to get an advantage over someone by doing something better, more extreme, etc. than them, or by saying that you did or will do something like this

They were trying to one-up each other over who had had the worst year. 他们试图在这一年谁最惨的比拼上压对方一头。

Every governor is one upping the other governor on how tough they can get. 每个统治者都在和别的统治者较劲看谁在（手段）强硬上更胜一筹。