The rock/paper/scissors system does face a bit of a challenge here. In theory, the order would be:
Pikes > Horses > Archers > Swords > Pikes
In order to achieve this, we would probably need both an attack and defence stat in melee as well as ranged as discussed above. However, one quickly runs into a dilemma here: If Swords need to be vulnerable to Archers, they need to have low (ranged) defence, but that will also make them completely useless during sieges, which is contrary to what one would want, because swords would seem to be the desired melee unit to bring to sieges.
Another scheme would be like this:
Pikes > Horses > Archers > Pikes
Swords > Cities
Siege Weapons > Cities
The rock/paper/scissor of Pikes/Horses/Archers comes pretty naturally from Archers being able to attack without taking damage, Horses being able (theoretically at least) to move in, attack, move out; and Pikes getting their bonus against Horses.
Taking out Swords from the regular rock/paper/scissor and focusing them more against Cities would be good in many ways, but leaves the dilemma of how Swords should interact with other units. I guess having it somewhat even with the Archer (both are vulnerable to each other), having it at slight advantage towards the Pike (Sword has the higher attack) and a slight disadvantage towards the Horse (Horse has higher attack and movement) could work.