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(Lets assume all these players are in their 30s and have been playing chess for many years, with no recent changes in how they train.)

Say we have 7 players who play each other in an isolated club round robin with 3 games a month, so each player plays the same person 6 times a year.

Three of these players also play in a local inter club match once a month, playing a different (with similar rating to themselves) player from a different club each month, playing each player no more then once a year.

Some of the players from the other clubs can be assumed to play in many tournaments, with the players in the isolated club not doing so. The other clubs can be assumed to have a turn over of members unlike the isoloated club.

Would the rating system be improved by using a lower K when a person has played the same opponent multiple times in the last year?

Would the answer changes if the players in the isolated club are old with a slowly reducing skill level and some of the players in other clubs are young with an rapid increasing skill level?

Does any rating systems do it?

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Would the rating system be improved by using a lower K when a person has played the same opponent multiple times in the last year?

No.

None of your setup is of any relevance whatsoever. The underlying basis of rating is statistical.

In general a player's level of play will vary over time. The aim of the rating system is to try and have the rating accurately reflect that level.

A low K factor will mean that in general the rating will take longer (more games) to adjust to a change in playing level but will also lead to relatively stable ratings.

A high K factor will do the opposite. It will change the rating by larger amounts when results come in that are away from the expected result. On the one hand this means that when a player's level is changing it will take fewer games for the rating to catch up. On the other hand it means that a stable player's rating will oscillate by a bigger margin.

It is for these reasons that FIDE gives a high K factor to juniors, who are normally improving rapidly, and to new players, since it takes 30 or 40 games for the estimate of a new player's level to become more accurate.

Similarly, very strong player (defined as rated over 2400) are expected to change their playing level very slowly. Generally speaking, the stronger you are the more work you have to do to improve and, for that matter, the more work you have to do just to maintain your level. Hence a low K factor is appropriate.

Who you are, young or old, and what your playing level is, weak or strong, is much more important and relevant than who you play.

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