The price of fairness with the extended Perles-Maschler solution

In Nash bargaining problem, due to fairness concern of players, instead of maximizing the sum of utilities of all players, an implementable solution should satisfy some axioms or characterizations. Such a solution can result in the so-called price of fairness, because of the reduction in the sum of utilities of all players. An important issue is to quantify the system efficiency loss under axiomatic solutions through the price of fairness. Based on PerlesMaschler solution of two-player Nash bargaining problem, this paper deals with the extended Perles-Maschler solution of multi-player Nash bargaining problem. We give lower bounds of three measures of the system efficiency for this solution, and show that the lower bounds are asymptotically tight.