Now imagine that your neighbor's party kept you up late, causing you $50 worth of damage. Sure, you could sue, but your neighbor doesn't believe that you will pay even $500 in legal expenses to sue them for $50; even if the court made them pay your legal expenses, you'd have to be unrealisitically confident that you would win to make it worth your while. Thus your neighbors have no incentive to settle out of court, and hence no legal incentive to not throw loud parties. Thus our legal system fails for small harms. Yes, less formal social incentives often substitute for law, but probably not often enough.
Here's a way to fix this: let people risk their lawsuits, double or nothing. After that damaging party, you would write out a simple complaint, including who hurt you when and how, and then take this complaint to the official lawsuit randomizing office, who would then randomly declare it worthless (50% chance) or double it (50% chance). If your suit were doubled, and you took it to court and won, so that the court said your neighbor caused you $X in damages, they would really owe you twice $X. And if you gambled and lost your neighbor would get a record of this, to defend against your trying to sue again over the same complaint.
More generally, you could keep doubling your suit so that it is worth any power of two times $X, or nothing. So if you gambled your $50 party suit ten times, for a factor of 1024, then 1023 times out of 1024, you'd get nothing. But that last time you'd get a suit worth $51,200 if you won. Then it would be worth your while to actually go to court to win it. And since your neighbor would know this, they should want to settle rather than go to court. And since they know, before you gamble your suit, that this is a possibility, they should be willing to settle up front for near the $50.
Thus double or nothing suits should create legal incentives for people to avoid hurting others by small amounts, as well as by large amounts.
Should there be a limit on how far people could double, so that the accused isn't at risk of losing their life savings over some small harm? Not only would it be hard to set such a limit, but I'm not sure we need it. Someone who doubles so far as to take near the total assets of someone else risks not getting paid the full amount of their suit. Better to get $50,000 for sure than half a chance of being owed $100,000, and only being paid $70,000. Also, I'd let the person being sued also risk the suit, double or nothing, as long as they showed they were good for the maximum damage amount.