get one up on someone
The public will soon get tired of this one-upmanship.
Behind the competition and one-upmanship that had always existed between them, there was solid friendship.
One-upmanship was the deciding factor when it came to how high they went.

SMART Vocabulary: related words and phrases

Competing and contending (non-sporting)
as a dare
be one up on someoneidiom
cat
catchup
challenge someone to something/do something
dare
get one up on someoneidiom
give someone/something a run for their moneyidiom
go in for something
go toe to toeidiom
hyper-competitive
jungle
pit
sharp elbowsidiom
sharp-elbowed
sharpen (something) up
sharpen your elbowsidiom
showing
snap
toe

one-upmanship | American Dictionary

one-upmanship
noun[ U ]
us/ˈwʌnˈweɪ/(alsoone-upsmanship, us/ˈwʌnˈʌps·mənˌʃɪp/)
an effort to show that you are better than someone you are competing with:
We heard the usual boasting and one-upsmanship before the boxers entered the ring.