Fair Division of an Archipelago
arXiv: Combinatorics(2018)
摘要
An archipelago of $m$ islands has to be divided fairly among $n$ agents with different preferences. What fraction of the total archipelago value can be guaranteed to each agent? Classic algorithms for fair cake-cutting can give each agent a share worth at least $1/n$ of the total value, but this share might be disconnected (spread over multiple islands). When each agent insists on getting a single connected piece (contained in a single island), it is shown that $1/(n+m-1)$ of the total value can be guaranteed, and this fraction is tight. When each agent insists on getting at most $k$ connected pieces, where $1 leq k leq m$, it is possible to guarantee at least $k/(nk+m-k)$ and impossible to guarantee more than $k/(n+m-1)$. The paper presents several cases in which the upper bound can be attained. Whether it can always be attained remains a mystery. Archipelago division has an application to a geometric problem --- fair division of a two-dimensional land estate shaped as a rectilinear polygon, where each agent must receive a rectangular piece.
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