Aggregation Theorems and the Combination of Probabilistic Rank Orders

There are many situations where we wish to combine multiple rank orders or other preference information on a fixed set of options to obtain a combined rank order. Two of the most common applications are determining a social rank order on a set of options from a set of individual rank orders on those options, and predicting (or prescribing) an individual’s overall rank order on a set of options from the rank orders on a set of component dimensions of the options. In this paper, I develop solutions to this class of problems when the rank orders can occur probabilistically. I develop aggregation theorems that are motivated by recent theoretical work on the combination of expert opinions and I discuss various models that have the property that the representations are ‘of the same form’ for both the component and overall rank order probabilities. I also briefly discuss difficulties in actually using such probabilistic ranking models in the social choice situation.