Efficient voting via the top-k elicitation scheme: a probabilistic approach

Top-i voting is a common form of preference elicitation due to its conceptual simplicity both on the voters' side and on the decision maker's side. In a typical setting, given a set of candidates, the voters are required to submit only the k-length prefixes of their intrinsic rankings of the candidates. The decision maker then tries to correctly predict the winning candidate with respect to the complete preference profile according to a prescribed voting rule. This raises a tradeoff between the communication cost (given the specified value of k), and the ability to correctly predict the winner. We focus on arbitrary positional scoring rules in which the voters' scores for the candidates is given by a vector that assigns the ranks real values. We study the performance of top-k elicitation under three probabilistic models of preference distribution: a neutral distribution (impartial culture); a biased distribution, such as the Mallows distribution; and a worst-case (but fully known) distribution. For an impartial culture, we provide a technique for analyzing the performance of top-k voting. For the case of arbitrary positional scoring rules, we provide a succinct set of criteria that is sufficient for obtaining both lower and upper bounds on the minimal k necessary to determine the true winner with high probability. Our lower bounds pertain to any implementation of a top-k voting scheme, whereas for our upper bound, we provide a concrete top-k elicitation algorithm. We further demonstrate the use of this technique on Copeland's voting rule. For the case of biased distributions, we show that for any non-constant scoring rule, the winner can be predicted with high probability without ever looking at the votes. For worst-case distributions, we show that for exponentially decaying scoring rules, k = O(log m) is sufficient for all distributions.