Envy-Free Division of Land

TL;DR

This paper introduces algorithms for fair division of multi-dimensional land resources, ensuring envy-freeness and proportionality with geometric shape constraints, addressing practical land and advertisement space division.

Contribution

It develops constructive algorithms for fair land division that incorporate geometric shape constraints, a novel extension of classic cake-cutting methods.

Findings

Algorithms achieve envy-freeness with geometric constraints.

Partial proportionality guarantees for land division.

Compatibility of envy-freeness with partial proportionality in natural settings.

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 multi-dimensional 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 multi-dimensional resource wherein the allocated piece must have a pre-specified geometric shape. We present constructive division algorithms that satisfy the two most prominent fairness criteria, namely envy-freeness and proportionality. In settings where proportionality cannot be achieved due to 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.