On the probability of a Condorcet winner among a large number of alternatives

TL;DR

This paper analyzes the probability of a Condorcet winner in elections with many alternatives and voters, providing asymptotic results for uniform distributions and identifying the minimal probability across all distributions.

Contribution

It determines the asymptotic probability of Condorcet winners under uniform rankings and identifies the minimal probability achievable with any distribution.

Findings

Asymptotic probability of Condorcet winner for uniform distribution

Minimal probability of Condorcet winner across all distributions

Explicit construction of distributions achieving minimal probability

Abstract

Consider $2k-1$ voters, each of which has a preference ranking between $n$ given alternatives. An alternative $A$ is called a Condorcet winner, if it wins against every other alternative $B$ in majority voting (meaning that for every other alternative $B$ there are at least $k$ voters who prefer $A$ over $B$). The notion of Condorcet winners has been studied intensively for many decades, yet some basic questions remain open. In this paper, we consider a model where each voter chooses their ranking randomly according to some probability distribution among all rankings. One may then ask about the probability to have a Condorcet winner with these randomly chosen rankings (which, of course, depends on $n$ and $k$, and the underlying probability distribution on the set of rankings). In the case of the uniform probability distribution over all rankings, which has received a lot of attention and is often referred to as the setting of an "impartial culture", we asymptotically determine the probability of having a Condorcet winner for a fixed number $2k-1$ of voters and $n$ alternatives with $n\to \infty$. This question has been open for around fifty years. While some authors suggested that the impartial culture should exhibit the lowest possible probability of having a Condorcet winner, in fact the probability can be much smaller for other distributions. We determine, for all values of $n$ and $k$, the smallest possible probability of having a Condorcet winner (and give an example of a probability distribution over all rankings which achieves this minimum possible probability).