An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with m≤n, and let $$T^n = \{ x \in \mathbb{R}^n |\Sigma _{i = 1}^n x_i = 1\}$$ be a subset of the n-dimensional Euclidean space ℝ n . For each i=1, . . . , m, there is a class $$\{ M_i^j {\text{| }}j = 1,...,n\}$$ of subsets M i j of Tn . Assume that $$\cup _{j = 1}^n M_i^j = T^n ,$$ for each i=1, . . . , m, that M i j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that $$x \in T^n$$ and its jth component xj≤B(i, j) imply $$x\not \in M_i^j$$ . Then, there exists a partition $$(\Pi (1),...,\Pi (m))$$ of {1, . . . , n} such that $$\Pi (i) \ne \emptyset$$ for all i and $$\cap _{i = 1}^m \cap _{j \in \Pi (i)} M_i^j \ne \emptyset .$$ We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.