# Voting Rules that are Unbiased but not Transitive-Symmetric

## TL;DR

This paper investigates the relationship between two symmetry properties of voting rules, demonstrating that unbiasedness does not necessarily imply transitive-symmetry through two distinct constructions.

## Contribution

It provides the first explicit and probabilistic constructions of voting rules that are unbiased but lack transitive-symmetry, clarifying their independence.

## Key findings

- Unbiased voting rules can lack transitive-symmetry.
- Explicit constructions of such rules are provided.
- Probabilistic methods demonstrate the same independence.

## Abstract

We explore the relation between two natural symmetry properties of voting rules. The first is transitive-symmetry -- the property of invariance to a transitive permutation group -- while the second is the "unbiased" property of every voter having the same influence for all i.i.d. probability measures. We show that these properties are distinct by two constructions -- one probabilistic, one explicit -- of rules that are unbiased but not transitive-symmetric.