Fair Cake Division Under Monotone Likelihood Ratios

This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation of cake cutting wherein each agent must receive a contiguous piece of the cake. Although multiple hardness results exist in this setup for finding fair/efficient cake divisions, we show that, if the value densities of the agents satisfy the monotone likelihood ratio property (MLRP), then strong algorithmic results hold for various notions of fairness and economic efficiency. Addressing cake-cutting instances with MLRP, first we develop an algorithm that finds cake divisions (with connected pieces) that are envy free, up to an arbitrary precision. The time complexity of our algorithm is polynomial in the number of agents and the bit complexity of an underlying Lipschitz constant. We obtain similar positive results for maximizing social, egalitarian, and Nash social welfare. Many distribution families bear MLRP. In particular, this property holds if all the value densities belong to any one of the following families: Gaussian (with the same variance), linear, Poisson, and exponential distributions, linear translations of any log-concave function. Hence, through MLRP, the current work obtains novel cake-cutting algorithms for multiple distribution families.

Constraints in fair division

The fair allocation of resources to interested agents is a fundamental problem in society. While the majority of the fair division literature assumes that all allocations are feasible, in practice there are often constraints on the allocation that can be chosen. In this survey, we discuss fairness guarantees for both divisible (cake cutting) and indivisible resources under several common types of constraints, including connectivity, cardinality, matroid, geometric, separation, budget, and conflict constraints. We also outline a number of open questions and directions.

Graphical Cake Cutting via Maximin Share

We study the recently introduced cake-cutting setting in which the cake is represented by an undirected graph. This generalizes the canonical interval cake and allows for modeling the division of road networks. We show that when the graph is a forest, an allocation satisfying the well-known criterion of maximin share fairness always exists. Our result holds even when separation constraints are imposed; however, in the latter case no multiplicative approximation of proportionality can be guaranteed. Furthermore, while maximin share fairness is not always achievable for general graphs, we prove that ordinal relaxations can be attained.

Towards Fairly Apportioning Sale Proceeds in a Collective Sale of Strata Property

Cake-cutting is a longstanding metaphor for ‘a wide range of real-world problems that involve’ the division of anything of value. Unsurprisingly, where owners of a strata scheme wish to end the strata scheme and collectively sell their development, one of the most contentious issues may be the apportionment of sale proceeds. In Singapore, this problem is compounded in mixed developments which have both commercial and residential elements as well as in developments with different sized units, often with disproportionate strata share values; even differing facings and the state of one’s unit may attract disenchantment when trying to apportion proceeds. This article critically analyses how New South Wales (‘NSW’) and Singapore allocate proceeds pursuant to a collective sale of strata property. In this respect, the Strata Schemes Development Act 2015 (NSW) and Strata Schemes Management Act 2015 (NSW) are significantly clearer than Singapore’s Land Titles (Strata) Act (Singapore, cap 158, rev ed 2009) as the latter does not prescribe any statutory formula for apportionment. In examining the jurisprudence and respective strata frameworks, this article proposes how proceeds in a collective sale could be more fairly apportioned.

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.

Envy-Free Division of Land

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.

Fair Division of Time: Multi-layered Cake Cutting

We initiate the study of multi-layered cake cutting with the goal of fairly allocating multiple divisible resources (layers of a cake) among a set of agents. The key requirement is that each agent can only utilize a single resource at each time interval. Several real-life applications exhibit such restrictions on overlapping pieces, for example, assigning time intervals over multiple facilities and resources or assigning shifts to medical professionals. We investigate the existence and computation of envy-free and proportional allocations. We show that envy-free allocations that are both feasible and contiguous are guaranteed to exist for up to three agents with two types of preferences, when the number of layers is two. We further devise an algorithm for computing proportional allocations for any number of agents when the number of layers is factorable to three and/or some power of two.