What do 'command' points do?

It's pretty obvious that 'morale' points for your general work by the troops they are commanding having more morale, which in turn makes them persevere more, but when a General has 'command' points what does it actually do? Sure, they have lovely stars to look at, but what does it help with? I'm commanding the forces, so what difference does it make, along with infantry and cavalry 'command' points?

I think the guy with more stars gets to see the other guy's deployment before deploying his own troops.

Apparently, higher Command generals will do significantly better in AI controlled battles.

Oh... so that's the reason I steam rolled Europe both as Arthur Wellesley with four cannon and as the French Marshals with half stacks. It all makes sense now.