I can't think of a single "word" for it, but I can offer a dual idiom. You could say:
Person A is holding onto the power position by keeping (or playing) his cards close to the vest.
The power position is where you have an upper hand in negotiations. From a review of a book on negotiating:
Negotiation doesn't begin at the table. It starts with preparation. Gosselin prepares you for success by showing you how to identify your underlying needs and those of your opponent, how to develop objectives and establish a position, how to use currencies and concessions, and how to assess your power position in negotiation situations.
The idiom playing cards close to the vest (or chest) refers to being careful not to reveal information or motivation in a negotiation setting. Alice Haverland1 says:
The literal reference is to holding your playing cards close enough to your chest so that no one else can see them. The idea is to prevent others in the game from gaining a strategic advantage over you. You do this by not allowing them – or their possible confederates standing behind you – even a glimpse of your cards.
The metaphorical use, then, is a natural extension: if you play it close to the vest/chest, you plot your course of action cautiously, carefully, and with cunning; and – most importantly – you keep your intentions and all relevant information hidden.
I don't think either idiom fully captures everything you want on its own, but, in concert, they seem to capture the gist of what you're asking about.
1When using this reference, click on the link that says "Impatient?" to read Haverland's essay.