Envy-Free Division of Land

Abstract

Classic cake-cutting algorithms enable people with different preferences to divide among them a heterogeneous resource (“cake”) such that the resulting division is fair according to each agent’s individual preferences. However, these algorithms either ignore the geometry of the resource altogether or assume it is one-dimensional. In practice, it is often required to divide multidimensional resources, such as land estates or advertisement spaces in print or electronic media. In such cases, the geometric shape of the allotted piece is of crucial importance. For example, when building houses or designing advertisements, in order to be useful, the allotments should be squares or rectangles with bounded aspect ratio. We, thus, introduce the problem of fair land division—fair division of a multidimensional resource wherein the allocated piece must have a prespecified geometric shape. We present constructive division algorithms that satisfy the two most prominent fairness criteria, namely envy-freeness and proportionality. In settings in which proportionality cannot be achieved because of the geometric constraints, our algorithms provide a partially proportional division, guaranteeing that the fraction allocated to each agent be at least a certain positive constant. We prove that, in many natural settings, the envy-freeness requirement is compatible with the best attainable partial-proportionality.