It is out of respect that he does it.
Just like in the Batman section...even though I have ideas for Elimination Games I do not just go and make the next round myself. Doc Ock does the games in Bat-World and we keep it that way. That is what the suggestion box is for.
But the person who started it gets the respect to be the only one to start the others...unless he gives permission to someone because they are going to be somewhere.
In The Comics -> DC Comics thread there is a game where you choose who would won in a fight between 2 superheroes/supervillains..it is the same thing. That is the only one that I recall other people starting new ones though because the original person could not at the moment.