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

## TL;DR

This paper investigates the manipulability of tournament rules, disproves a conjecture about the limits of Condorcet-consistent rules, and introduces new rules with improved fairness and resistance to manipulation.

## Contribution

It refutes a conjecture on the manipulability bounds of Condorcet-consistent tournament rules and proposes a new rule that is more resistant to manipulation and fair.

## Key findings

- No Condorcet-consistent rule is $k$-SNM-$1/2$ for sufficiently large $k$
- A new tournament rule is $k$-SNM-$2/3$ for all $k$
- Introduces Randomized-King-of-the-Hill, which is $2$-SNM-$1/3$ and cover-consistent

## 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-$\alpha$ if no set of $\leq k$ teams can fix the $\leq \binom{k}{2}$ matches among them to increase their probability of winning by $>\alpha$ and asks: for each $k$, what is the minimum $\alpha(k)$ such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) $k$-SNM-$\alpha(k)$ tournament rule exists? A simple example witnesses that $\alpha(k) \geq \frac{k-1}{2k-1}$ for all $k$, and [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-$1/3$([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 \emph{cover-consistent} (the winner is an uncovered team with probability $1$).