At least 4 players are dropped into an arena. None of them can see any one's username, in chat the names will be garbled out or changed to something generic like "Greg" or "John". However, one person can see the username's of the others, they're the ones in possession of the Death Note.
The Death Note is presented as a Book and Quill. The Death Note User would write someone's username down in it and close the book. In 40 seconds that player will die. The objective of the game is to find and kill the Death Note User.
Each game lasts a varying amount of time, and the end of that time limit the Death Note User will die and the game will end. The way time would be determined would depend on the amount of players, with one minute and ten seconds added for each non-Death Note User present. Meaning that if there were five players in a round, the round would last 4 minutes 40 seconds.
All players are in possession of a bow that kills with one fully charged shot. To prevent players from simply killing everybody, anyone that kills someone who is not the Death Note User will die. Not only that but a random non-Death Note User will die as well. The penalty would prevent a four player round from ending by one player simply killing the other and themselves, leaving the non-Death Note User and the Death Note User alive.
This means that the only way to catch the Death Note User is to catch them while they're writing. While this does seem to put the user at a huge advantage, it can be avoided by having the group stay together at all times. Although that seems like it puts the User at a massive disadvantage, the game is lost and all non-Users immediately die if the User is not killed by the end of the round.
I don't really like this idea. I prefer the old request for a Death Note mod where there's one or two copies in the world with invisible mobs called shinigami that eats apples, for SMP and with the requirements of both seeing the player before managing to kill him and knowing it's name to actually write it in the Death Note. Of course, such mod would be hard to create since you would need to modify the entire chat engine and hide all the player names.