Pure Bargaining Problems and the Shapley Rule

Pure Bargaining Problems with a Coalition Structure

“…Pure bargaining problems, introduced in [5], constitute a natural setup and, at the same time, a simple case of both Nash's bargaining model and the cooperative game model (as they can be identified with quasi-additive games). Their simplicity allows us to better capture the meaning of certain notions, most of which are translated from the cooperative game theory.…”

“…Given u ∈ E n+1 , for each i ∈ N the i-coordinate f i [u] of vector f [u] would give the share of u N that is allocated to agent i according to f . The proposal made in [5] is the Shapley rule, denoted here as ϕ and defined by…”

“…We refer to [5] for a rationale on this rule and any additional information. Thus, the Shapley rule solves each problem in the following way: (a) first, each agent is allocated his individual utility; (b) once this has been done, the remaining utility-the surplusis equally shared among all agents.…”

“…A consistent alternative to the classical proportional rule, the Shapley rule, has been suggested in [5] 1 as a much better solution for pure bargaining problems. It derives from the classical Shapley value for cooperative games [11] (also in [10]) because any pure bargaining problem can be linearly identified in a one-to-one way with a quasi-additive cooperative game.…”

“…In Section 2 we briefly recall the notions of pure bargaining problem and Shapley rule introduced in [5] and include a new axiomatic characterization of this rule. In Section 3 we consider pure bargaining problems with a coalition structure (CS-problems) and discuss the main options available to the agents: individual behavior (I), cooperative behavior (C), isolated unions behavior (U), and bargaining unions behavior (B).…”