Some probabilistic models of best, worst, and best-worst choices

Over the past decade or so, a choice design in which a person is asked to select both the best and the worst option in an available set of options has been gaining favor over more traditional designs, such as where the person is asked, for instance, to: select the best option; select the worst option; rank the options; or rate the options. In this paper, we develop theoretical results for three overlapping classes of probabilistic models for best, worst, and best-worst choices, with the models in each class proposing specific ways in which such choices might be related. The models in these three classes are called random ranking and random utility, joint and sequential, and ratio scale. We include some models that belong to more than one class, with the best known being the maximum-difference (maxdiff) model, summarize estimation issues related to the models, and formulate a number of open theoretical problems.