My kids are older now, the two youngest are teens, the two eldest young adults, so things have a changed a lot.
When they were young, I would do a lot of research on games before I would buy them or allow them to buy them. There used to be sites where parents would rate or grade the games and I would look for ones that were fun for the kids, challenging, and had some sort of value (although not necessarily academically oriented). It worked pretty good and the kids were happy for the most part. I remember games like Picman and Harvest Moon, Zelda games (they still love those), Kirby, Spyro…etc. These were are not necessarily games that will teach them something concrete, but overall, they were fun, engaging, low violence, and had some degree of problem solving…if you look hard enough at times. lol So…often I would see games they wanted and do the research and tell them…or sometimes, they would hear of a game from a friend or see it on TV/internet. So a mix of the two I would say. When they were wee little ones, I would pick them.
They have since graduated to Final Fantasy, Minecraft, Portal, and other similar games.
My son just reminded me that one his favorite games as a kid was The Incredible Toon Machine, made by Sierra. He remembers it very fondly. He loved the problem solving (He said that without any prompting by me! LOL…I am so proud!)
Anyway…I think I answered your last question. Because I helped them choose good, quality games early on, I think now that is what they prefer. Yes, there is a little more “action” in the games they play now. My daughters are currently into Minecraft and Assassin’s Creed. My son, between college semesters, plays Elite Dangerous.
So yeah, it changes at they get older. I say what they cannot bring into the house, and they tell me what they want. We discuss the games, decide if they are okay. I am not okay with extreme violence, blood and gore. I know they will play them someday, maybe…but not at my house. It is a line I draw.
Still though…I have some input, either because I tell them I want to have input, or more than likely now, because they know I will be reasonable without our limits. It has worked out well.
Also…I did and still do for my younger ones, limit game play time. So games that are ultra addicting or where you couldn’t save whenever you wanted, drove me nuts.