Even though it sounds rather one-sided being the 2nd in charge of a heavy cavalry regiment, but to me it seems logical that heavy cavalry is, in the view of utilitarianism, the best cavalry (or 'was', either way). This because of the advantages they have. heavy cavalry would in most occasions beat infantry and cavalry alike when engaged 1on1 (of course in an ideal situation; a perfect 1on1 situation). Of course one could argue that having light cavalry is more useful because of faster movement & hit and run situations, adding a tactical advantage, and that lancers have the advantage that they can deal heavy blows in a charge vs cavalry and infantry. But in the end the heavy cavalry is that which can fend most for itself, granted not entirely, you'd always need support in a cavalry charge, but they can take the hardest hit, were in most cases most intimidating and dealt the heaviest blows because of their main (obvious) aspect, being heav(ily armed).
Of course it can always depend on the situation that you have, when odds aren't ideal you might require other cavalry. But I can only imagine that if you had a perfect 1on1 situation, heavy cavalry would be the most effective.
Having said that, I am aware that an ideal tactical situation nearly never occurred, so historically speaking i'm probably wrong, practically speaking though, that is my reasoning behind it.
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