Before The Last of Sheila

The story so far: Joe Belfiore created the Game after being inspired by Midnight Madness, which was inspired by Don Luskin's Games from the 70s, which were inspired (at least "in part") by The Last of Sheila. So what inspired The Last of Sheila?

Tony Perkins and Steven Sondheim wrote [the film] and they were ardent game players in real life. If you went to their house, if you had dinner there, there was always a big, complicated type of game. It wasn't charades; it was a very complicated version of charades. There were puns... anyway, it was always complex and it tested you. Everything always tested you. So they wrote this script that is actually a game. - Richard Benjamin, commentary on The Last of Sheila

Pretty cool, except that it even goes beyond that.

The movie was inspired by an irregular series of elaborate, real-life scavenger hunts [Stephen] Sondheim and [Anthony] Perkins arranged for their show business friends . . . in Manhattan in the late 1960s and early 1970s. The climax of one hunt was staged in the lobby of a seedy flophouse, where participants heard a skipping LP record endlessly repeating the first line of the Harold Arlen/Johnny Mercer standard "One for My Baby (and One More for the Road)" ("It's quarter to three...") The winning team eventually recognized the clue -- 2:45 -- and immediately headed for room 245 of the hotel, where bottles of Champagne awaited them. - Wikipedia

So a composer and a Psycho actor held puzzle hunts! From what I can tell, Sondheim was the major driver, though.

There's a story about Stephen Sondheim's affinity for puzzles and puzzle hunts, such as one he threw where players had to visit this house hosted by an old lady (Stephen's mother). The lady would serve you cake, and if you ate it, you lost: there was a clue in the frosting. - Dan Sanderson, Brainlog

Where did his inspiration come from? Sondheim did have a facination with puzzles: He wrote crosswords for the New York magazine, and is considered to be the one who brought British-style cryptics to America. Perhaps he was inspired by other musicians?

I was brought up, you know, by Oscar Hammerstein in my early teens, and he liked anagrams, but it was the decorous kind. And then, when I met Leonard Bernstein, he and his family played cut-throat anagrams, and that's how I got into that. - Stephen Sondheim, Academy of Achievement Interview

I'd like to go back further, to find the true source of Sondheim's inspiration for puzzle hunts. There's a gap in what I can research, though, but the Game is a type of treasure hunt, which is generally considered to have been created by Elsa Maxwell:

[Elsa] is proud of having invented such games as Treasure Hunt and Scavenger Hunt, because of their psychological importance. Not unmindful of science (she once devoted most of a column to the fact that she has never had to blow her nose), she says: "Let's break them down scientifically. In the Treasure Hunt . . . intellectual men were paired off with great beauties, glamor with talent. In the course of the nights escapades anything could happen." - Time Magazine, Elsa at War

I can't seem to nail down when she "invented" the treasure hunt, but I can make an educated guess as to what might have inspired her. Going all the way back to 1881:

Though it's not where the premise first appeared, Robert Louis Stevenson's Treasure Island was instrumental in planting the treasure map concept in the popular consciousness. - Cecil Adams, The Straight Dope on Pirate Treasures