one-upmanship
Pronunciation:
/(ˌ)wʌnˈʌpmənʃɪp//wənˈəpmənˌʃɪp/noun
Definitions
The art or practice of successively outdoing a competitor.
Example: They are bent on one-upmanship. I think it's hopeless to try to stop them.
A succession of instances of outdoing a competitor.