The militias are fighting each other, they are fighting against the government and they are fighting against the coalition troops in Iraq.

Also, this:

Adding or removing American troops in this situation changes little for the Iraqi fighters because their war goals are not peace based on the common Iraqi good. They are fighting for settlements that redress historical grievances and secure power for their own factions.

contradicts this:

The issue here is not a traditional one of helping a legitimate government to defeat rebellions against its authority. No faction in Iraq is rebelling against a government’s authority.

as well.

They should be trying to “secure power for” the government.

Not for their own factions.

No faction in Iraq is rebelling against a government’s authority. Each faction is trying to fill a power vacuum within its own community and, eventually, a power vacuum in the entire country.

I think the second statement here kind of contradicts the first. I think a more accurate way to phrase that first part would be “Each of the various factions is rebelling against a government’s authority.” The central govt is weak and each faction is trying to fill that power vaccuum, as you said, but in doing so they are fighting against the govt authority (weak though it is.) In point of fact, they’re fighting against the IDEA of a central govt authority- or at least the model of a central govt that unifies all of the factions.