Ethereal
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Added 52 minutes later
If you believe this reference, here's the explanation:
The AI doesn't take double shots into account, and the other combat stats of regular and greater archers are the same, so the AI simply attacks the lower stack (to win). But then why didn't it switch to the greater archers when it realized it was losing? Because there were wounded units in the regular stack?
http://handbookhmm.ru/offline-spravochniki-po-hmm
The AI always finishes off wounded units. Moreover, it prioritizes ranged stacks, then flying units, and finally melee units. As for archers and crossbowmen, their stats are the same, but generally, un-upgraded units have lower stats and are easier to kill. Regular archers are an un-upgraded unit for the AI, so it's happy to try.