Games run until a winner emerges, usually about 3 weeks, and new games start when there's enough demand. Normally there will be 2 games running, numbered 3 and 4 for complicated reasons.
Game 3 is "open", with players allowed to join without any checks. This is good for ease of getting into it, but bad in that some players will run multiple positions and/or drop out without playing any turns, and possibly fake each others' orders to sabotage rival positions.
Games 4 & 5 are "closed", with players only allowed to join if they have an account with w++ Games, and orders are only accepted with the correct password. It's best to try out game 3 before going to the effort of opening an account.