For the exact same reasons that they don't really do it for movies or music. Bad publicity.
Games have pretty much always been gotten in one of three ways: Usenet, Torrents, and IRC. I am not familiar with how Usenet handles things, but Torrents and IRC are pretty hard to take out. You stop Suprnova or PirateBay, another twenty sites take their place. And the legal fees involved in suing a private tracker just aren't worth the hassle.
So that leaves taking out Little Billy and his Grandma. And while that does make people think twice, it also makes you look pretty bad. And since it would actually have to be EA or Activision doing the lawsuits (and not a blanket organization like the RIAA or MPAA), that actually matters.
I don't follow your argument. You are saying they don't go after filesharing sites because of bad press? Surely it looks worse for a corporation the size of Activision suing kids than issuing C&D to torrent sites?
I agree there is nothing that can be done about IRC but then not many people these days use it or are even aware of its existence. Remember I am taking about just making it hard for people to get the torrents not making it impossible. The casual pirate only does it because its so bloody easy to do. All a larger company would need is to take some of that money they use for these utterly pointless lawsuits and invest it in online police teams. If Activision had a team of 5-6 people checking torrent and Usenet sites daily looking for any Activision games being torrented and issuing C&D orders on them it would be far more productive. This torrent sites only exist because they claim they don't check what is being shared (although we all know this is bare faced lie), if companies took measures like this then they would be made aware and would have to remove the torrents. In the same way YouTube has had to remove copyright videos for the same reason.
Eventually (after a long time granted but eventually) people would simply stop bothering uploading torrents knowing that they will be taken down almost as quickly. Files may even be pre-checked for content like YouTube does now. The harder you make it for people to get the cracks the more the casual pirate (the ones who are the real problem) is going to think "Ah sod this I will just buy it".
My point is a simple one. If you stop making the torrents available then people cant pirate as easily and over all piracy rates would decrease. Far more than the current system where people consider it an acceptable risk as realistically you have a one in a million chance of being caught.
Well, keep in mind that they don't want to alienate people. So if we see in the news "Activision sues Pirate Bay" or something, it is going to make a lot of people angry. Then you will have the idiots who pretend this is about net neutrality and that everyone on that site was just exchanging Linux distros :p
Then you have the morons who spin it as "Activision doesn't want you to try out their games before you buy them!".
So just massive headaches. When the RIAA or MPAA do it, it is pretty neutral as far as the individual brands/companies/whatevers are concerned.
As for having a special task group whose job is to go patrol the torrent sites and the like: Four words for you "Big Brother is Watching". Sure it has jack all to do with that and is really just a matter of protecting a company's interests, but really, how often do any of the things people call "Orwellian" actually have anything to do with the crap in the book? :p
So yeah, they are basically screwed on the virtue of public relations. That being said, I think EA are part of something that does this for "abandonware" games.