Definition of one-downmanship in English:

one-downmanship

Pronunciation /wʌnˈdaʊnmənʃɪp/

noun

(also one-downsmanship)
  • The art or practice of being or appearing to be at a disadvantage. Compare "one-down", "one-upmanship".

Origin

1960s; earliest use found in The Times. From one-down + -manship, after one-upmanship.