The Anarchy System consists of making the decision of one Twitch user something viable in the game.
This said, let's say everyone wants to go up so they can enter a Gym. Then the Main Character will go up, but if just one person wanted it to go down, the Main Character will also go down. This is small compared to the reality, where there are more than 10,000 users wanting to do different actions and having different objectives (Grind, Heal, Buy, PC, etc).
The Anarchy Crisis
The Anarchy System is problematic because the decisions of the Hive Mind are different, making it impossible to reach or make certain goals possible without having problems, like Releasing Pokémon in the PC, falling off The Ledge, etc. This of course was not a problem until Team Rocket HQ Maze, where after more than 24 hours, RED was unable to get to Giovanni succesfully. Because of this, the counter part of Anarchy was created, Democracy.