Nobody really wants there to be a completely RPS solution, but as long as there's going to be a clear advantage based on weapon type, it should be something like this IMO:
Two-handers (axes, swords, etc.): Poor against charging cavalry, good/strong against shielded infantry (sword/axe respectively), great against defensive polearms, equal to offensive polearms.
Shielded one-handers (most/all of them): Bad against charging cavalry, generally poor against two-handers, great against defensive polearms, poor against offensive polearms.
Defensive polearms (long spear, pike, halberd): Great against charging cavalry, bad against shielded one-handers, poor against two-handers/offensive polearms.
Offensive polearms (bec, battlefork, longmaul): Poor against charging cavalry, good (great for the maul) against shielded one-handers, mostly equal to two-handers, and good against defensive polearms.
Archers/crossbowmen: Overall great against unshielded infantry, bad against shielded infantry, bad against charging cavalry.
Cavalry: Overall superiority when charging (except for defensive polearms), overall inferiority when stationary. Falls to teamwork but is good against stragglers and uncoordinated teams.
Throwing (with a shield): Great against archers/crossbowmen and unshielded infantry, okay against shielded infantry, poor against charging cavalry. Has less ammo capacity than bows/crossbows.
This doesn't take into account difference in armor or pierce/cut/blunt damage, just how the weapons would fight each other.
When stationary, a cavalryman doesn't have the same mobility as he would have on foot and when he falls off his horse, is completely vulnerable, hence them being generally weak to everything when still.
Just my suggestions, they might not be good for everyone.