A family of ordinal solutions to bargaining problems with many players

A solution to bargaining problems is ordinal when it is covariant with respect to order-preserving transformations of utility. Shapley has constructed an ordinal, symmetric, efficient solution to three-player problems. Here, we extend Shapley's solution in two directions. First, we extend it to more than three players. Second, we show that this extension lends itself to the construction of a continuum of ordinal, symmetric, efficient solutions. The construction makes use of ordinal path-valued solutions that were suggested and studied by O'Neil et al. [Games Econ. Behav. 48 (2004) 139-153].