One of the principles of interesting computer games is that sometimes a simple action by the player leads to a lot of response from the game. This has an obvious application to why hurting people might be fun.

Without empathy, anything you do with other people is pretty much a game.

No it isn't. Why don't you try to crawl out of your typical mind space for a moment?

Killing people in a computer game is fun for many people.

That's because it usually has good consequences for the player, the violence is cartoony, and NPCs don't really suffer. You could be an incredibly unempathethic person, and still not find hurting real people fun even in the gut level because it has so many other downsides than your mirror neurons firing.

I myself possess very little affective empathy, and find people suggesting that I therefore should be a sadist pretty insulting (and unempathetic). I'm also a doctor, so you people should tremble in fear for my patients :)

(By the way, this discussion partially assumes that psychopaths are exactly like neurotypical people, just minus the empathy. Which may be wrong. Which may make some of our conclusions wrong.)

It's wrong.