Private Good Economies Research Articles

Strategy-proofness in private good economies with linear preferences: An impossibility result

We consider the problem of allocating a social endowment of private goods among a group of n agents with linear preferences. We search for rules that satisfy three standard requirements. First is (Pareto-)efficiency. Second is the fairness requirement of “equal treatment of equals” (in welfare terms). Third is the incentive requirement of “strategy-proofness”: no agent should ever benefit from misrepresenting their preferences. We show that no such rule exists. A variety of domains have been explored and similar negative results have been proved but one would have hoped that the domain of linear preferences is sufficiently narrow to allow for some positive results. Our result dashes such hopes.

On Strategy-proofness and the Salience of Single-peakedness in a Private Goods Economy

We consider strategy-proof rules operating on a rich domain of preference profiles in a set up where multiple private goods have to be assigned to a set of agents with entitlements where preferences display satiation. We show that if the rule is in addition desirable, in that it is tops-only, continuous, same-sided and individually rational with respect to the entitlements, then the preferences in the domain have to satisfy a variant of single-peakedness (referred to as semilattice single-peakedness). We also provide a converse of this main finding. It turns out that this domain coincides with the one already identified in a general set up with a public good. Finally, we relate semilattice single-peaked domains to well-known restricted domains under which strategy-proof and desirable rules do exist.

Equality of Power and Fair Public Decision-Making

Ronald Dworkin's equality of resources, and the closely related concept of envy-freeness, are two of the fundamental ideas behind fair allocation of private goods. The appropriate analog to these concepts in a public decision-making environment is unclear, since all agents consume the same bundle of resources (though they may have different utilities for this bundle). Drawing inspiration from equality of resources and the Dworkin quote below, we propose that equality in public decision-making should allow each agent to cause equal cost to the rest of society, which we model as equal externality. We term this equality of power. The first challenge here is that the cost to the rest of society must be measured somehow, and it is generally impossible to elicit the scale of individual utilities (in the absence of monetary payments). Again drawing inspiration from foundational literature for private goods economies, we normalize each agent's utility so that every agent's marginal utility for additional power is the same. We show that for quadratic utilities, in the large market limit, there always exists an outcome that simultaneously satisfies equal power, equal marginal utility for additional power, and social welfare maximization with respect to the normalized utilities.

Strategy Proofness and Unanimity in Private Good Economies with Single-Peaked Preferences

In this paper we establish the link between strategy-proofness and unanimity in a domain of private good economies with single-peaked preferences. We introduce a new condition and we prove that if this new property together with the requirement of citizen sovereignty are held, a social choice function satisfies strategy-proofness if and only if it is unanimous. As an implication, we show that strategy-proofness and Maskin monotonicity become equivalent. We also give applications to implementation literature: We provide a full characterization for standard Nash implementation and partially honest Nash implementation and we determine a certain equivalence among these theories.

Nash implementation in private good economies with single-plateaued preferences and in matching problems

In this paper we explore the problem of Nash implementation providing two new properties called I-monotonicity and I-weak no-veto power. These properties form together with unanimity a new sufficient condition for a social choice correspondence to be implementable in Nash equilibria with at least three agents. As applications, we prove that: (1) In private good economies with single-plateaued preferences, each solution of the problem of fair division that has a full range is Nash implementable if and only if it satisfies Maskin monotonicity, (2) In matching problems, Maskin monotonicity is a necessary and sufficient condition for implementation.

Compensated Lindahl Equilibrium: Efficiency Theorems and Gains from Trade with Individual Production and Public Goods

In general equilibrium theory for private good economies, possible discontinuities in budget correspondences create obstacles to decentralizing Pareto efficient allocations, as well as to existence of Walrasian equilibrium with gains from trade. Then McKenzie's (1957) notion of compensated demand, and the associated notion of compensated Walrasian equilibrium, are important tools for surmounting such obstacles. Here, for economies that include public goods along with both individual and public production, we provide analogous results for a compensated notion of Lindahl equilibrium.

