I guess I don't have a problem with that, but that's now how I read just2kicku's comment.
"In a word....NO! I would not let them run, I would deal out my own brand of punishment. Even with the laws, I would still do the same. I would do everything in my power to make it as painful as I possibly could before putting them down."
I read 'putting them down' as killing the person. My apologies to both of you if that's not what was meant.
Even if that's not what was meant, I'm envisioning some of those responding here to be picturing some kind of martial-arts movie kind of scenario, where the good guy, whose family has just been brutally done in, stands and delivers, choosing to use less-lethal-but-more-painful techniques to exact revenge and inflict pain, before heroically delivering the coup de grace just before the police arrive, skidding to a stop with lights and sirens blaring on the dark, wet, backstreet near the factory, etc.
Makes for great movies.
Whilst I've never had a family member gunned down in a hail of bullets and had to tear my shirt off and get all medieval on some punks from out of town, I have had to chase down and tackle bad guys, occasionally armed bad guys (mostly I found that out later). It really doesn't work like all of that, in my experience. And that was 20 years ago, when I had no MA training but was in good physical condition. No protracted stand-up punch-em-up, just a wild swing or two, everybody falls down (tripping, sliding, and falling over accidentally happen a LOT more than a MMA-style takedown, in my experience), hurt knees and elbows, glasses broken, nose bleeds, and I finally get the guy over on his belly and put the hand irons on. Shirt torn, hat missing, baton twenty feet away, taste of copper in my mouth, panting like a steam train, handset screaming at me with 'what's your 20?' questions from fellow officers, and wondering why the heck I chose that career path.
My guess is that in the OP's 'scenario' that's pretty much how it would go, presuming a) I could catch the guy, and b) I could subdue him once I did. Granted that as a civilian, I'd have no hand-irons and no backup - so I might have to run his head into the pavement a few times to get him to quit fighting to get up. I think revenge, anger, extracting vengeance, and so on would be out of my concious thought until much, much, later. This is primitive mind stuff - survival, immediate needs, stuff like that. I'd be more into finding a place to pee than thinking about how to put a dramatic flourish on my ax-kick to the back of his noggin.
No offense meant to anyone who has a more dramatic notion of how things would go, but since rolling around in the dirt with drunken idiots used to be my job, I kind of remember how it goes.