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.

The maximin share (MMS) guarantee is a desirable fairness notion for allocating indivisible goods. While MMS allocations do not always exist, several approximation techniques have been developed to ensure that all agents receive a fraction of their maximin share. We focus on an alternative approximation notion, based on the population of agents, that seeks to guarantee MMS for a fraction of agents. We show that no optimal approximation algorithm can satisfy more than a constant number of agents, and discuss the existence and computation of MMS for all but one agent and its relation to approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for 2/3 of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee the value that an agent receives by partitioning the goods into 3n/2 bundles, improving the best known guarantee when goods are partitioned into 2n-2 bundles. Finally, we provide empirical experiments using synthetic data.

Fair division is a subfield of multiagent systems that is concerned with object distribution. When objects are indivisible, the Maximin Share Guarantee (MMS) is a desirable fairness notion; however, it is not guaranteed to exist. While MMS allocations may not always exist, a relaxation of MMS is guaranteed to exist. We show that there exists a family of instances for which this relaxation fails to guarantee the MMS value for all but a small constant number of agents.

In fair division of indivisible goods,  ℓ-out-of-d maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into d bundles and choosing the ℓ least preferred bundles. Most existing works aim to guarantee to all agents a constant fraction of their 1-out-of-n MMS. But this guarantee is sensitive to small perturbation in agents' cardinal valuations. We consider a more robust approximation notion, which depends only on the agents' ordinal rankings of bundles. We prove the existence of ℓ-out-of-⌊(ℓ + 1/2)n⌋ MMS allocations of goods for any integer ℓ ≥ 1, and present a polynomial-time algorithm that finds a 1-out-of-⌈3n/2⌉ MMS allocation when ℓ=1. We further develop an algorithm that provides a weaker ordinal approximation to MMS for any ℓ > 1.

The maximin share (MMS) guarantee is a desirable fairness notion for allocating indivisible goods. While MMS allocations do not always exist, several approximation techniques have been developed to ensure that all agents receive a fraction of their maximin share. We focus on an alternative approximation notion, based on the population of agents, that seeks to guarantee MMS for a fraction of agents. We show that no optimal approximation algorithm can satisfy more than a constant number of agents, and discuss the existence and computation of MMS for all but one agent and its relation to approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for 2 3 of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee MMS 3n/2 , i.e., the value that agents receive by partitioning the goods into 3 2 n bundles, improving the best known guarantee of MMS 2n−2 . Finally, we provide empirical experiments using synthetic data.