Group Strategy-Proofness in Private Good Economies Without Money: Matching, Division and House Allocation

We observe that three salient solutions to matching, division and house allocation problems are not only (partially) strategy-proof, but (partially) group strategy-proof as well, in appropriate domains of definition. That is the case for the Gale-Shapley mechanism, the uniform rule and the top trading cycle solution, respectively. We embed these three types of problems into a general framework. We then notice that the three rules, as well as many others, do share a common set of properties, which together imply their (partial) group strategy-proofness. This proves that the equivalence between individual and group strategy-proofness in all these cases is not a fortuitous event, but results from the structure of the functions under consideration.

On Partially Honest Nash Implementation in Private Good Economies with Restricted Domains: A Sufficient Condition

AbstractIn this article, we study the problem of Nash implementation in private good economies with single-peaked, single-plateaued, and single-dipped preferences in the presence of at least one minimally honest agent. We prove that all solutions of the problem of fair division satisfying unanimity can be implemented in Nash equilibria as long as there are at least three agents participating in the mechanism (game form). To justify this result, we provide a list of solutions which violate the condition of no-veto power.

Nash Implementation in Private Good Economies when Preferences are Single-Dipped with Best Indifferent Allocations

In this paper we study the problem of the Nash implementation in private good economies when preferences are single-dipped by allowing multiple best indifferent elements. We give a full characterization in this domain by showing that a social choice correspondence (SCC) is Nash implementable if and only if satisfies Maskin monotonicity. To support this result, we give examples of some well-known SCCs that satisfying or not satisfying Maskin's proprieties.

Nonmyopia and incentives in the piecewise linearized MDP procedures with variable step-sizes

The paper formulates a piecewise linearized version of the procedure developed by Sato (1983) and analyzes its properties. In so doing, Fujigaki’s (1981) private goods economy is extended to involve a public good; and the intertemporal game of Champsaur and Laroque is piecewise localized by dividing the time interval in their game and by using variable step-sizes to formalize the piecewise linearized procedure, called λMDP Procedure, that can possess similar desirable properties shared by continuous-time procedures. Under Nonmyopia assumption, each player’s best reply strategy at any discrete date is to reveal his/her anticipated marginal rate of substitution for a public good at the end of a current time interval of the λ MDP Procedure.JEL Classification:H41.

The relation between monotonicity and strategy-proofness

The Muller–Satterthwaite Theorem (J Econ Theory 14:412–418, 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–Satterthwaite (J Econ Theory 14:412–418, 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 preference 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 preference 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 preference domains” (e.g., Proposition 3).

A characterization of the uniform rule without Pareto-optimality

We consider private good economies with single-peaked preferences. We show that the uniform rule is the only allocation rule satisfying Ω-continuity, no-envy, and one-sided resource-monotonicity. This result strengthens a characterization of the uniform rule due to Thomson (Soc Choice Welf 11:205–223, 1994).

Population monotonic rules for fair allocation problems

We consider the property of population monotonicity in the context of fair allocation problems in private good economies. It is already known that the property is compatible with the equal division lower bound. We show that if the equal division lower bound is replaced with no envy, the compatibility does not hold. We also show that the incompatibility persists even though no envy is weakened to no domination or e-no domination. Hence, among the fairness criteria mentioned above, only the equal division lower bound is compatible with population monotonicity.

Existence and Regularity of Equilibria in a General Equilibrium Model with Private Provision of a Public Good

We prove existence and generic regularity of equilibria in a general equilibrium model of a completely decentralized pure public good economy. Competitive rms using private goods as inputs produce the public good, which is privately provided by households. Previous studies on private provision of public goods typically use one private good, one public good models in which the public good is produced through a constant returns to scale technology. As two distinguishing features of our model, we allow for the presence of several private goods and non-linear production technology. In that framework, we use an homotopy argument to prove existence of equilibria and we show that economies are generically regular. 1 Introduction In this paper, we prove the existence and regularity of equilibria in a general equilibrium model of a completely decentralized pure public good economy. The model studied extends the standard pure exchange model with private goods by allowing households to make voluntary purchases of (or “privately provide”) a public good that is produced by competitive rms using private goods as inputs. The interest in a general equilibrium model with private provision of public goods lies in the fact that it serves as a benchmark extension of an analysis of completely decentralized private good economies to public good economies. Moreover, there are some relevant situations in which public goods are in fact We would like to thank the Editor and three anonymous referees for their helpful comments. In particular, we are very grateful to one of the referees for very stimulating and thoughtful comments and suggestions that helped improve the paper greatly. We would also like to express our gratitude to late Murat Sertel for a helpful discussion on the paper yCorresponding author. Department of Mathematics for Decision Theory (Di.Ma.D), University of Florence, via Lombroso 6/17, 50134 Firenze, Italy; e-mail: villanac@uni .it. zDepartment of Economics, Bo1⁄4 gazici University, 80815 Bebek, Istanbul, Turkey; e-mail: zenginob@boun.edu.tr.

