Debasis Mishra (ISI Delhi)

Centre for Development Economics
and
Department of Economics, Delhi School of Economics

ANNOUNCE A SEMINAR

Implementation in
Multidimensional Dichotomous Domains

by

Debasis Mishra
ISI Delhi

Thursday, February 9, 2012 at 3 p.m.

Venue : AMEX Conference Room (Second Floor)
Department of Economics, Delhi School of Economics

All are cordially invited

Abstract

We study multidimensional mechanism design in private value and quasi-linear environments, e.g. auction domains, matching problems with transfers, choosing a public good among multiple public goods with transfers etc. We restrict attention to deterministic implementation in dominant strategies. Our focus is on domains where agents have dichotomous preferences over alternatives. A dichotomous type of any agent is characterized by a positive real number, which we call the value of the agent at this type, and a non-empty subset of alternatives, which we call the acceptable alternatives. The interpretation is that an agent of dichotomous type gets (the same) utility equal to his value from each alternative in his acceptable set, but gets zero utility on any alternative that is not acceptable. Note that both the value and the set of acceptable alternatives are private information of the agent. This makes such type spaces multidimensional. We call a type space a dichotomous domain if every type belonging to it is a dichotomous type. We characterize the set of implementable allocation rules in dichotomous domains using a condition called “generation monotonicity”. Generation monotonicity is a new (non-trivial) simplification of the “cycle monotonicity” condition of Rochet in dichotomous domains.

Our most striking result comes in a particular class of dichotomous domains. We show that for a large class of dichotomous domains, which we refer to as “rich dichotomous domains”, a significantly weaker condition than generation monotonicity characterizes implementability. We refer to this weaker condition as 2-generation monotonicity, and show it to be equivalent to 3-cycle monotonicity. 3-cycle monotonicity is significantly weaker than cycle monotonicity but stronger than 2-cycle monotonicity, a condition used to characterize implementability in convex domains. A dichotomous domain is not convex, but still multidimensional. While most of the earlier results in the literature found domains where 2-cycle monotonicity is necessary and sufficient for implementability, to our knowledge, this paper is the first to identify multidimensional domains where we see K-cycle monotonicity (K > 2) is necessary and sufficient for implementation. We show, by way of an example, that 2-cycle monotonicity is not sufficient for implementability in rich dichotomous domains. We demonstrate the usefulness of our characterizations by deriving a revenue maximizing mechanism for the one-sided matching problem where agents have dichotomous preferences over alternatives.