In a round-robin tournament, each player (or team) plays each other competitor (usually once, sometimes twice in a double round-robin tournament). In some competitions (e.g., World Cup in soccer and cricket), there is a preliminary round-robin phase where members of a pool play each other to determine which go through to the later elimination stage of the competition.
"The term round-robin is derived from the French term 'ruban', meaning 'ribbon'. Over a long period of time, the term was corrupted and idiomized to 'robin'."
The phrase round-robin had earlier been used to describe "correspondence to a single address authored or signed by numerous individuals (as found in a petition). ... The modern use of the term dates from the 17th Century French ruban rond (round ribbon). This described the practice of signatories to petitions against authority (usually Government officials petitioning the Crown) appending their names on a document in a non-hierarchical circle or ribbon pattern (and so disguising the order in which they have signed) in order that none may be identified as a ring leader."