ThunderGr: No idea what other tasks are done on "game end". Same goes for the other "do you know" questions. I don't say that this is implementable code. This is a description, pseudo-code.
To implement correctly one certainly has to consider side effects.

When I say section (not session) over, I mean the section as a whole in the tournament. NOT individual games within the section. Those should continue. All we want is for the tournament to progress.

"For" loops can be costly, but those I described shouldn't be. they are on the number of players in a section and one on the number of sections in a round.