From Longman Dictionary of Contemporary Englishone-upmanshipone-up‧man‧ship /wʌnˈʌpmənʃɪp/ noun [uncountable] BETTERattempts to make yourself seem better than other people, no matter what they do
Examples from the Corpus
one-upmanship• Watching the children together gave reassurance that childhood is not all one-upmanship and go-for-the-jugular. 