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arXiv · 1807.03743

Algebraic Voting Theory & Representations of $S_m \wr S_n$

Hélène Barcelo·Megan Bernstein·Sarah Bockting-Conrad·Erin McNicholas·Kathryn Nyman·Shira Viel
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Abstract

We consider the problem of selecting an $n$-member committee made up of one of $m$ candidates from each of $n$ distinct departments. Using an algebraic approach, we analyze positional voting procedures, including the Borda count, as $\mathbb{Q}S_m \wr S_n$-module homomorphisms. In particular, we decompose the spaces of voter preferences and election results into simple $\mathbb{Q}S_m \wr S_n$-submodules and apply Schur's Lemma to determine the structure of the information lost in the voting process. We conclude with a voting paradox result, showing that for sufficiently different weighting vectors, applying the associated positional voting procedures to the same set of votes can yield arbitrarily different election outcomes.

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Hélène Barcelo, Megan Bernstein, Sarah Bockting-Conrad, Erin McNicholas, Kathryn Nyman, Shira Viel. 2018-07-06. Algebraic Voting Theory & Representations of $S_m \wr S_n$. https://arxiv.org/abs/1807.03743

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