A game in which there are two separate but interconnected layers of gameplay.
In the first, people are trying to complete a series of non-violent, not-mutually exclusive objectives within the time limit. They aren't competing with each other, they're just running around the map doing stuff. These are the civilians. The second layer is two teams playing a standard FPS deathmatch, with an exception: all of them are penalized for killing civilians. Some roles could have objectives that cross over, for example a doctor character who is supposed to treat all wounded on either side.
An example of one way this could go:
The map is a city street that contains eight houses, a police station, a bank, a hospital and a gun store. There are 8 civilians, 4 policemen, and 4 hardcore criminals. On the civilian layer, most players are ordinary civilians with one being the doctor and a third of the remainder being petty criminals. There are two shifts. Each civilian will spend one shift working at one of the three stores, and another completing an objective: either going to one of the stores and receiving a service if they are a law abiding citizen, or robbing a building if they are a petty criminal. If they work at a place they are responsible for grabbing things and giving them to people if they bought them, but also being alert to petty criminals trying to steal or fool them out of money/guns/painkillers. All citizens can call 911 to call a doctor or report crimes in progress (including the appearance of the criminal); petty criminals can spend a few seconds lockpicking doors but don't have guns unless they steal them from the gun store.
Cops start with body armor and a pistol, as well as their choice of a taser, baton, or shotgun. They can arrest anyone if they get close enough, and they are informed whenever a crime is reported. Career criminals are usually bank robbers who have pistols, but they can also be junkies who have knives. Robbers want to break into the bank vault, junkies want to break into the hospital and steal painkillers. Both objectives take some time, but the criminals can take hostages to encourage the police to leave them alone.
If there are less people buildings and roles get subtracted in order of importance. Rounds are reasonably fast (no more than ten minutes certainly). At the end everyone is scored; civilians and robbers get points for completing objectives, police get points for every crime stopped and criminal arrested, everyone gets less points for killing people (depending on their role and who they killed), anyone who lives gets points, and police are penalized for all crimes that happened (more if they were unpunished). Everyone on the side of the law gets points if they report a crime/injury and more if it was successfully dealt with by the appropriate service. Everyone working at a building gets penalized if it is robbed during their shift, everyone gets penalized if their house gets robbed. There is some system in place to make people want points.
Probably too complicated and massively vulnerable to abuse. Plus it would be hurt a lot if people didn't voice chat. Still I think it could work. The idea behind that scenario is that civilians might have a boring day, but they might also get caught up in a crime/shooting based medical emergency and have to deal with that. You could get emergent stuff like police asking civilians where a criminal went, or a hostage in a bank robbery getting shot in the legs trying to escape and the doctor has to get inside to treat them. Also it would be an FPS where people have to be careful who they shoot.