Cats definitely do initiate games. One of mine plays peek-a-boo from behind the TV. She disappears behind it, I express dismay about having lost her, and she pops her head out from behind, usually with a big trill. At times, the others have initiated a game of fetch or "chase me," (which they usually win).
Mine also sit in front of the TV when I'm trying to watch a program. I think every cat I've ever had has done that. They don't often watch what's on, but sit and look at me for attention.
Mine also sit in front of the TV when I'm trying to watch a program. I think every cat I've ever had has done that. They don't often watch what's on, but sit and look at me for attention.