If you want to punish a common or garden psychopath (also, sociopath), it is advisable to move out of their direct sphere of influence first. Then you can decide how the punishment is to be meted out. It is better to let the authorities deal with them wherever possible, but the best way to punish a psychopath is through exposure. They need to be carefully unmasked so that society can take a good hard look at what lies beneath. And the emphasis is definitely on lies—their stock-in-trade. They aren’t necessarily all that proficient (after all, it’s difficult even for the most accomplished confidence trickster not to tangle themselves up in the end), but they are at their most comfortable when bullshitting—virtually polygraph-proof. Conscience, guilt, remorse... forget all that. Not one genuine drop of those sentiments will you wring out of them. Don't waste your efforts.
You might have to employ some underhand tactics of your own in order to trap your psychopath. It may require you to do some things you wouldn’t ordinarily be comfortable with. You will need to keep a keen eye on your target, get a feel for their patterns of behaviour and try to set a snare for them.
Intelligence, imagination and intuition are your 'particular set of skills', plus patience. Lots of patience. Be smart, but also listen to your instincts. When something they say or do feels off, follow your hunch. But don't just follow your hunch, follow them, literally. Find out as much about their activities as possible: crosscheck their stories with others, track their movements and so on.
You need to catch them smack bang in the headlights. If they are doing something illegal, inform the authorities, even if it is minor. Better still, help the authorities to catch them lying. They will naturally lie to them like they do everyone else. This is the real jackpot. The authorities have a certain degree of power to penalise their compulsive lying; so every time they perjure themselves, they are, in effect, committing an act of self-harm. This is the most satisfactory method of punishing a psychopath short of torture.