The notifications about finished games, tournaments, etc. come in our inbox as well as the personal messages. They often say something that we already know and are not something that I want to keep, so that I am deleting those messages very regularly, in order to keep the view of the personal messages sent to me. This is a repetitive and boring task.
Those notifications are useful, but it would be much better if they came in a separate, new mailbox, called something like "Notifications". Then either we could leave them pile in there, either have a "Purge" button emptying this mailbox, either they could be scheduled for deletion after they are more than one month old, for instance. In the latter case, we could still choose to keep any notification we want, by archiving it.