How were real medieval armies composed?

[econ21]

In MTW, the general strain of composition of armies has been kind of like rock-paper-scissors, with melee infantry beating spear infantry, spear infantry beating cavalry, and cavalry beating melee infantry. But from what I've read, this isn't true- rather, apparently most infantry carried spears with them as well as some kind of close-range melee weapon like a sword or mace. Is this true?

That's my perception. I suspect that by and large most medievel combat units were not characterised by, or standardised in terms of, their weapons until towards the end of the period. I have not found any clear examples of non-spear Catholic melee infantry units in the early medieval period. As I will explain, I think the rise of standardised non-spear melee infantry came about partly in response to technology (especially the introduction of plate and the pike). But it is also surely not a coincidence that greater standardisation of weapons happened when armies were getting less feudal and more professional. (In an earlier age, compare the weapon standardisation of a professional legion with a warband, for example.)

And how did halberdiers, two-handed swordsmen, and other non-spear infantry fit into the real-life equation of medieval armies? I'm guessing that mass gave them some advantage, but wouldn't spears be better, since they can keep attackers from a frontal attack, both on foot and on horseback?

I think these units were partly a response to the introduction of plate - halberds in particular seem to have been the weapon of choice in close combat in the War of the Roses, for example. A spear is not that useful against a man in full or half plate, but a halberd or two hand sword could bash through helmets or armour limbs etc. The halberd also had similar stand-off properties to the spear, but inferior to the pike obviously.

I think the two handed sword was less common than the halberd and the units I have heard of which specialised in it seem to have been a response to the pike. A unit of "zweihanders" or halberdiers hacking into the flank of a pike would have been devastating. The phalanxes would also tend to use such weapons to protect their flanks. The sword and buckler men were similarly designed, although contrary to their depiction in M2TW, I gather they did not always beat the pikemen.