Introduction

We come back here to the Egalitarian solution. We will show this solution to be the only solution to satisfy Weak Pareto-Optimality, Symmetry, Continuity, Monotonicity, and a weakening of the Stability condition introduced in Chapter 7, which we refer to as Weak Stability.

Recall the condition of (Multilateral) Stability. Starting from some problem T involving some group Q, consider some subgroup P and the subproblem S consisting of all the points of T at which the utilities of all the agents in Q/P are fixed at their values at χ ≡ F(T). Stability requires that S be solved at a point that coincides with the restriction of χ to the subgroup P, that is, that F(S) = χP. The condition of Weak Stability says that the solution outcome of S should dominate, instead of being equal to, the restriction of χ to the subgroup P. Any solution satisfying Weak Stability and Weak Pareto-Optimality satisfies Stability (and, of course, Pareto-Optimality) on the subdomain of problems whose weak Pareto-optimal and Pareto-optimal boundaries coincide. Therefore, when used in conjunction with Weak Pareto-Optimality, as will be the case here, the two conditions of Weak Stability and Stability say nearly the same thing. In the instances in which they differ, the weaker requirement does not say, as Stability does, that no member of a subgroup would ever want to renegotiate; it simply says that renegotiations by subgroups would benefit all agents in the subgroup.