for all players until the end of the year of their 18th birthday, as long as their rating remains under 2300.

If the number of games (n) for a player on any list for a rating period multiplied by K (as defined above) exceeds 700, then K shall be the largest whole number such that K x n does not exceed 700.

• What if they are current top 10? All rated above 2750. Also "i.e nobody plays 2 more games than anyone else" why this criteria? Rest I understand. May 9 at 22:51
• @Indigo1729 According to that quote, it looks like K = 10 symmetrically for the top adult players, unless someone plays over 70 games within a given time period. May 10 at 0:26
• See also this old answer of mine, which says that \sum_{p \in players} elo_p / K_p is "almost" conserved. May 10 at 17:58
• @EmilKarlsson Well, "carve-out" is a word too, and it means "exception in a law or rule," so it seems to fit here. (It's an interesting coincidence that "caveat" and "carve-out" have somewhat similar spellings as well as somewhat similar meanings.) May 11 at 13:43
• @Michael Well, there's eggcorn—a word or phrase that's created by mistake based on the sound and meaning of an existing word or phrase. May 12 at 17:18

You don't mention what country is under consideration.

Some jurisdictions may apply "participation points" to encourage play, and "feedback points" to offset accelerations enjoyed by your opponents. These policies are not conservative.

At one time (unsure if this is still the case) the Chess Federation of Canada used participation points. The Elo system used by the North American Scrabble Players Association uses both acceleration and feedback points.