Quote Originally Posted by SadieKate
Is that the right term?
from merriam-webster (so, it's not Oxford)

"one-upmanship"

Pronunciation: "w&n-'&p-m&n-"ship

Variant(s): also one-ups·man·ship /-'&ps-m&n-/

Function: noun

: the art or practice of outdoing or keeping one jump ahead of a friend or competitor <engaged in a round of verbal one-upmanship>