I also assume that both P and A prefer some deal to the status quo; that is, [x.
At the approval stage P and A (if given veto power) accept or reject x.
At the approval stages, both P and A (if A is given veto power) never reject a proposal that they (weakly) prefer to the status quo; otherwise, their strategies are not subgame perfect.
Unlike the complete information version, however, the alignment of preferences between P and A matters when F has incomplete information about P's preferences.
As an anonymous referee for this journal observes, if we relax the assumptions that both P and A prefer a negotiated outcome to the status quo or that F's ideal point is not acceptable to P and A in comparison to the status quo, P's incentive to give veto power to A weakens.