Bounties. From the Wikipedia article about Jesse James:
Crittenden had made capture of the James brothers his top priority; in his inaugural address he declared that no political motives could be allowed to keep them from justice. Barred by law from offering a sufficiently large reward, he had turned to the railroad and express corporations to put up a $5,000 bounty for each of them.
Though this isn’t a perfect free market example, it shows that capturing serial murderers, master thiefs, etc. could be rather efficient in a market system where businesses are allowed to place bounties on aggressors.