Decomposing Random Mechanisms

Abstract

Ordinal random mechanisms have been used in real-life situations for reasons such as sustaining fairness or preventing collusion. Two examples of such domains are voting and matching. We investigate whether desirable properties of a random mechanism survive decomposition as a lottery over deterministic mechanisms that also hold such properties. To this end, we introduce a framework in which we can represent a random mechanism and its desirable properties such as strategy-proofness or individual rationality using linear constraints. Using the theory of totally unimodular incidence matrices from combinatorial integer programming, we introduce a sufficient condition for decomposability of linear constraints on random mechanisms. As two examples, we show that individual rationality is totally unimodular in general, and that strategy-proofness is totally unimodular in a certain individual choice model. However, strategy-proofness, unanimity, and feasibility together are not totally unimodular in collective choice environments in general. Thus, we introduce a direct constructive approach for such problems. Using this approach, we prove that feasibility, strategy-proofness, and unanimity, with and without anonymity, are decomposable on non-dictatorial single-peaked voting domains.

Keywords: random mechanisms, ordinal mechanisms, total unimodularity, single-peaked preferences, voting, individual rationality, strategy-proofness, unanimity, anonymity, generalized median mechanisms

JEL Classification: C60, D71, D72