The Fair Division of a Fixed Supply Among a Growing Population

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Abstract
We reconsider the traditional problem of fair division. Division principles should be general enough to accommodate changes in what is to be divided as well as variations in the number of agents among whom the division is to take place. In the usual treatment of the question, this number is assumed to be fixed. Here, we explicitly allow it to vary. Agents are assumed to have von Neumann–Morgenstern utility functions and division problems are defined as subsets of the utility space. Our approach is axiomatic. Apart from the familiar axioms of Pareto-optimality, scale invariance, continuity and anonymity, we formulate and impose an axiom of monotonicity with respect to changes in the number of agents, stating that if sacrifices have to be made to support an additional agent, then everybody should contribute. There is a unique division principle that satisfies them all: it is the natural extension to the n-person case of the two-person solution proposed by Kalai and Smorodinsky.

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