One of the suggestions submitted by Miguel Balauag from Stanford:

For small-sized games, I've found that starting with nightfall and killing a player often tips the game too much in Mafia's favor, but starting with daybreak isn't productive because there aren't any leads. One suggestion is that the first nightfall is actually a "botched" murder, and that the victim stays alive, and the attempted victim is still announced at daybreak. This can start the game off with some leads, and lets a player who would otherwise be out for the rest of the game stay for a while (it also helps that people know the player is innocent).

However, the Mafia could be sneaky and do a botched murder on one of their own to make them seem innocent; to prevent that, you can either rule that illegal, or the moderator can choose to NOT let the players know beforehand that the first murder will fail.

The moderator could easily do something like flip a coin to determine if the murder is botched or not. Thus the Mafia would have to take a risk if they want to be sneaky.