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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.13

Single-Elimination Brackets Fail to Approximate Copeland Winner

Authors: Reyna Hulett

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract

Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature.

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Reyna Hulett. Single-Elimination Brackets Fail to Approximate Copeland Winner. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hulett:LIPIcs.APPROX-RANDOM.2019.13,
  author =	{Hulett, Reyna},
  title =	{{Single-Elimination Brackets Fail to Approximate Copeland Winner}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{13:1--13:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.13},
  URN =		{urn:nbn:de:0030-drops-112283},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.13},
  annote =	{Keywords: Voting theory, mechanism design, query complexity, approximation}
}

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Single-Elimination Brackets Fail to Approximate Copeland Winner

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