Two derivations of the uniform rule and an application to bankruptcy

We consider the problem of allocating a single infinitely divisible commodity to agents with single-peaked preferences, and establish two properties of the rule that has played the central role in the analysis of this problem, the uniform rule. Among the efficient allocations, it selects (1) the one at which the difference between the largest amount received by any agent and the smallest such amount is minimal, and (2) the one at which the variance of the amounts received by all the agents is minimal. We also show that an important solution for bankruptcy problems, the constrained equal-award solution, can be characterized by analogous minimization exercises, subject to different constraints.