You are assuming that everyone purchasing a game, would not purchase it if they could download a pirated version. Call me naive, but I like to think that a lot of people that have the money to purchase a game in the first week, would still purchase it even if a cracked version is available. Just look at the billions of revenue earned by Spotify and iTunes before it, even if most music is available on release date (or even before the release date!) online without even needing cracks. I like to think that most (!) people pirating stuff, would not purchase the pirated product at all if no pirated version were available.

On the other hand, you seem to be missing the number of people just sick of DRM and all the hoops it makes you go through before actually being able to play a game. Some people may just start pirating games because the hoops of running a cracked version are easier than the hoops of running a legit version with all the DRM restrictions. Or they may just give on games altogether and decide to spend their valuable time and money on something else entirely.

So adding DRM may just as well be a net negative for game publishers.

I don't have the data to say which is correct, but just assuming that DRM is profitable because most sales happen in the first week, before a crack is available, seems a bit too simple.