What kinds of literature could be assigned as companions with The Hunger Games?"The Lottery" by Shirley Jackson is one I have. What others could work in a potential text set?
If you are looking for short stories or other examples of literature to help you, I think that you want to find works that play with the theme of how the individual responds to a bizarre world. Ursula Leguin's "The Ones Who Walk Away from Omelas" is an excellent example of this idea. It is something that can get the reader to think about how bizarre the world of Omelas is and what parallels can be seen in our world today. Another similar example of such a story would be Bradbury's "The Veldt." In Collins' work, children are set to be the hunted. Bradbury's short story flips this in articulating what happens when the parents are the hunted? Exploring this inversion of power, contrasting it to Collins' work, and examining how violence is used by those in the position of power would all be interesting ideas to emerge. Lowry's book, The Giver, is another great example of the bizarre nature of a world where the protagonist has to make a choice as to whether they will instill the idea of justice in a world that lacks it. I think that these might be good starting points for you.
I think Ray Bradbury's book, Fahrenheit 451, could also fit in nicely with that group, because it is also in a futuristic setting that is different that our modern one.
I am actually doing the same thing at the moment! Im allowed to pair my novel with a film; so I chose to do;
Its a rather gory film, so if you cant handle that then I wouldnt suggest it. However it explores very similar themes to that of the Hunger Games. :)