Which sociological approach would view sports as an agent for defining people's social positions as players, coaches, and referees as a result of their performances and reputations?

Answer: interactionism or interactionist Perspective

Explanation:In sociology,

interactionism is a perspective which focuses on the follwing social processes conflict, identity formation and cooperation as all part of human interaction. Interactionist study how society act as they interact with each other. According to this perspective even gender stratification is due to the meaning created through interaction with each other. So in each an every interaction between people, meaning of certain situations are invented to make sense of the world around us.

"defining people's social positions as players, coaches, and referees as a result of their performances and reputations" our interaction with sport will lead to defining each person position based on their performance as the way of making meaning of the game and the people involved in it; and this is all part of identity formation.