# Flexible Level-1 Consensus Ensuring Stable Social Choice: Analysis and Algorithms

## TL;DR

This paper investigates the rarity of Level-1 Consensus in preference profiles, introduces a weaker but more probable variant called Flexible Consensus, and provides algorithms and theoretical analysis for these properties.

## Contribution

It presents an algorithm to detect Level-1 Consensus, demonstrates its scarcity, and proposes Flexible Consensus as a more attainable stability condition with strong theoretical backing.

## Key findings

- Level-1 Consensus is very rare in large preference profiles.
- Under impartial culture, probability of Level-1 Consensus approaches zero as group size increases.
- Flexible Consensus is significantly more probable and maintains stability properties.

## Abstract

Level-1 Consensus is a property of a preference-profile. Intuitively, it means that there exists a preference relation which induces an ordering of all other preferences such that frequent preferences are those that are more similar to it. This is a desirable property, since it enhances the stability of social choice by guaranteeing that there exists a Condorcet winner and it is elected by all scoring rules. In this paper, we present an algorithm for checking whether a given preference profile exhibits level-1 consensus. We apply this algorithm to a large number of preference profiles, both real and randomly-generated, and find that level-1 consensus is very improbable. We support these empirical findings theoretically, by showing that, under the impartial culture assumption, the probability of level-1 consensus approaches zero when the number of individuals approaches infinity. Motivated by these observations, we show that the level-1 consensus property can be weakened while retaining its stability implications. We call this weaker property Flexible Consensus. We show, both empirically and theoretically, that it is considerably more probable than the original level-1 consensus. In particular, under the impartial culture assumption, the probability for Flexible Consensus converges to a positive number when the number of individuals approaches infinity.