When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2017.35
URN: urn:nbn:de:0030-drops-81605
URL: https://drops.dagstuhl.de/opus/volltexte/2017/8160/
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### Condorcet-Consistent and Approximately Strategyproof Tournament Rules

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### Abstract

We consider the manipulability of tournament rules for round-robin tournaments of n competitors. Specifically, n competitors are competing for a prize, and a tournament rule r maps the result of all n(n-1)/2 pairwise matches (called a tournament, T) to a distribution over winners. Rule r is Condorcet-consistent if whenever i wins all n-1 of her matches, r selects i with probability 1.

We consider strategic manipulation of tournaments where player j might throw their match to player i in order to increase the likelihood that one of them wins the tournament. Regardless of the reason why j chooses to do this, the potential for manipulation exists as long as Pr[r(T) = i] increases by more than Pr[r(T) = j] decreases. Unfortunately, it is known that every Condorcet-consistent rule is manipulable. In this work, we address the question of how manipulable Condorcet-consistent rules must necessarily be - by trying to minimize the difference between the increase in Pr[r(T) = i] and decrease in Pr[r(T) = j] for any potential manipulating pair.

We show that every Condorcet-consistent rule is in fact 1/3-manipulable, and that selecting a winner according to a random single elimination bracket is not alpha-manipulable for any alpha > 1/3. We also show that many previously studied tournament formats are all 1/2-manipulable, and the popular class of Copeland rules (any rule that selects a player with the most wins) are all in fact 1-manipulable, the worst possible. Finally, we consider extensions to match-fixing among sets of more than two players.

### BibTeX - Entry

```@InProceedings{schneider_et_al:LIPIcs:2017:8160,
author =	{Jon Schneider and Ariel Schvartzman and S. Matthew Weinberg},
title =	{{Condorcet-Consistent and Approximately Strategyproof Tournament Rules}},
booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
pages =	{35:1--35:20},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-029-3},
ISSN =	{1868-8969},
year =	{2017},
volume =	{67},