On Level-1 Consensus Ensuring Stable Social Choice

Level-1 consensus is a property of a preference profile. Intuitively, it means that there exists some preference relation such that, when ordering the other preference-relations by increasing distance from it, the closer preferences are more frequent in the profile. This is a desirable property, since it enhances the stability of the 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 back this empirical findings by a simple theoretical proof 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 retaining the stability implications. The weaker level-1 consensus is considerably more probable, both empirically and theoretically. In fact, under the impartial culture assumption, the probability converges to a positive number when the number of individuals approaches infinity.