Introduction

In this chapter we present our first example of an axiom relating solution outcomes across cardinalities, and we use it in conjunction with several familiar axioms to characterize the Kalai–Smorodinsky solution.

The axiom expresses a form of solidarity among agents in circumstances in which their number varies while the opportunities available to them remain unchanged. Imagine that a solution has been selected and consider a particular problem involving some group of agents. After the solution has been applied to the problem, a new agent enters the scene and is recognized to have claims as legitimate as everyone else's. Accommodating these claims will typically require sacrifices from some of the agents originally present. However, at the new solution outcome, some others could be better off. It is this possibility that the axiom will prohibit: All agents should share in the new responsibilities of the group, thus the term solidarity. If a solution satisfies it, it will be said to be population monotonic.

The axiom, which will be called the Population Monotonicity axiom, is illustrated in Figure 3.1. There S is a problem involving the group P ≡ {1,2}, and T is the problem faced by the enlarged group Q ≡ {1,2,3,}. The problem has the particular feature that its projection onto (or equivalently, since we consider only comprehensive problems, its intersection with) the coordinate subspace relative to P coincides with S.