noun: In conversation, the act of purposefully avoiding the pronouns "her," "him," "she," and "he," to hide the gender of the person being spoken about from the listener.
Common substitutes include: they; they're; someone, this person; my friend

The particular phrase "pronoun game" comes from Kevin Smith's film "Chasing Amy" though the idea is pretty common.
by wayfarer March 25, 2006
"playing the pronoun game" is when a character uses a pronoun such as "he" or "they" in place of a name forcing another character to ask "who?". Often used in films to give an excuse for exposition or as a cheap way to add mystery.
interviewee: he's coming
interviewer: who? who's coming? you can't just say "he" and expect everyone to know who you're talking about don't play the pronoun game with me
by fringnubs December 14, 2016