By definition, illegal copies of games don't have DRM protection. I'm not sure what you mean by this.
The analysis studies pre-crack and post-crack sales, and specifically observed the dip in sales after the crack. The dip was larger, the closer to release the game was cracked. A theoretical day 1 crack caused a 20% drop in sales.
I'm also not sure what you mean by games that are cracked almost immediately are a better sample. You can't measure sales before and after the crack was released because you only have the latter. Sure, if we could somehow measure how the game would have sold in an alternate universe where it wasn't cracked that would be a more robust finding. But obviously that's not possible.
The study focused on denuvo protected games because those are essentially the only games that go for extended periods of time without being cracked. They're the only games that actually offer any insight into how games sell without a crack available.
> By definition, illegal copies of games don't have DRM protection. I'm not sure what you mean by this.
Games released without DRM are less of an inconvenience to legitimate purchasers. They don't get negative sentiment from past customers complaining about the DRM causing problems for people who actually paid for it and thereby deter others from buying it.
This can even cause the effect you're seeing: The game comes out with onerous DRM, people buy it initially having not realized this yet. The DRM being more onerous to legitimate purchasers makes it both more likely to be cracked (people spend more effort to crack it so they can play the game without the DRM causing problems) and more likely to have sales decline as DRM problems for legitimate purchasers become known and sour customers on buying it, so you get a correlation between how fast the game gets cracked and how fast sales fall off.
In general it assumes there is no existing correlation between how quickly a game gets cracked and the rate at which would sales decline regardless, e.g. it assumes that more anticipated games don't both get cracked sooner and have more front-loaded sales, but relationships like that are pretty plausible.
In addition to that, once the crack is available, you're stuck with DRM if you pay but not if you use the cracked version, so then the cracked version is better. It outcompetes paying not just on price but also on utility, whereas if the paying got you no DRM to begin with then the cracked version would have only one advantage instead of two.
You also unconditionally lose 100% of the sales to people who simply never buy games with DRM.
Losing X% of sales to pissing off customers with DRM in order to avoid losing Y% of sales to pirates is only worth it if X is less than Y, but they're only even attempting to measure Y, and probably overestimating it.