Asymptotic efficiency of majority rule relative to rank-sum method for selecting the best population

The ranking and selection problem has been well-studied in the case of continuous responses. In this paper, we address the situation in which continuous responses are replaced by discrete orderings. When individuals in the population provide exhaustive rank-orderings of the alternatives, two common decision rules are the majority rule and the rank-sum method. In the former case, the alternative receiving the most first-place votes is declared superior, while in the latter, the alternative with the smallest rank-sum is deemed the best. Both the Pitman efficiencies and the lower bounds on Bahadur efficiencies of the majority rule relative to the rank-sum method are derived, assuming that the rank data are generated from either the Plackett-Luce or the translative strengths models. In addition, finite sample properties of the two methods are compared with the maximum likelihood approach through simulation studies. Our results suggest two things. First, when it is substantially more difficult to obtain a complete rank-ordering than simply the top choice, the majority rule performs adequately and efforts would be better spent asking many voters to provide top choice rather than fewer voters to provide complete orderings. Second, the rank-sum rule compares favorably to, and is substantially more robust than, the maximum likelihood approach.