Settings: Linking Twitter with your Game Center

Your Game Center can be linked with Twitter so that "tweets" are sent out before games start and also during games as they are being played (with updates between periods). This applies to all games listed in your account.

Enter your Game Center Manager and go to Settings:

Activate the Link Twitter function ...

... and then configure/authorize it with a Twitter account of choice. 

The tweets will then send automatically based on your games.