So many things I would like to say here, but I am a little new to BYC.
the term "game" can mean ALOT of different things.
1) Game-- one of the breeds of chicken bred for pit-fighting, for example, Hatch, Roundhead, Kelso, Claret, Whitehackle, etc. This is usually called "dead game" or "pit game" meaning that the roosters will fight to the death (but not all of them, some will quit, they cant stand the pain.)
2) Any of the breeds of chicken that can survive and thrive running out.
Loosely speaking, you can actually call any of the landrace breeds "game" because of their survivor instincts, and ability to raise with very little help from us. The hens set eggs and raise babies very well.
You can say that any of the following are games.... Fayoumi, Sumatra, Buckeyes, Old English Game Bantam(which do not fit the above category), even Silkies fit into this category, ignore all the fluff and glamor and you will see it.
I have hens that are very mixed up that I call "game" They are from an Asil X Sumatra hen and a Phoenix rooster, some are from Denizli X Ohiki rooster and an Asil X Pitgame hen. it is the mindset of the bird.
Leghorns dont qualify They lay too many eggs and never set them
RIRs dont qualify, they dont LOOK the part, and set to rarely
Pure Aracaunas DO qualify. They are a landrace breed.
3) Wild-Type fowl, like: Any of the Junglefowl, any F1 crosses from them, any Pheasant, etc (Gamebirds)
I wanted to add that all games are not mean.
In fact, all Pit Games are not overbearing. Some of my most loyal and productive and gentle mothers are birds I got from Cockers. If they werent good mothers, they wouldnt reproduce so well running out in the yard. I have two hens that have raised THREE batches of babies so far this year, and I expect at least one more before cold weather gets here.
Hope this helps