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

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).