Aggregating Partially Ordered Preferences

Abstract

Preferences are not always expressible via complete linear orders: some- times it is more natural to allow for the presence of incomparable outcomes. This may hold both in the agents' preference ordering and in the social order. In this paper we consider this scenario and we study what properties it may have. In par- ticular, we show that, despite the added expressivity and ability to resolve con icts provided by incomparability, classical impossibility results (such as Arrow's theorem, Muller-Satterthwaite's theorem, and Gibbard-Satterthwaite's theorem) still hold. We also prove some possibility results, generalizing Sen's theorem for majority voting. To prove these results, we dene new notions of unanimity, monotonicity, dictator, triple-wise value-restriction, and strategy-proofness, which are suitable and natural generalizations of the classical ones for complete orders.