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The police car is rarely a neutral presence in a game. Whether the player steals one, or is in legitimate possession of one, or is trying to get away from police officers driving one, a police vehicle is often a symbol for authority and power.
When a player is involved in criminal activities, such as in the Grand Theft Auto
series, spotting a police car is the sign for the player to at least act like a law abiding citizen, to steal the vehicle and get a few bonuses, or if it's too late for that, it's best for them to run until the trail is cold (or go to jail).
It could be argued that most games rarely allow for players to interact with police cars and their drivers in the way that most players do in real life. Many public figures often cite the fantasy violence in games, often having to do with police and other authority figures, as endemic with what is dangerous about video games, while players themselves tend to scoff at this, no matter what side of the wheel they're usually on.
Other nicknames for police vehicles include black and white, prowler or cruiser, often used by law enforcement. "Cop Car" is a common, neutral phrase used by civilians.