Hm, I thought you play a game. And you lead a game, not a play. I always thought "a" play is something you connect to music as in "The play of Daniel" from Pro Cantione Antiqua directed by Mark Brown. But my english isn't that good.
In German <you play a game> is "Du spielst ein Spiel" so it's the verb and the Noun using the same word. Maybe you can say <you play a play> but it isn't something I heard of 'till now.
Game Master == Spielmeister, translated literally.
Play Master sounds really odd but would mean the same.
That's a good reason to stick with MASTER == MEISTER or HERR.