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If you equalize the technologies - which you would need to do for a meaningful comparison - then ultimately, the Romans at their height would have won.
Why? The Romans were principally noted for their heavy infantry - but also for their ability to adapt tactics. They fought other horse cultures - sometime losing, sometimes winning. Disciplined, heavy infantry is capable of defeating cavalry and has done so on many occaisions.
More than that, they used lots of auxilla to supplement their forces - including lots of cav.
Last, but not least, the Romans at their height, would have been fighting from much reduced lines of communication, with far greater population densities. They may have lost lots of battles, but they would have won the war.
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Nemo Me Impune Lacessit - Scottish Motto
"They that approve a private opinion, call it opinion; but they that dislike it, heresy; and yet heresy signifies no more than private opinion” Thomas Hobbes - Leviathan
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