Given a set of $p$ players we consider problems concerning envy-free allocation of collections of $k$ pieces from a given set of goods or chores. We show that if $p\le n$ and each player can choose $k$ pieces out of $n$ pieces of a cake, then there exist a division of the cake and an allocation of the pieces where at least $\frac{p}{2(k^2-k+1)}$ players get their desired $k$ pieces each. We further show that if $p\le k(n-1)+1$ and each player can choose $k$ pieces, one from each of $k$ cakes that are divided into $n$ pieces each, then there exist a division of the cakes and allocation of the pieces where at least $\frac{p}{2k(k-1)}$ players get their desired $k$ pieces. Finally we prove that if $p\ge k(n-1)+1$ and each player can choose one shift in each of $k$ days that are partitioned into $n$ shifts each, then, given that the salaries of the players are fixed, there exist $n(1+\ln k)$ players covering all the shifts, and moreover, if $k=2$ then $n$ players suffice. Our proofs combine topological methods and theorems of Füredi, Lovász and Gallai from hypergraph theory.
An analysis of a fair division protocol for drawing legislative districts
Landau，Reid和Yershov [A Fair Division Solution to the Problem of Redistricting，\textit{Social Choice and Welfare}，2008]提出了一种基于两个玩家公平划分过程来绘制立法选区的协议，其中每个玩家都有权为州的一部分绘制选区。我们称之为\textit{LRY protocol}。Landau和Su [Fair Division and Redistricting，arXiv：1402.0862，2014]提出了一种衡量州的地区公平性的方法，称为\textit{geometric target}。在本文中，我们证明了在LRY协议下，一方可以赢得的地区的数量最多可以比他们的几何目标少两个，假设没有几何约束的地区，并提供例子来证明这个界是紧的。我们还表明，如果LRY协议被应用在一个状态与几何约束，结果可以是任意远离几何目标。
CNRS, Université Claude Bernard Lyon 1, INSA Lyon, LIRIS（法国国家科学研究中心、里昂第一大学、里昂高等师范学院）；McGill University（麦吉尔大学）；Massachusetts Institute of Technology（麻省理工学院）
