Envy-free division of multi-layered cakes
Ayumi Igarashi, Fr\'ed\'eric Meunier

TL;DR
This paper addresses envy-free division of multi-layered cakes, providing existence results under certain conditions and designing an FPTAS for approximate solutions with monotone preferences.
Contribution
It establishes the existence of envy-free, non-overlapping, and contiguous multi-layered cake divisions for prime power number of agents and layers, and develops an FPTAS for approximate solutions.
Findings
Existence of envy-free divisions for prime power agents and layers.
Development of an FPTAS for two-layered cake division among three agents.
Extension of results to groups of nearly equal size and arbitrary group sizes with monotone preferences.
Abstract
We study the problem of dividing a multi-layered cake under non-overlapping constraints. This problem, recently proposed by Hosseini et al. (IJCAI, 2020), captures several natural scenarios such as the allocation of multiple facilities over time where each agent can utilize at most one facility simultaneously, and the allocation of tasks over time where each agent can perform at most one task simultaneously. We establish the existence of an envy-free multi-division that is both non-overlapping and contiguous within each layer when the number of agents is a prime power and the number of layers is at most the number of agents, providing a positive partial answer to an open question by Hosseini et al. To achieve this, we employ a new approach based on a general fixed point theorem, originally proven by Volovikov (1996), and recently applied by Joji\'{c}, Panina, and \v{Z}ivaljevi\'{c}…
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Taxonomy
TopicsGame Theory and Voting Systems · Auction Theory and Applications · Economic theories and models
