So how can I prevent them to send me game requests? I imagine there must be an arrangement in Facebook to let me stop sending me game requests.
You cannot block requests from all games in general. You can ignore requests from a partiicular app or a specific friend from your requests list.
Or, if you don't use any apps on Facebook (even those that are simply about allowing you to connect your FB account with another website), you can go to Applications tab in your privacy settings and disable Facebook Platform at all. After doing that you will not be able to use any of FB apps.
Or you can ask the annoying person to stop sending them. But sometimes they don't stop because FB is constantly nagging them to do so. Then your best bet is to block all app requests from that individual. Go to Privacy Settings -> App Settings -> Blocking, then add their name to the list under Block App Invites.