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The primary value of cavalry is pursuing defeated Romans in pitched battles or when your beseiging army defeats the forces that sally and/or attempt to relieve. Given that cavalry is worth 1/2 in attacking cities and (as I recall) worth zero defending in camps, cavalry must establish their value in pursuing, or not at all.

My first attempt was disappointing: in an introductory level game 12 factors of cavalry pursued 6 factors of Roman cavalry with the result of 1 hit to each. At least one writer has suggested this anecdotal result was atypical, so I kept track the next game.

My assumption is that in pursuit a cavalry rated "2-1" gets two die rolls in pursuit or in defense of pursuit and is therefore worth "2". No pursuits were done with the Hannibal Inspired Unit card applied to a Carthaginian cavalry. The game was at normal level.
"C" is Carthage, "R" is Rome. Hits are recorded in a single factor loss. Actual units lost is shown in parenthesis.

Pursuit #1: 16 C vs. 8 R , C took 2 hits (1) R took 2 hits (2)
Pursuit #2: 14 C vs. 6 R, C took 1 hit (1) R took 3 hits (3)
Pursuit #3: 12 C vs. 7 R , C took 1 hit (1) R took 1 hit (1)
Pursuit #4: 10 C vs. 0 R, C took 0 hit R took 4 hits (4)

Although the sample is too small for serious analysis, I remain unimpressed with cavalry pursuits.

You are correct in saying that pursuing cavalry get a chance to score a hit for each attack factor. A cavalry unit with an AF of 2 has twice as many chances to score a hit as one with an AF of 1. This is the same basic procedure as in "ordinary" combat.

Cavalry pursuit is much more effective when there is no defending cavalry. If there is even one defending cavalry unit, the chances for scoring hits is reduced. But you can wear down the Romans' cavalry if you have clear superiority during pursuit.

Cavalry that have an inherent attack bonus (indicated by the circle around their AF) keep their bonus during pursuit. So the Companion cavalry for instance are more effective in pursuit.

Also note that losses come from routed units first, and routed units all have a defense factor of 1. 4 hits from 4 routed units would not be too bad, I think, to use one of your examples. If you can rout the entire enemy army you can do well with a good number of pursuing cavalry.

Cavalry are not completely worthless in camp battles. They have the same values as in city battles. Light cavalry such as the Numidian cavalry are not worth much in either battle, however. That is a reflection of the way that they were armed.

Of course, cavalry units have the same value and function in pursuit after a camp battle as after a pitched battle, a sally battle, or a relief battle.

I hope that this helps. [:)]

My two cents: as with every units in game, you have to work around cavalry's strengths and weaknesses. As mercenarius mentione, cavalry becomes much more valuable in pursuit when unopposed by enemy cavalry screen. The results can be devastating for the routing army. Basically, a difference between the army retreating with a couple of casualties and an army being annihilated (especially during first turns).

Myself, I watch enemy cavalry during the battle. If the enemy is outnumbered, he will try to retreat after two turns. If I see that AI has substantial cavalry, I rout my cavalry units instead of infantry and seldom give a pursuit, which may cost me my cavalry units (it's all about preserving your own strength). If not, I make a point of saving my cavalry for pursuit, often finishing off the enemy general for good.

A thing to remember (and this is more to the benefit of other players), cavalry is not supposed to be a super unit as it was used to be (or claimed to be) during Medieval ages. Romans were not smart to neglect cavalry as much as they did, but they did have valid reasons for doing so. More often than not, cavalry was useful only during pursuits.

A good point about the use of cavalry is given by Syracusans when they debate the preparations against the Athenian expedition of 415-413 BC. The point was that with superior cavalry numbers. Syracusans should not fear defeat in battle for the Athenians won't be able to inflict crushing casualties on them. And the same numbers will assure that any Athenian defeat will be crushing, since the Athenian meager cavalry won't be able to do much to save their fleeting army.