I tried to summarise the information presented by Dr. Wiedrock into a text format, maybe it would be easier to understand when presented two ways.
Preamble:
Command penalties decrease the CV of your units participating in combat. They are percentage-based.
They come from two sources, the effects of which are applied together additively:
a) units subordinated to different HQs
b) units subordinated directly to high-level HQs
1. "Different HQ" CV penalty
1.1. Why is there a penalty
To work together, the HQs need to find a common HQ they both answer to.
But information passing through intermediary levels of command creates delays and inefficiency.
1.2. How is the penalty calculated
One HQ (the one with the most CV of directly subordinated units participating) is chosen as the overall commander.
Then, all other HQs try to find a common HQ with the commanding HQ.
To do that, they go up along their command chains (you do not go down the command chain) until they find an HQ that can give orders to all the participating HQs.
An HQ can give orders to themselves, which is why different units from the same HQ do not take this penalty.
And because all HQs answer to High Command eventually, any two units will have a common HQ.
1.3. The percentages
Any "step up the chain" costs a number of % of inefficiency, depending on the level of the higher HQ.
The level of the lower HQ does not matter: a Corps -> High Command move is as bad as a Front/AG -> High Command move.
The penalty is 3% for an Army, 5% for a Front/AG and 10% for High Command.
2. "High-level HQ direct control" penalty
2.1. Why is there a penalty
High-level HQs usually do operational and strategic work and are less suited to commanding units in the field on the tactical level.
2.2. How is the penalty calculated
If a unit is directly subodinate to a Front/AG, they will take an extra 10% penalty, which goes up to 20% for units directly under High Command.
Note that this penalty applies even if all units in the combat are from the same HQ.
3. Examples
See Wiedrock's post:
https://www.matrixgames.com/forums/view ... 2#p5211962