The Letter Game is an improv writing exercise in which two people develop a story by writing a series of letters back and forth.
The game is simple enough: player #1 writes a letter from Character A's point of view, addressed to Character B. This letter will likely be responsible for laying out the basic setting and characters of the story, but might not give any strong clues as to the plot. Player #2 then writes a response, from the point of view of Character B, addressed to Character A.... and so on and so forth. The two writers should not discuss the story outside of these letters, and once a letter has been sent, it cannot be changed.
This is an interesting exercise, because it gives you practice in an art that usually is looked down upon -- writing around another person's story. This is usually the sort of skill needed by writers of fan fiction, and is rarely needed by 'more serious' authors. However, some collaborative fiction of this sort has made it into publication. A very brief list includes:
The Letter Game may also be referred to as Persona Letters or Ghost Letters. It is a subset of collaborative writing. It is somewhat related to Exquisite Corpse Poems, Eat poop you cat, and Improve Acting. Somewhat.