An Ordinal Bargaining Solution with Fixed-Point Property

Abstract

Shapley's impossibility result indicates that the two-person bargaining problem has no non-trivial ordinal solution with the traditional game-theoretic bargaining model. Although the result is no longer true for bargaining problems with more than two agents, none of the well known bargaining solutions are ordinal. Searching for meaningful ordinal solutions, especially for the bilateral bargaining problem, has been a challenging issue in bargaining theory for more than three decades. This paper proposes a logic-based ordinal solution to the bilateral bargaining problem. We argue that if a bargaining problem is modeled in terms of the logical relation of players' physical negotiation items, a meaningful bargaining solution can be constructed based on the ordinal structure of bargainers' preferences. We represent bargainers' demands in propositional logic and bargainers' preferences over their demands in total preorder. We show that the solution satisfies most desirable logical properties, such as individual rationality (logical version), consistency, collective rationality as well as a few typical game-theoretic properties, such as weak Pareto optimality and contraction invariance. In addition, if all players' demand sets are logically closed, the solution satisfies a fixed-point condition, which says that the outcome of a negotiation is the result of mutual belief revision. Finally, we define various decision problems in relation to our bargaining model and study their computational complexity.