Collective Decision Making under Incomplete Knowledge: Possible and Necessary Solutions

Most solution concepts in collective decision making are defined assuming complete knowledge of individuals' preferences and of the mechanism used for aggregating them. This is often unpractical or unrealistic. Under incomplete knowledge, a solution advocated by many consists in quanrtifying over all completions of the incomplete preference profile (or all instantiations of the incompletely specified mechanism). Voting rules can be `modalized' this way (leading to the notions of possible and necessary winners), and also efficiency and fairness notions in fair division, stability concepts in coalition formation, and more. I give here a survey of works along this line.

Incomplete Preferences in Single-Peaked Electorates

Incomplete preferences are likely to arise in real-world preference aggregation scenarios. This paper deals with determining whether an incomplete preference profile is single-peaked. This is valuable information since many intractable voting problems become tractable given singlepeaked preferences. We prove that the problem of recognizing single-peakedness is NP-complete for incomplete profiles consisting of partial orders. Despite this intractability result, we find several polynomial-time algorithms for reasonably restricted settings. In particular, we give polynomial-time recognition algorithms for weak orders, which can be viewed as preferences with indifference.