Guide to the Negotiation
Composure (typically purple cards) is defense for your resolve and your arguments.
Influence comes from green cards, and lasts as long as it's not attacked and destroyed by the opponent's arguments, or if it's not spent by card abilities. To do this, you have to look at the red numbers above the icon (signalling the opponents attack), and make sure you have the necessary amount of composure to defend it.
I didn't use dominate much in my playthrough, but I think it also lasts as an argument, as long as the opponent doesn't destroy it, and increases the attack of red/hostility cards...
However, you also want to keep your resolve up (which serves as hp), so you have to defend that too, along with the other argument icons, but only if it's being targeted by the opponent... If the opponent attacks something that has a resolve of 2, but you add 4 composure, it'll take 6 damage from the enemy to destroy. If the enemy targets it with an attack of 8, the argument will be destroyed, and the left over 2 damage attacks your resolve, unless you have composure there to defend it.
In most cases, the opponent's intents and possible damage is shown, unless they have a special argument skill that hides their intent, which is when you target that argument and try to destroy it with green or red attacks... Otherwise, you use the shown information to defend your side of the negotiation, though sometimes it's okay to take some damage if it means attacking and destroying the opponent's possible arguments.