Decomposing Random Mechanisms
University of California, Los Angeles (UCLA)
M. Utku Ünver
Boston College - Department of Economics
December 13, 2012
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 inspect whether desirable properties of a random mechanism survive decomposition as a lottery over deterministic mechanisms each of which also holds such properties. In cases where desirable properties survive decomposition, without loss of generality we can focus our mechanism design efforts on deterministic mechanisms. 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 an example, we show that individual rationality is totally unimodular and hence decomposable together with totally unimodular feasibility constraints. However, strategy-proofness, unanimity, and feasibility are not totally unimodular together in voting or matching domains. Thus, we introduce a direct constructive approach for such problems. Using this approach, we prove that feasibility, strategy-proofness, unanimity with and without anonymity are decomposable on the non-dictatorial single-peaked voting domains.
Number of Pages in PDF File: 31
Keywords: random mechanisms, ordinal mechanisms, total unimodularity, single-peaked preferences, voting, individual rationality, strategy-proofness, unanimity, anonymity, generalized median mechanisms
JEL Classification: C60, D71, D72working papers series
Date posted: December 14, 2012 ; Last revised: January 22, 2013
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