It would really depend on the child at hand and his personality and particularity his social skills. I would be more worried about gaming is an 'out' or easier past time for a child then playing outside, bullying not withstanding of course.
As how strict would depend on how influential a child is, pretty much anything that bases a story on more complex term such as murder, revenge, betrayal or actions such as intentional murder with a particular sense of 'murder', say hitman esque' games. Gratuitous violence or games particularly narrow-minded and stereotypical to the point of bigotry to anything from sexism to anti-gay or anti-<insert foreign religion/people> here should be monitor at pre-teen to teen to young adult even(yes even after puberty the brain still develops quite a bit) and denied at very impressionable ages.
When a child shows a strong sense of self and isn't impressed with behavior or world views i my self would never want to see in another human being let alone my own child, these of course topics of looking down on people, masochistic, racist not self expression trough non oppressive actions regard to dress, taste in music and food. That is personality, not ignorance and hatred.
In such cases, i would be more inclined to let them view more diverse content, including the aforementioned subjects in order to for them to explore and reckonize it, and hopefully i will be blessed with children who will take time to discuss such things and become strong well manner thinkers and respectfully people.
i gotta stop making walls of text.
tl;dr, at a young age definitely, depending on how impressionable a child is monitoring will relax accordingly.
