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arXiv · 2608.25558

On the Dimension of the Best-Worst Choice Polytope

Keivan Mallahi-Karai·Majid Salamat
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Abstract

For a set of $n$ alternatives, the best--worst choice polytope is the convex hull of the deterministic best--worst choice vectors induced by all linear rankings of the alternatives. The question of determining the dimension of this polytope as a function of $n$ has been raised by Doignon \cite{Doignon2023}. In this paper we answer this question by finding an explicit formula for this dimension. Our proof is motivated by a representation-theoretic interpretation of the relevant rank problem for the symmetric group, but is presented entirely in elementary linear-algebraic terms. The connection with representation theory will be explored elsewhere.

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Keivan Mallahi-Karai, Majid Salamat. 2026-08-26. On the Dimension of the Best-Worst Choice Polytope. https://arxiv.org/abs/2608.25558

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