I came across this on a google search and since it's an older post I wanted to add how I've worked with Facebook on stuff like this:
I create a simple group for people to join using their regular public profile. I develop an excel spreadsheet to track stats, do dice rolls, etc. and then usually design some simple maps on Power Point.
The game works like this:
I post a Poll to the group with a long description of the who/what/when of the game and provide them with a number of options to select in the poll (you can allow them to add options, have multi-option polls). They discuss the situation in the comments and put in their votes. After a day or so I see who has/hasn't seen the post, who has voted, and close the poll. I roll for their selected option and post the next description + poll.
The format is sort of a group based, choose your own adventure but it can be as simple or as complex as you'd like to make it. You can style it as a small group, the actions of one person, or as a civilization/government taking actions.
The one's I've run that have been successful were a small stone aged tribe in which players were the "Council Elders," and decided what the tribe would do. We did a sort of senate-style one for a bronze age civilization and then one in which we were guiding a roman-style legion during a wider civil war.
Included in that, you could allow for individual characters and rolls in FB messenger chat (which I have seen done in other games) but come back to the main "poll" as sort of what your party decides to do if you really want to provide the single player character experience.
Thought I'd add that since it can be done in a variety of ways as folks have mentioned. It works for our group on FB because we have a lot of casual people and nearly everyone checks FB regularly, so we can get through 1 or 2 polls a day and move it along. If someone misses a vote the game goes on but the group page makes it easy to go back down the threads and posts to catch up.