Scholars Can't Not Rule

Merchants can't rule because they can't tell the truth. You can always trap a merchant into self-harm by holding hostage the lies their friendships depend upon. All their short-term incentives encourage shameless irresponsibility, and as a result they grow up to lack discipline. This is a tragedy as the basic merchant interaction is positive-sum, unlike the other castes. Disqualified.

Warriors can't rule because the correct way to choose the warrior chief is through violence. If you can beat up the old chief you get to be the new chief. In a warrior hierarchy having the weak lead the strong makes no sense at all. Did you notice the treachery? The correct way to become head warrior is to betray the head warrior. Which means the chief wants to betray all his competent subordinates. He wants to cripple his own organization so there's a band gap between himself and his underlings. A warrior organization is inherently a non-society. The test for membership is willingness to betray. The way to lead it is to undermine leadership. A human warrior hierarchy can't declare peace on itself without renouncing Martial virtue. 

The correct warrior strategy is to forgo crippling your own organization only long enough to destroy every other warrior organization. This means warriors can't declare peace on each other, either. Disqualified.

By process of elimination, always and everywhere the scholar caste is in charge. Your priests, your brahmin, your university professors, your shaman, or whatever you happen to call them locally. Neither the emotional left heart nor the strong right arm can make decisions - only the brain can make decisions. 

This is still a problem. The scholar caste having a zero-sum challenge is better than the warrior caste's negative-sum, but not by much. The top scholar wants to keep all wisdom away from any of his challengers, surrounding himself with idiots and taking all the society's wisdom with him when he dies. At best every human scholar must reinvent the wheel. As in, literally starting at reinventing oscillatory transportation. 

That said, of all the castes, only the scholar caste can be expected to appreciate long-term incentives. It is not bound by its mindless biology.