We consider the situation that two players have cardinal preferences over a finite set of alternatives. These preferences are common knowledge to the players, and they engage in bargaining to choose an alternative. In this they are assisted by an arbitrator (a mechanism) who does not know the preferences. Our main positive result suggests a satisfactory-alternatives mechanism wherein each player reports a set of alternatives. If the sets intersect, then the mechanism chooses an alternative from the intersection uniformly at random. If the sets are disjoint, then the mechanism chooses an alternative from the union uniformly at random. We show that a close variant of this mechanism succeeds in selecting Pareto efficient alternatives only, as pure Nash equilibria outcomes. Then we characterize the possible and the impossible with respect to the classical bargaining axioms. Namely, we characterize the subsets of axioms can be satisfied simultaneously by the set of pure Nash equilibria outcomes of a mechanism. We provide a complete answer to this question for all subsets of axioms. In all cases that the answer is positive, we present a simple and intuitive mechanism which achieves this goal. The satisfactory-alternatives mechanism constitutes a positive answer to one of these possibility cases (arguably the most interesting case). Our negative results exclude the possibility of an efficient mechanism with unique equilibrium outcome, and exclude the possibility of an efficient symmetric mechanism which is invariant with respect to repetition of alternatives.
