The basic idea of this competition is that there are 4+ selected leaders, each is in charge of a team. Each team has a thread where they keep up with their roster and can communicate with each other, unless they want to do it privately elsewhere. Players may apply to the teams they like through the threads, but they may ultimately only be on one team if they are accepted by more than one. When applying, players should specify what games they are willing to play. In the end, each team will have the appropriate number of players for each game, and they will compete in them against other team's players. The leader himself/herself may also compete in the games. Whatever team has the most wins in the end wins the whole competition. As for the games, I have a few in mind, but it really depends on popularity. If there are only 3 Team Fortress players interested, then it's probably a no go. There are probably several other games that could be played that I haven't heard of or have neglected to mention. This is not a definite list:
Chess: lichess | Free Online Chess
HaxBall: HaxBall - Play
Pokemon Online (download required): Pokémon Online
Team Fortress 2 (download required): Team Fortress 2
Transformice: Transformice !
[Your Idea Here]:
I also think it may be important for each leader to create his/her team with a sort of theme, just for fun and flavor. For example: The Bronies, Team Rocket, The Ninja Alliance.