Realised I had missed this out somehow as I worked through the Hugo winners, so went back to find it. History does not record what other works were in the frame, though other classic sf from 1957 which I have read includes The Door into Summer, The Black Cloud, Citizen of the Galaxy and The Midwich Cuckoos (and On the Beach is on my shelf, recently acquired but as yet unread). I would rate The Midwich Cuckoos more highly than The Big Time, but it was probably too British to be considered by whatever mechanism the Worldcon was using that year.
I’ve always liked this one, apart from one silly moment at the end – the psychological drama is resolved when two male characters decide to trust each other because they attended the same Cambridge college, several centuries apart. I hereby give notice that if the Right Honorable Peter Lilley MP, let alone some time-travelling avatar of Lord Cornwallis, both of whom are fellow Clare graduates, should ever try this on me they will get a rude response.
Apart from that, I love the setting – an enclosed space beyond space and time, a rest station in the ongoing Change Wars between Snakes and Spiders, two time-travelling factions changing the history of Earth (and, we understand, of many other worlds) for thei own ends, with little regard to the human and other lives that are put at stake. The story is rather theatrical in presentation, and one can easily imagine it being put on stage. Not as mature as his other Hugo-winning novel, The Wanderer, and with as I said a somewhat silly ending, but very entertaining all the same.