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# Approximately Strategyproof Tournament Rules: On Large Manipulating Sets and Cover-Consistence

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LIPIcs.ITCS.2020.3.pdf
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## Acknowledgements

The authors are extremely grateful to Mikhail Khodak and Jon Schneider, who contributed both with many helpful discussions as well as code to help test the non-manipulability of tournament rules. The authors would also like to thank the anonymous reviewers for their feedback on extensions, clarifications and relevant references unknown to the authors.

## Cite As

Ariel Schvartzman, S. Matthew Weinberg, Eitan Zlatin, and Albert Zuo. Approximately Strategyproof Tournament Rules: On Large Manipulating Sets and Cover-Consistence. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 3:1-3:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ITCS.2020.3

## Abstract

We consider the manipulability of tournament rules, in which n teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all binom{n}{2} matches. Prior work defines a tournament rule to be k-SNM-α if no set of ≤ k teams can fix the ≤ binom{k}{2} matches among them to increase their probability of winning by >α and asks: for each k, what is the minimum α(k) such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) k-SNM-α(k) tournament rule exists? A simple example witnesses that α(k) ≥ (k-1)/(2k-1) for all k, and [Jon Schneider et al., 2017] conjectures that this is tight (and prove it is tight for k=2). Our first result refutes this conjecture: there exists a sufficiently large k such that no Condorcet-consistent tournament rule is k-SNM-1/2. Our second result leverages similar machinery to design a new tournament rule which is k-SNM-2/3 for all k (and this is the first tournament rule which is k-SNM-(<1) for all k). Our final result extends prior work, which proves that single-elimination bracket with random seeding is 2-SNM-1/3 [Jon Schneider et al., 2017], in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is 2-SNM-1/3 and cover-consistent (the winner is an uncovered team with probability 1).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Algorithmic mechanism design
##### Keywords
• Tournament design
• Non-manipulability
• Cover-consistence
• Strategyproofness

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## References

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