"Games" are game birds and RIR is Rhode Island Red.
I have an RIR in my mixed flock and she's not a problem, but then mine free range so they all have plenty of room to avoid each other if necessary and she isn't top hen anyway.
You might also want to consider blue, green and/or olive egg layers if you are interested in having a bonny coloured egg box. Legbars, Easter Eggers(EEs), Araucanas lay coloured eggs and Marans and Welsummers lay beautiful darker brown eggs (toffee and terracotta coloured).
I would say that probably most of us hobby chicken keepers have a mixed flock and there are certainly benefits to it..... some breeds are more inclined to become broody and raise chicks, some are egg laying machines, some will lay right through winter and others will stop and then as previously mentioned, some lay really pretty coloured eggs. Also some breeds are really friendly whilst others are very flighty but also perhaps better at evading predators, if you are thinking of free ranging.
Lots to consider when thinking of which breeds to start with. I love the fact that my flock of birds look pretty and when they are all different, you can tell them apart easier.
Good luck with choosing your flock.
Barbara