Accommodating Cardinal, Ordinal and Mixed Preferences: An Extended Preference Domain for the Assignment Problem

Abstract

We extend the preference domain of the assignment problem to accommodate ordinal, cardinal and mixed preferences together and thereby allow the mechanism designer to elicit different levels of information about individuals' preferences. Given a fixed preference relation over a finite set of alternatives, our domain contains preferences over lotteries which are monotonic, continuous and satisfy independence axiom. Under a natural coarseness relation, the stochastic dominance relation is the coarsest element of the domain and represents fully ordinal preferences. Any von Neumann Morgenstern expected utility preference is a finest element and represents fullycardinal preferences. The extended domain can be characterized by an expected multi-utility representation. We provide a preference reporting language enabling agents to use the extended domain. We generalize the pseudo market mechanism of Hylland and Zeckhauser (1979) for the extended domain and show that the family of pseudo market mechanisms are efficient and weakly envy-free while they fail strategy-proofness. The extended domain naturally admits the impossibility results of the cardinal domain and the ordinal domain. We show, however, that one can find a mechanism which is efficient and weakly strategy-proof for the agents with ordinal preferences.