Surprisingly Popular Voting for Concentric Rank-Order Models

An important problem on social information sites is the recovery of ground truth from individual reports when the experts are in the minority. The wisdom of the crowd, i.e. the collective opinion of a group of individuals fails in such a scenario. However, the surprisingly popular (SP) algorithm~\cite{prelec2017solution} can recover the ground truth even when the experts are in the minority, by asking the individuals to report additional prediction reports--their beliefs about the reports of others. Several recent works have extended the surprisingly popular algorithm to an equivalent voting rule (SP-voting) to recover the ground truth ranking over a set of $m$ alternatives. However, we are yet to fully understand when SP-voting can recover the ground truth ranking, and if so, how many samples (votes and predictions) it needs. We answer this question by proposing two rank-order models and analyzing the sample complexity of SP-voting under these models. In particular, we propose concentric mixtures of Mallows and Plackett-Luce models with $G (\ge 2)$ groups. Our models generalize previously proposed concentric mixtures of Mallows models with $2$ groups, and we highlight the importance of $G > 2$ groups by identifying three distinct groups (expert, intermediate, and non-expert) from existing datasets. Next, we provide conditions on the parameters of the underlying models so that SP-voting can recover ground-truth rankings with high probability, and also derive sample complexities under the same. We complement the theoretical results by evaluating SP-voting on simulated and real datasets.