Implementation of the Walrasian Correspondence: The Boundary Problem

Consider exchange economies in which preferences are continuous, convex and strongly monotonic. It is well known that the Walrasian correspondence is not Nash implementable. Maskin monotonicity (Maskin, 1999) is violated for allocations at the boundary of the feasible set. We derive an impossibility result showing that it is in fact not implementable in any solution concept. Next, we construct a sequential mechanism based on price-allocationannouncements that fits the very description of Walrasian Equilibrium. Imposing an additional domain restriction, we show that it fully implements the Walrasian correspondence in subgame perfect and strong subgame perfect equilibrium. We thus take care of the boundary problem that was prominent in the Nash implementation literature.microeconomics ;

Equal-Budget Choice Equivalent Solutions in Exchange Economies

Given a family of linear budget sets, an allocation is equal opportunity equivalent (Thomson, 1994) if there exists a common budget set such that each agent is indi¤erent between the bundle that he gets and the best bundle he can obtain in the choice set. We first study therobustness properties of equal opportunity equivalent correspondences with respect to change in preferences. We impose independence to irrelevant preference changes and connect this property with the implementation of rules via some game-theoretic solution concept. We provide an equivalence result with the equal-income Walrasian rule. Next, we study robustness with respect to change in the number of agents and derive a haracterization of the equal-income Walrasian rule. Our results provide additional justifications for the equal-division of resources as a first step toward fairness.microeconomics ;

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackson and Palfrey, 2001).microeconomics ;

Maximal Domains for Strategy-proof or Maskin Monotonic Choice Rules

Domains of individual preferences for which the well-known impossibility Theorems of Gibbard-Satterthwaite and Muller-Satterthwaite do not hold are studied. First, we introduce necessary and sufficient conditions for a domain to admit non-dictatorial, Pareto efficient and either strategy-proof or Maskin monotonic social choice rules. Next, to comprehend the limitations the two Theorems imply for social choice rules, we search for the largest domains that are possible. Put differently, we look for the minimal restrictions that have to be imposed on the unrestricted domain to recover possibility results. It turns out that, for such domains, the conditions of inseparable pair and of inseparable set yield the only maximal domains on which there exist non-dictatorial, Pareto efficient and strategy-proof social choice rules. Next, we characterize the maximal domains which allow for Maskin monotonic, non-dictatorial and Pareto-optimal social choice rules.Strategy-proofness; Maskin monotonicity; Restricted domains; Maximal domains

Maximal Domains for Strategy-Proof or Maskin Monotonic Choice Rules

Domains of individual preferences for which the well-known impossibility theorems of Gibbard-Satterthwaite and Muller-Satterthwaite do not hold are studied. To comprehend the limitations these results imply for social choice rules, we search for the largest domains that are possible. Here, we restrict the domain of individual prefer ences of precisely one individual. It turns out that, for such domains, the conditions of inseparable pair and of inseparable set yield the only maximal domains on which there exist non-dictatorial, Pareto-efficient and strategy-proof social choice rules. Next, we characterize the maximal domains which allow for Maskin monotone, non-dictatorial and Pareto-efficient social choice rules.mathematical economics;

The Relation between Monotonicity and Strategy-Proofness

The Muller-Satterthwaite Theorem (Muller and Satterthwaite, 1977) establishes the equivalence between Maskin monotonicity and strategy-proofness, two cornerstone conditions for the decentralization of social choice rules. We consider a general model that covers public goods economies as in Muller and Satterthwaite (1977) as well as private goods economies. For private goods economies we use a weaker condition than Maskin monotonicity that we call unilateral monotonicity. We introduce two easy-to-check domain conditions which separately guarantee that (i) unilateral/Maskin monotonicity implies strategy-proofness (Theorem 1) and (ii) strategy-proofness implies unilateral/Maskin monotonicity (Theorem 2). We introduce and discuss various classical single-peaked domains and show which of the domain conditions they satisfy (see Propositions 1 and 2 and an overview in Table 1). As a by-product of our analysis, we obtain some extensions of the Muller-Satterthwaite Theorem as summarized in Theorem 3. We also discuss some new "Muller-Satterthwaite domains" (e.g.,Proposition 3).Muller-Satterthwaite Theorem; restricted domains; rich domains; single-peaked domains; strategy-proofness; unilateral/Maskin monotonicity

Priorities in the Location of Multiple Public Facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number k of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We provide a characterization of the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among "interest groups". We characterize each of the subclasses of priority rules that respectively satisfy anonymity, hiding-proofness and strategy-proofness. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Alternatively, any such rule can be viewed as a collection of fixed-populations generalized peak-selection median rules (Moulin, 1980), that are linked across populations, in a way that we describe.Multiple public facilities; Priority rules; Hierarchical rules; Object-population monotonicity; Sovereignty; Anonymity; Strategy-proofness; Generalized median rules; Hiding-proofness