Player rating for tournaments across all installments of Heroes.

I am more inclined towards the formula B = N / (R(i) - M(i)). The values will be small here.

Typo. It should be the other way around - (M(i) - R(i)). And by R(i), you mean not the rating, but the position in the ranking, right? There is an inverse relationship here; the higher the rating, the smaller the number representing the player's position... :) Because the rating itself can be any number in the range of 0-1000.

But if we want to reward those who make a sharp breakthrough, then it makes sense to use a different formula - B = N / (M(i) / R(i)). Where N is the number of players, R(i) is the player's rating, and M(i) is the player's position in the tournament.

I think it would be more correct to write the formula as follows: B = N * R(i) / M(i). The bonus will be in the range of [1; N^2], which is very critical.

Example: 50 people participated in the tournament. The tournament winner - M(i) = 1 - is participating in tournaments for the first time, which means his current rating is 0, i.e., R(i) = 50. And what will he get as a bonus? B = 50 / (1/50) = 2500 rating points... :) What is the initial mistake? It is that you assume that the current rating indicates the player's strength. But this is not the case. And we have been talking about this for a long time - the current rating has nothing to do with the player's strength. It shows the player's activity, taking into account his strength, which, I agree, is not the same thing... Any newcomer, as well as any inactive player, has a zero rating. This does not mean that they play poorly, and therefore the bonus should not have such sharp jumps...

The method of calculating the bonus from the average rating of all tournament participants, which I proposed earlier, will not lead to such an imbalance because it takes into account all players at the same time, and not just one... Before this idea was voiced, it was, of course, repeatedly tested... :)

Therefore, I ask everyone who wants to demonstrate their calculation method to first test it for extreme situations, see how the function will behave, and assess the probability of these extreme situations occurring... The example above shows that in an extreme situation, a player will receive an unattainable bonus, and the probability of this situation occurring is 100% :) Because the tournament winner may very well be participating in a tournament for the first time... :) Also, a strong player can easily skip a few tournaments on purpose to lower his current rating in order to then increase his rating after a brilliant performance. Rating calculation is, first of all, logic, and then mathematics.