2022/09/03 by Ayumi Igarashi, Igarashi, Ayumi · 2 citations
Decision Sciences · Economics, Econometrics and Finance · #Auction Theory and Applications #Computer Science and Game Theory (cs.GT) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Game Theory and Voting Systems
paper · pdf · doi:10.48550/arxiv.2209.01348
openalex publication_date 2022/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Cake-cutting is a fundamental model of dividing a heterogeneous resource, such as land, broadcast time, and advertisement space. In this study, we consider the problem of dividing a discrete cake fairly in which the indivisible goods are aligned on a path and agents are interested in receiving a connected subset of items. We prove that a connected division of indivisible items satisfying a discrete counterpart of envy-freeness, called envy-freeness up to one good (EF1), always exists for any number of agents n with monotone valuations. Our result settles an open question raised by Bilò et al. (2019), who proved that an EF1 connected division always exists for the number of agents at most 4. Moreover, the proof can be extended to show the following (1) secretive and (2) extra versions: (1) for n agents with monotone valuations, the path can be divided into n connected bundles such that an EF1 assignment of the remaining bundles can be made to the other agents for any selection made by the secretive agent; (2) for n+1 agents with monotone valuations, the path can be divided into n connected bundles such that when any extra agent leaves, an EF1 assignment of the bundles can be made to the remaining agents.