open access

https://hdl.handle.net/2115/30120

Chambers of arrangements of hyperplanes and Arrow's impossibility theorem

英語

Arrow's impossibility theorem

2007-09-10

Elsevier Inc.

Advances in Mathematics

214(1)

366-378

Let A be a nonempty real central arrangement of hyperplanes and Ch be the set of chambers of A. Each hyperplane H defines a half-space H+ and the other half-space H−. Let B={+,−}. For H ∈ A, define a map ∈+H : Ch → B by ∈+H(C) = + (if C ⊆ H+) and ∈+H(C) = - (if C ⊆ H−). Define ∈-H = -∈+H. Let Chm=Ch×Ch×･･･×Ch (m times). Then the maps ∈±H induce the maps ∈±H : Chm → Bm. We will study the admissible maps Φ : Chm → Ch which are compatible with every ∈±H. Suppose |A| ≥ 3 and m ≥ 2. Then we will show that A is indecomposable if and only if every admissible map is a projection to a component. When A is a braid arrangement, which is indecomposable, this result is equivalent to Arrow's impossibility theorem in economics. We also determine the set of admissible maps explicitly for every nonempty real central arrangement.

学術雑誌論文

Accepted Manuscript

0001-8708

1090-2082