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

Authors Ariel Schvartzman, S. Matthew Weinberg, Eitan Zlatin, Albert Zuo

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Author Details

Ariel Schvartzman
  • Department of Computer Science, Princeton University, NJ, USA
S. Matthew Weinberg
  • Department of Computer Science, Princeton University, NJ, USA
Eitan Zlatin
  • Department of Computer Science, Princeton University, NJ, USA
Albert Zuo
  • Computer Science Department, Stanford University, CA, USA


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.

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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)


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
  • Tournament design
  • Non-manipulability
  • Cover-consistence
  • Strategyproofness


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