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Cardinal scores (numeric ratings) collected from people are well known tosuffer from miscalibrations. A popular approach to address this issue is toassume simplistic models of miscalibration (such as linear biases) to de-biasthe scores. This approach, however, often fares poorly because people'smiscalibrations are typically far more complex and not well understood. In theabsence of simplifying assumptions on the miscalibration, it is widely believedby the crowdsourcing community that the only useful information in the cardinalscores is the induced ranking. In this paper, inspired by the framework ofStein's shrinkage, empirical Bayes, and the classic two-envelope problem, wecontest this widespread belief. Specifically, we consider cardinal scores witharbitrary (or even adversarially chosen) miscalibrations which are onlyrequired to be consistent with the induced ranking. We design estimators whichdespite making no assumptions on the miscalibration, strictly and uniformlyoutperform all possible estimators that rely on only the ranking. Ourestimators are flexible in that they can be used as a plug-in for a variety ofapplications, and we provide a proof-of-concept for A/B testing and ranking.Our results thus provide novel insights in the eternal debate between cardinaland ordinal data.

Your 2 is My 1, Your 3 is My 9: Handling Arbitrary Miscalibrations in Ratings