If you put equal number and equal tier cavalry against equal number and tier archers, then archers will lose. The only way players can lose their cavalry against archers is if they use inferior numbers and/or inferior quality. But given AI does not use armies full of t5 units, it's likely the first case.
Also people don't realize it, but you can order your cavalry in to shieldwall and then tell them to charge. They will attack and keep their shields up which will protect them against head on shoots. Might prevent them from couching lance but that's something I did not test yet.
Cavalry shouldn't be able to roll over formed infantry. That's why I don't want to see Warband cavalry. Fact that in Bannerlord cavalry can push through any infantry formation and will not get stuck is already large advantage. But I can live with that given it compensates for simplistic AI.
You can do that only if enemy have inferior cavalry or your own cavalry can keep them busy. Depending on the order, cavalry in Bannerlord is programmed to target certain units in certain order. If I remember right, heavy cavalry will target horse archers first, then other cavalry, then archers, then infantry last.
If AI have cavalry superiority, they will mess up your archers. And not just by directly attacking them, they will also mess their aim, because your archers will be shooting at the cavalry instead of what you want them to shoot at. And archers have a hard time hitting cavalry due to fast speed.
Of course if player brings army of Battanian fian champions against AI that is mostly made of t3-t4 cavalry, then that's not exactly fair comparison.
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