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When Root sets up a virtual battleground inside a laptop, Samaritan had won in every battle-simulation and the Machine had a winning count of zero simulations.
However, during the final battle after being uploaded to a satellite, the Machine's code defeated the copy of Samaritan abruptly.
How was this possible?
While we are never given a direct answer to this in the show there are some factors to consider:
The Machine was always a leaner system, capable of functioning to a very high level on much lower specification hardware than Samaritan (as seen "Root_Path_(/)" when in order to get Samaritan running Decima has to steal a bleeding edge high performance CPU and then start building lots of them). By comparison the Machine has been shown to run on relatively low-end hardware, since satellites tend to have relatively low amounts of computational power this could perhaps be considered the Machine's "home turf" and it may be able to operate better with extremely limited resources than Samaritan. While the training bouts were done on laptops rather than server farms Finch would likely have been trying to simulate a battle between the two as there were out in the world at the time - fully functional and prepared.
The Machine created the copy specifically to go to that satellite and fight Samaritan - whereas the latter had sent one of dozens of compressed copies there with the plan merely to wait out the Ice-9 infection. While the Machine no doubt compressed it's own copy it likely had the superior compression/decompression algorithm (obtained in YHWH from former POI Caleb) which may have allowed it to either send more of itself and/or a version that was specifically optimised to run on that satellite's hardware allowing it to come out "swinging" on a relatively unprepared, bare-bones version of Samaritan.
These are both supported by a "Word of God" answer on this very issue from Jonathan Nolan and Greg Plagman:
Nolan: I think a part of it was all that relentless training, in a sense. Finch and Root trying to scheme out how this would work but also running these simulations with the Machine in order to figure out how it could win in its most paired down form. So you've got this decommissioned Soviet satellite and these things were up in there in their most compact fighting form. Sort of like how armies used to march out and select one champion to represent them, and the fight would be decided with that champion and the rest of the army would abide by that outcome. But at this point, the armies have been decimated and destroyed and only the champions remain. The sparest algorithmic versions of these ASIs uploaded into the satellite like two strands of DNA having a kung fu battle. So kind of fun, but also kind of hard to visualize.
Plageman: Also, keeping in mind that Finch unleashed this virus that could hobble Samaritan enough to put the two ASIs on more equal footing.
Nolan: Both of them have been reduced to their respective essence, and in that form, the Machine was going to kick Samaritan's ass.
This answer is part of an interview that can be found here
