I just finished Archer Jones' The Art of War in the Western World, which is a dandy book (Archer Jones is one of my favorite military historians, and I recommend his books to anyone with an interest in the subject. He is always clear and incisive, and makes complex situations highly understandable without needing to simplify).
In his latest book, he points out that, starting with the Macedonians, ancient warfare consisted of four combat arms: Light Infantry (missile-armed), Heavy Infantry, Light Cavalry (missile-armed), and Heavy Cavalry. These have a very specific interrelationship, which has been demonstrated in numerous battles throughout history (up till the invention of the pistol).
"Offensively Superior" means that the combat arm is able to defeat its rival in an attack -- though the "attack" may not look much like one. The classic example is Light Cav vs. Heavy Cav. The Light Cav constantly retreats during an action, so it can't be caught, but it is offensively superior because it can force an action upon the Heavy Cav, who can't get away from the faster, more loosely-organized horse archers.
"Defensively Superior" means that the combat arm's rival is not able to successfully overcome its rival in an assault.
Light Infantry is offensively superior to Heavy Infantry.
Light Cavalry is offensively superior to Heavy Cavalry
Heavy Cavalry is offensively superior to Light Infantry
Light Infantry is defensively superior to Light Cavalry
Heavy Infantry is defensively superior to Heavy Cavalry
When fortifications are involved, of course, these relationships change. Cavalry becomes useless, and the offensive superiority of light infantry is canceled out.
Also, if heavy cavalry can get to the flank of heavy infantry, they can triumph. This was one of Alexander's techniques -- by pinning down the enemy hoplites with his own, and thus keeping them from maneuvering, he could take his Companions and charge the enemy mass from the side or rear.
Note that at Hastings (for instance), the Norman knights were unable to penetrate the Saxon infantry line until they'd broken it up with repeated small charges and sustained archery, thus slowly wearing down the Saxons.
NOTE: light infantry and light cavalry normally take much longer to reach a decision when offensively superior than do the heavies. This is because light troops have to run away when the heavies charge (usually they can do this, because they're faster), then run back when the heavies try to flee. This continual ebb and flow takes quite a while, but the decision is no less certain than when heavy cavalry smash into a batch of slingers -- just slower.
There are exceptions to this general rule, but not nearly as many as you might think. Some well-known sample exceptions are below:
"At Marathon, Athenian heavy inf defeated Persian light inf." The Athenians were able to close with the Persians, whose back was up against the sea. Under normal circumstances, the Persians should have been able to flee while keeping up harassing fire upon the Athenians. There were many cases in which peltasts were able to destroy unsupported hoplites. Obviously, if lights (cavalry or infantry) can be trapped so as to prevent retreat, the heavies have it all over them. But this is the exception.
"At Crecy, English light infantry (longbowmen) defeated heavy cavalry." In fact, the French did not attack the longbowmen, but instead charged the main English line, basically heavy infantry. The infantry was well able to resist the charge, especially after the archers had sowed confusion. On those occasions when cavalry charged longbowmen, the longbowmen ran away, were massacred, sought shelter among accompanying heavy infantry, or were saved by a countercharge from friendly cavalry.
"The Romans managed to rule an entire empire, relying entirely on their excellent heavy infantry". The Roman infantry was excellent. But the Romans also had significant numbers of auxiliaries serving as light infantry. In addition, they invariably had locally-recruited cavalry forces, normally the equivalent of heavy cavalry. They did generally lack light cavalry, and this showed in their Parthian campaigns.
NOTE: "heavy" cavalry does not necessarily mean heavy armor. It just means they're trained to fight in large masses in a melee rather than in the Parthian style. For example, during the Crusades, the Crusaders learned to hold off Saracen horse archers by using crossbowmen (using light infantry to defeat light cavalry) for their heavy infantry and cavalry to shield behind. The Saracen light cavalry was trained to fight both with bow and sword, and thus could "switch" over to a heavy-style melee combat. On occasions that the Crusaders crossbowmen did not have support from nearby heavy infantry or cavalry, the Saracen cavalry was able to charge and, despite their light arms and armor, disperse and kill the crossbowmen. In effect, they could act as light or heavy cavalry. Of course, if they had to face genuine Crusader knights, the Saracens couldn't hold up in melee with these horsemen. Instead, they evolved their own form of genuine heavy cavalry, which was still not as good as the Crusaders, but a heck of lot better than nothing.
