But in the larger context of improv, game has a slightly different meaning - but can be applied to both short-form and long-form improv. The game of the scene is what the main through-line of the whole scene is. Improviser and blogger Erik Voss has a great blog post over on Splitsider that tries to break down the definition of the game of the scene.
The game is any pattern that emerges within a scene that the improvisers may follow while exploring the relationship between the characters.
In short, the game is what makes a scene interesting - from one of the quotes on the blog post: "I want to see that improviser do that more." There might be lots of ideas that are thrown at the wall at the beginning of the scene, but the game of the scene is the one idea that sticks to everybody and makes the scene interesting - interesting to watch and interesting to perform. It doesn't have to be absurd or silly - often times I catch my high school students "going for the funny" in an attempt to define the game of the scene, and it falls flat. Other times, they have had an outstanding dramatic scene because they listen to each other, work with a group mind, and collaboratively arrive on the one throughput for the scene and stick to that.
Erik's own blog Vossprov has a great image to summarize it:
Lots of times high school improvisers can't find or don't trust that a single game of the scene can carry them through the rest of the scene. That's what Erik's If, Then refers to in the diagram - If the game of the scene (the first unusual thing) happened, Then what would logically happen next?
So feel free to take some time to muck around with some scene ideas. Set up the reality of the scene, characters, relationships, etc. But once you find the game, go with it. If you stay true to this exercise, then your troupe will also develop the ability to develop and find better and better things to make the game of the scene, and your scenework will continue to improve. Give it a shot.