The Midpoint-Constrained Egalitarian Bargaining Solution

A payoff allocation in a bargaining problem is midpoint dominant if each
player obtains at least one n-th of her ideal payoff. The egalitarian solution of
a bargaining problem may select a payoff configuration which is not midpoint
dominant. We propose and characterize the solution which selects for each
bargaining problem the feasible allocation that is closest to the egalitarian
allocation, subject to being midpoint dominant. Our main axiom, midpoint
monotonicity, is new to the literature; it imposes the standard monotonicity
requirement whenever doing so does not result in selecting an allocation which
is not midpoint dominant. In order to prove our main result we develop a
general extension theorem for bargaining solutions that are order-preserving
with respect to any order on the set of bargaining problems.