I think it would be very cool if NPCs had certain objects of interest they could see or find in player structures that they themselves would want. Say you leave the door to your house open for a while, you may return and have some items missing! Now, this would be super annoying if you return and your place is empty, but I think it would be awesome to chase burglars away or track one down to get back your precious goldfish. I'm not too sure about players stealing from NPCs though, because have a heart! We're all just trying to make a living out there!
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As you said, it would be very annoying. There's no real point to it, as chasing NPCs would quickly become a chore (especially if they get away and you're left with missing items). I don't think this is something I'd like to see in the game, sorry. But, your mention of players stealing from NPCs might be worth a look. Combined with a good stealth mechanic, this could lead to some good thievery quests.
