Computing Possible and Necessary Winners from Incomplete Partially-Ordered Preferences

There are many situations where we wish to represent and reason with preferences. We consider here how to combine the preferences of multiple agents despite incompleteness and incomparability in their preference orderings. An agent’s preference ordering may be incomplete because, for example, we are in the middle of eliciting their preferences. It may also contain incomparability since, for example, we might have multiple criteria we wish to optimize. To combine preferences, we use social welfare functions, which map a profile, that is, a sequence of partial orders (one for each agent), to a partial order (the result). For example, the Pareto social welfare function orders A before B iff every agent orders A before B, else if there is some disagreement between agents declares A and B to incomparable. Since agents’ preferences may be incomplete, we consider all the possible ways they can be completed. In each possible completion, we may obtain different optimal elements (or winners). This leads to the idea of possible winners (those outcomes which are winners in at least one possible completion) and necessary winners (those outcomes which are winners in all possible completions) [5]. Possible and necessary winners are useful in many scenarios including preference elicitation [3]. In fact, elicitation is over when the set of possible winners coincides with that of the necessary winners [4]. In addition, as we argue later, preference elicitation can focus just on the incompleteness concerning those outcomes which are possible and necessary winners. We 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.