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On Approximately Strategy-Proof Tournament Rules for Collusions of Size at Least Three

David Mikšaník, Ariel Schvartzman, Jan Soukup

TL;DR

This work addresses the challenge of designing tournament rules that are Condorcet-consistent (CC) while resisting collusion among up to $k$ teams, quantified via $k$-SNM-$\alpha$. It introduces two explicit, CC, monotone rules: a generalized RdSEB built on $d$-ary trees with a provable manipulability bound $\alpha_{d,k} \le 1 - \frac{2 (d)_k}{d^{k+1}}$ and a notably improved SignificantOnly rule that achieves $2$-SNM-$\tfrac{1}{3}$ and $3$-SNM-$\tfrac{1}{2}$ (tight). The authors also propose a scalable reduction framework that extends small-$n$ top-cycle consistent, $k$-SNM-$\alpha$ rules to larger $n$ with an explicit bound $\alpha' \le \alpha\left(1 - \frac{(k-1)^2}{n}\right) + \frac{(k-1)^2}{n}$. Together, these results advance explicit, monotone, approximately strategy-proof tournament rules and provide a tractable path from small to large-scale settings via the top-cycle extension method.

Abstract

A tournament organizer must select one of $n$ possible teams as the winner of a competition after observing all $\binom{n}{2}$ matches between them. The organizer would like to find a tournament rule that simultaneously satisfies the following desiderata. It must be Condorcet-consistent (henceforth, CC), meaning it selects as the winner the unique team that beats all other teams (if one exists). It must also be strongly non-manipulable for groups of size $k$ at probability $α$ (henceforth, k-SNM-$α$), meaning that no subset of $\leq k$ teams can fix the matches among themselves in order to increase the chances any of it's members being selected by more than $α$. Our contributions are threefold. First, wee consider a natural generalization of the Randomized Single Elimination Bracket rule from [Schneider et al. 2017] to $d$-ary trees and provide upper bounds to its manipulability. Then, we propose a novel tournament rule that is CC and 3-SNM-1/2, a strict improvement upon the concurrent work of [Dinev and Weinberg, 2022] who proposed a CC and 3-SNM-31/60 rule. Finally, we initiate the study of reductions among tournament rules.

On Approximately Strategy-Proof Tournament Rules for Collusions of Size at Least Three

TL;DR

This work addresses the challenge of designing tournament rules that are Condorcet-consistent (CC) while resisting collusion among up to teams, quantified via -SNM-. It introduces two explicit, CC, monotone rules: a generalized RdSEB built on -ary trees with a provable manipulability bound and a notably improved SignificantOnly rule that achieves -SNM- and -SNM- (tight). The authors also propose a scalable reduction framework that extends small- top-cycle consistent, -SNM- rules to larger with an explicit bound . Together, these results advance explicit, monotone, approximately strategy-proof tournament rules and provide a tractable path from small to large-scale settings via the top-cycle extension method.

Abstract

A tournament organizer must select one of possible teams as the winner of a competition after observing all matches between them. The organizer would like to find a tournament rule that simultaneously satisfies the following desiderata. It must be Condorcet-consistent (henceforth, CC), meaning it selects as the winner the unique team that beats all other teams (if one exists). It must also be strongly non-manipulable for groups of size at probability (henceforth, k-SNM-), meaning that no subset of teams can fix the matches among themselves in order to increase the chances any of it's members being selected by more than . Our contributions are threefold. First, wee consider a natural generalization of the Randomized Single Elimination Bracket rule from [Schneider et al. 2017] to -ary trees and provide upper bounds to its manipulability. Then, we propose a novel tournament rule that is CC and 3-SNM-1/2, a strict improvement upon the concurrent work of [Dinev and Weinberg, 2022] who proposed a CC and 3-SNM-31/60 rule. Finally, we initiate the study of reductions among tournament rules.
Paper Structure (9 sections, 18 theorems, 28 equations, 1 table)

This paper contains 9 sections, 18 theorems, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation
  4. Analysis for the RdSEB Rule
  5. Analysis for the SignificantOnly Rule
  6. Almost Optimal Rules for Large $n$ from Optimal Rules for Small $n$
  7. Conclusion and Future Directions
  8. Missing Proofs from Section \ref{['sec:sigOnly']}
  9. Missing Proofs from Section \ref{['sec:reduction']}

Key Result

Theorem 1

Let $2 \leq k \leq d$. The $\textsc{R}d\textsc{SEB}$ rule is Condorcet-consistent, monotone and $k\textsc{-SNM-}\alpha_{d, k}$ for where $(d)_k = \prod_{i=0}^{k-1} (d-i)$ is the falling factorial of $d$ with $k$ terms.

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 3
  • ...and 33 more