Incompleteness and Incomparability in Preference Aggregation

We consider how to combine the preferences of multiple agents despite the presence of incompleteness and incomparability in their preference orderings. An agent’s preference ordering may be incomplete because, for example, there is an ongoing preference elicitation process. It may also contain incomparability, which can be useful, for example, in multi-criteria scenarios. We focus on the problem of computing the possible and necessary winners, that is, those outcomes which can be or always are the most preferred for the agents. Possible and necessary winners are useful in many scenarios. For example, preference elicitation need only focus on the unknown relations between possible winners and can ignore completely all other outcomes. Whilst computing the sets of possible and necessary winners is in general a difficult problem, we identify sufficient conditions where we can obtain the necessary winners and an upper approximation of the set of possible winners in polynomial time. Such conditions concern either the language for stating preferences, or general properties of the preference aggregation function.