An Ordinal Bargaining Solution with Fixed-Point Property

Abstract

Shapleys 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

A Multi-demand Adaptive Bargaining based on Fuzzy Logic

“…In such a problem, it is hard to elicit numerical utilities and do quantitative analyses (Zhang, 2010). Thus, some researchers tried to express bargainers' preferences in an ordinal scale (Shubik, 2006;Zhang and Zhang, 2008). However, the information relevant to the bargainers' risk attitudes, a very important factor in bargaining (García-Gallego et al, 2012), is lost (Zhang, 2010).…”

A logic-based axiomatic model of bargaining

“…4 The reason is, as pointed out by many researchers, that the information about bargainers' attitudes towards risk, which in fact determines the negotiation power of a bargainer, is not describable by ordinal preferences [26,31,40]. 5 With the Nash bargaining model, bargainers' risk attitudes, combined with the preferences on the possible agreements, are represented by utility scales, i.e., cardinal utility, through the non-linearity or curvature of utility functions (for an intuitive example see [44]). However, such information is lost when the model is converted to an ordinal model through an order-preserving transformation.…”

“…However, there is no explicit representation of players' attitudes towards risk. Zhang and Zhang proposed a purely qualitative model of bargaining based on bargainers' ordinal preferences [44]. Similar to but different from Rubinstein et al's framework, the preference ordering of each player is defined on the player's demand items (instead of the lotteries of possible agreements).…”

“…Bargaining then is viewed as a procedure of conflict resolution over two sets of ranked demands (for two-player bargaining). A solution concept was proposed based on minimal changes but no axiomatic characterization was provided [44]. This paper will develop an axiomatic model of bargaining based on the ordinal preference structure proposed in [44].…”

“…A solution concept was proposed based on minimal changes but no axiomatic characterization was provided [44]. This paper will develop an axiomatic model of bargaining based on the ordinal preference structure proposed in [44]. We describe a bargaining situation in propositional logic and represent bargainers' preferences in total pre-orders.…”