Dept. Math, Hokkaido Univ. EPrints Server

Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem

Preprint Series # 799
Terao, Hiroaki Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem. (2006);

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Abstract

Let ${\mathcal A}$ be a nonempty real central arrangement of hyperplanes and ${\rm \bf Ch}$ be the set of chambers of ${\mathcal A}$. Each hyperplane $H$ makes a half-space $H^{+} $ and the other half-space $H^{-}$. Let $B = \{+, -\}$. For $H\in {\mathcal A}$, define a map $\epsilon_{H}^{+} : {\rm \bf Ch} \to B$ by $ \epsilon_{H}^{+} (C) = + ~\text{(if~} C\subseteq H^{+}) \, \text{~and~} \epsilon_{H}^{+} (C) = - ~\text{(if~} C \subseteq H^{-}).$ Define $ \epsilon_{H}^{-}=-\epsilon_{H}^{+}.$ Let ${\rm \bf Ch}^{m} = {\rm \bf Ch} \times{\rm \bf Ch}\times\dots\times{\rm \bf Ch} \,\,\,(m\text{~times}).$ Then the maps $\epsilon_{H}^{\pm}$ induce the maps $\epsilon_{H}^{\pm} : {\rm \bf Ch}^{m} \to B^{m} $. We will study the admissible maps $\Phi : {\rm \bf Ch}^{m} \to {\rm \bf Ch}$ which are compatible with every $\epsilon_{H}^{\pm}$. Suppose $|{\mathcal A}|\geq 3$ and $m\geq 2$. Then we will show that ${\mathcal A}$ is indecomposable if and only if every admissible map is a projection to a component. When ${\mathcal A}$ 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.

Item Type:Preprint
Additional Information:30
Uncontrolled Keywords:arrangement of hyperplanes, chambers, braid arrangements, Arrow's impossibility theorem
Subjects:32-xx SEVERAL COMPLEX VARIABLES AND ANALYTIC SPACES
91-xx GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES
52-xx CONVEX AND DISCRETE GEOMETRY
ID Code:1609