On the main screen, there's options to vote for the Senate and the World Council, which the game says will globally affect players. However, it doesn't specify how often you get to vote for a new initiative or president.
The descriptions are vague, saying that "frequent" votes take place for the Senate, and new presidents are appointed on a "regular" basis.
For example, this is the current Senate vote.
As you can see, some of the bonuses are quite good, but I'm wondering how long these bonuses will last before new ones are cycled in.
Best Answer
Interestingly, I have done a bit of research into the exact time of this, but I am unable to clamp down a time. I have only seen it mentioned as "I voted for x this week" - so I can logically deduce it's 7 days. However, Global Events end every 30 days, so either:
7 days OR 30 days.
Logically: 7 days because of the bonuses it provides and players may want something different in their Anno 2070 democracy that week.
Best guess: 7 days.