Strategy-proof allocation when preferences are single-plateaued

Abstract. We consider private good economies with single-plateaued preferences. A solution selects for each preference profile a non-empty set of allocations. An agent strictly prefers a set of allocations to another set of allocations if and only if he strictly prefers any allocation in the first set to any in the second set according to his single-plateaued preference relation. We characterize the class of sequential-allotment solutions by essentially single-valuedness, Pareto-indifference, and coalitional stragety-proofness. These solutions are the Pareto-indifferent extensions of the solutions studied by Barbera et al. 1997.

Double implementation of linear cost share equilibrium allocations

In this paper we consider the problem of double implementation of Linear Cost Share Equilibrium (LCSE) allocations by a feasible and continuous mechanism whose Nash allocations and strong Nash allocations coincide with Linear Cost Share Equilibrium allocations. The mechanism presented here allows preferences and initial endowments as well as coalition patterns to be privately observed, a feature missing from much recent work in implementation theory. Since LCSE contains Lindahl equilibrium and Ratio equilibrium as special cases, it doubly implements these two equilibria allocations. Further, if one reinterprets the commodity space, this mechanism also doubly implements Walrasian allocations for private goods economies. Thus, the mechanism given in the paper appears to represent a ‘generic’ mechanism to doubly implement market-type allocations in private and/or public goods economies.

A reconsideration of the problem of social cost: Free riders and monopolists

One version of the Coase Theorem is, If property rights are fully allocated, competition leads to efficient allocations. This version implies that the public goods problem can be solved by allocating property rights fully. We show that this mechanism is not likely to work well in economies with global externalities because the privatized economy is highly susceptible to strategic behavior: The free-rider problem manifests itself as a complementary monopoly problem in an associated private goods economy. Thus, our work relates the validity of the Coase Theorem to the literature on the incentives for strategic behavior in economies with complementarities.

Indifference and the uniform rule

We consider private good economies with single-plateaued preferences. We show that the uniform rule is the only allocation rule satisfying indifference (in terms of welfare), strategy-proofness and no-envy. Indifference (in terms of welfare) means that if one allocation is recommended, then another allocation is recommended if and only if all agents are indifferent between these allocations.

Implementation of Linear Cost Share Equilibrium Allocations

In this paper we investigate incentive aspects of the Linear Cost Share Equilibrium principle by designing mechanisms whose Nash allocations coincide with Linear Cost Share equilibrium allocations. The mechanisms presented here allow not only preferences but also initial indowments to be privately observed, a feature missing from much recent work in implementation theory. We give mechanisms for both cases of withholding and destruction of endowments. If one reinterprets the commodity space, our mechanisms also implements Walransian allocations to economics with purely private goods. Thus, our mechanisms are sufficiently general to cover Walrasian equilibria, Lindahl equilibria, and cost share equilibria. In other words, the mechanisms in this paper appear to represent "generic" mechanisms to implement competitive-type allocations in private and public goods economies. Journal of Economic Literature Classification Numbers: C72, D61, D78.

On equity with public goods

Existence theorems for envy-free and efficient allocations are derived for economies with public goods. For economies with an arbitrary but finite number of private and public goods, an adaptation of the existence proof of Svensson for private good economies is used. For economies with one private and one public good, a case often studied in theory and applications, two more direct proofs are given, using different conditions and taking advantage of the particularly simple structure of the set of envy-free allocations in this case. These proofs are also shown to apply to the case of excludable public goods with congestion.