I think that archers win Captain games as the DPS, but only if certain conditions are met.
1) They must remain unspoiled by cavalry
2) They must be as close as they can to the enemy to maximize accuracy and damage
3) They need a clear line of sight to the enemy, without any shields or friendly troops in their way
4) They must save their ammo for the opportune moment
Conversely, archers will fail if
1) They’re harassed by cavalry
2) Are too far away for accurate and deadly fire
3) Are blocked by friendly troops or shields
4) Wasted their ammo
While the archers must be protected by infantry if the enemy has cavalry, they cannot output maximum damage if their escort is in the melee. Therefore, to maximise damage, the archers should flank the melee and shoot to the exposed enemy side. However, this in turn exposes the archers to enemy cavalry attack. Alternatively the friendly infantry leads the enemy infantry away from the archers, but this, too, leaves the archers open for a cavalry attack. With some maps and factions, the vulnerability of the archers can be mitigated. Archers can be stationed on cliffs where enemy cavalry cannot easily manouvre, or against a wall which will also hamper the cavalry. Some archers, like the Palatine Guard, can be equipped with Menavlions as well. Theoretically a team could forgo everything else but archers, if every unit also has such heavy polearms.