How Likely Are Large Elections Tied?

Understanding the likelihood for an election to be tied is a classical topic in many disciplines including social choice, game theory, political science, and public choice. Despite a large body of literature and the common belief that ties are rare, little is known about how rare ties are in large elections except for a few simple positional scoring rules under the i.i.d. uniform distribution over the votes, known as the Impartial Culture (IC) in social choice. In particular, little progress was made after Marchant explicitly posed the likelihood of k-way ties under IC as an open question in 2001. We give an asymptotic answer to the open question for a wide range of commonly studied voting rules under a model that is much more general and realistic than i.i.d. models (especially IC) -- the smoothed social choice framework by Xia that was inspired by the celebrated smoothed complexity analysis by Spielman and Teng. We prove dichotomy theorems on the smoothed likelihood of ties under positional scoring rules, edge-order-based rules, and some multi-round score-based elimination rules, which include commonly studied voting rules such as plurality, Borda, veto, maximin, Copeland, ranked pairs, Schulze, STV, and Coombs as special cases. We also complement the theoretical results by experiments on synthetic data and real-world rank data on Preflib. Our main technical tool is an improved dichotomous characterization on the smoothed likelihood for a Poisson multinomial variable to be in a polyhedron, which is proved by exploring the interplay between the V-representation and the matrix representation of polyhedra and might be of independent interest.