Bad people are gamers too. So, absolutely!

You can also consider DnD games such as Neverwinter Nights, Baldurs gate etc. The user can typically commit bad acts, but it might have consequences later on.

I usually play a good character, but I like having the bad option available. The games that do it best seem to view it as a sandbox, where most things the AI can do the player can do as well, including good and bad.

Wasn't there a game called Overlord or something where you were the Badguy and your job was to influence other people to be bad?

It's a cop out. They have the main character be a good hero that went to kill the previous overlord. The overlord erased your memory and made you think you were him.

You can choose to be good or evil at the end as well.

Stubbs the zombie's a game where you play a zombie though.

should the player be able to say "gimme all your stuff or else!" to NPCs? IE rob them? the reply would depend on the strength of the player's and NPC's parties. anything from laughing in your face to "here - take everything - just please don't kill me!", followed by a transfer of items. unlike skyrim, a thief will only try to rob you if relative party strength is not against them. so one stupid guy on the road doesn't try to shake down you, your 3 followers, your warhorse, and your wardog. gangs of thieves are planned for Caveman.

There are very few games I've seen that do this well.

Nethack is great this way, if you steal from a shopkeeper and they realize it, they will come at you and the shopkeepers are usually well equipped. They also call for the cops who chase after you. You can pacify the shopkeepers by paying back what you stole plus a small amount they require. You don't really know what the shopkeeper can do, they may have strong weapons, spells of fireball, or even a rare wand of death. Depending on your character class and your alignment (lawful, neutral, or chaotic) it can also change your deity's feelings about you: a lawful knight stealing from a store will harm their relationship with their god, a chaotic character is rather expected to loot and plunder. If you are high enough level you can murder all the shopkeepers and loot their corpses and their stores, then deal with the cops fairly easily.

Some games keep it central, as you are the bad guy and are expected to do harm. GTA, The Godfather, and others come to mind.

Many role playing games have the hero doing particular criminal acts with impunity: go into everyone's one, break their vases and crates and pottery looking for coins and hearts and potions, stealing anything from people's homes that isn't nailed to the floor and then go sell it at the pawn shop. Many also have you killing guards and other people who are against you without sympathy or remorse or legal systems, they were in the way and are therefore doomed to die. Few RPG games take steps to deal with that, they operate as though whoever attacks you is acceptably doomed.

A few games put in fun situations where the player suffers consequences of bad acts. Chrono Trigger's fair, for example, if you steal or eat someone's lunch or cause other problems the jury will find you guilty. Of course in the game even if they all find you innocent you are still sent to execution, but that's a different problem. Some will have NPCs make comments late game about how robbers had come through their home or town, looting their homes while they were in them.

Like the comments above, I think there should be repercussions for attacking NPCs. At the very least, getting caught stealing in a city should invoke the guards or police for some time. Criminal acts should impact your purchasing power at shops, either having them call the police or raising prices since they're dealing with known brigands. This may open up new shopping outlets in the criminal underworld while closing off lawful stores.

Repercussions may do more than that. It may also invoke wanted posters and people hunting you down (new quests!) and perhaps even trigger end-of-game if you get caught and jailed once you are sufficiently outlaw. It may change end-of-game so instead of working for the law and getting rewarded by the king, outlaws may be be rewarded by the criminal underworld.