Regarding Phil's point about rules, I've started to draft some with a melee and movement structure that will allow poleweapons to be engaged at 1" whereas sword/mace etc will have to be in base to base contact. This means that billmen attacking say archers will have a free poke/hack before the archers can close to full contact and respond. I'm trying to achieve a situation where the archer will have the advantage at a distance (bows) and at base contact (swords) whereas billmen will have the advantage at near base contact. What do you think: realistic?
I'm not being intentionally awkward, but there's an assumption here that they fought as separate entities. It might be valid that bills stood at the back of the archers and then moved forwards when the enemy closed... the 'mixed' or 'combined' formation of some rule sets. Alternatively if your bills and men at arms were in one unit and your archers on the flanks, attacking archers with your bills, leaves your archers vulnerable to your opponent's heavy infantry.
I honestly don't have a clue how they lined up for battle, or what their tactics were. A man who knows what he's doing with a bill could well be more than a match for an experienced archer with sword and buckler, a man that doesn't is likely to be a liability to himself and his companions. Would the archers, if they felt themselves vulnerable to the bills, actually stand there patiently waiting to be charged, or would they fall back behind a main battle line of men at arms and bills?
We're pretty much back to the same answer as for troop ratios, whatever works for you is best.