Player rating for tournaments across all installments of Heroes.

To be honest, even I didn't quite understand what you calculated here. Do you mean that after playing one stage, a player will no longer participate in any of the other stages?

Quoting:

4. We obtain a new rating for each player, which we will call R.new (new rating).

R.new = (2/3)*R.old + R.tur

This formula indicates that the old rating R.old of each player decreases by one-third, even for those who did not play in this tournament but who previously participated in the tournament. For new players, R.old=0.
Therefore, for new players, the new rating will consist solely of R.tur.
For old players and those who played in this tournament, the new rating will consist of R.tur and R.old, reduced by one-third.

From this, I understand that the rating decay occurs regardless of whether the player participated in the current tournament or not. After each tournament, you simply reduce the old rating by one-third and add the new one. Logically, this means that my result in the first tournament (let's say 50 points) will be subject to the decay procedure 10 times by the end of the eleventh tournament. This means that only 0.86 will remain of the 50. From the result of the second tournament (let's say 50 again), 1.3 will remain. From the result of the third tournament, 1.95 points will remain... I won't continue, the point should be clear.

Let's say I'm the strongest player and played 12 tournaments in a year, finishing them with a victory - 50 points each. The total is 600. After the decay, it will be 148.84. You started playing in the middle of the year and also received 50 points for six tournaments, i.e., you shared the prize with me. Your total is 300, after applying the decay procedure, it is 136.8. (although you played twice as few). Let's say I missed the last tournament. As a result, I immediately lost 50 points - 98.8 points (!!!)... And if I had missed the first tournament, my result at the end of the year would hardly have changed and would be 148.26. Is that clear?

I am forced to repeat my words:
"Why play the first stage for such a small gain? You can just play the last two and that's it..."

And I can only hope not for winning the tournament, but for you to miss it... :)

Let's still use numbers for the example, and not theory... The "weight" of the result in the final formula should not depend on the time that has passed since it was achieved... Please, test this mechanism in Excel, and you will understand that by the end of the year, the "weight" of a victory achieved many months ago will dissolve even against the background of weak results from newly joined players... Imbalance.

If you want to see only the player's strength at a specific current point in time, and it doesn't matter how he played a few months ago, then what is the point of calculating an annual rating?

Do you really need to play several tournaments to be sure of this flaw?
Or maybe I'm not calculating it correctly?