in a short answer,its because -
why they make a unique civ choice-
A)the Byzantine empire, afthe fall of the western empire continue on its own unique culteral course, merging differnt aspects of Roman, and Hellenistic (greek) culture and civilizations, while doing it from the unique religious perspective of orthadox christianity- this culture ended being far and away more differnt then a continuation of Rome in the west would have produced, and as such in the culteral real, of the arts and society it uniques
B)in governemnt as well, the Byzantine structure was differnt, right down to how the next emperor- for example, the Byzantine empire, having differnt social needs, and wants, placed an emphisis on peaceful transition of power, with the next emperor being ratified by a three way vote of the army, the senate, and the
people!
c)this culteral diversion from main stay Roman culture was known to, approvied by, and activley persued by the populace of the empire, emperor Heraclius II, arguabley the BEST emperor the nation ever had, cleaning up the mess Justinin II "the great", and his wife
had left the empire in by the end on there reign, actvely persued it, and wnet so far as to change to official language of the empire from Latin to Greek- furthing this culteral blend
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now why do the deserve to be in the game-
A)for 1000 years, the byzantine empire served as the great wall of EUrope against middle eastern imperialist, protecting western europe, if unknowinglly, from the ravaging forces sure to be inflicted upon it with man,horse,steel, and blood
B)the Byzantines were the preservers of classical knowlrdge- knowledge theat would be passed, in some ways directlyy from Byzantium, and in other ways through the Arabic empires, to the western nations, knowldege that woul deventually lead to the Renaissance, and the re-birth of classical knowledge, and insperation in the west.
these two reasons alone should be more then enough, though I'll come back later if you need more convincing