Question: What definition does contemporary analytic philosophy give of 'concept'? And what is the difference between a concept of something and a conception of something? Then what's the difference between concept, conception and theory? 
I'm trying to establish an epistemological priority like conception>concept>theory.

Answer: 
The word "game" makes a good metaphor for the interaction between people in a society where there is gain and lose and where the interactions are not happening chaotically but in more or less systematic ways and according to social norms some of which are clear to articulate about while others are implicit. But I think that the use of the word "game" became widely used after the publication of the popular book "Games People Play: The Psychology of Human Relationships" in 1964 by the psychiatrist Eric Berne which became bestseller and of which millions of copies were sold since its publication.
It is probably a reference to mathematical "Game Theory" which studies conflict and cooperation between intelligent rational decision-makers seeking particular positive outcomes, subject to a fixed set of rules. If you think of the "set of rules" as those of a given society and of the "players" as the members of that society, then their interactions, decisions, aspirations, etc., which are the subject of Sociology, can be thought of as a "game" in the sense of Game Theory. The word "game" in this context is not meant to diminish the "seriousness" of the subject being studied or the seriousness with which the subjects being studied behave and act.