Fair Division of an Archipelago

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 of 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. In general, when each agent agrees to get up to $k$ connected pieces, where $1 < k < m$, the exact fraction that can be guaranteed is an open problem. The paper presents upper and lower bounds for this fraction. It also presents a potential application --- dividing a two-dimensional land estate shaped as a rectilinear polygon.

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