Existence of efficient envy-free allocations of a heterogeneous divisible commodity with nonadditive utilities

This paper studies the existence of Pareto optimal, envy-free allocations of a heterogeneous, divisible commodity for a finite number of individuals. We model the commodity as a measurable space and make no convexity assumptions on the preferences of individuals. We show that if the utility function of each individual is uniformly continuous and strictly monotonic with respect to set inclusion, and if the partition matrix range of the utility functions is closed, a Pareto optimal envy-free partition exists. This result follows from the existence of Pareto optimal envy-free allocations in an extended version of the original allocation problem.     

The equity-efficiency trade-off reconsidered

Envy-free and Pareto optimal allocations never exist in large economies with production, if there is enough variation with respect to preferences and innate abilities of the agents. If distributional equity is given priority only second best envy-free allocations may be considered. Those allocations are characterised in finite and continuum economies. In continuum economies envy-free allocations are type-egalitarian, i.e. agents with the same preferences but different abilities obtain the same consumption-labour bundle. Regarding implementation envy-free allocations can only be implemented by a tax depending on labour and income.I would like to thank two anonymous referees and the editor in charge of this paper for very helpful comments.     

A simple procedure for finding equitable allocations of indivisible goods

The paper investigates how far a particular procedure, called the “descending demand procedure,” can take us in finding equitable allocations of indivisible goods. Both interpersonal and intrapersonal criteria of equitability are considered. It is shown that the procedure generally fares well on an interpersonal criterion of “balancedness”; specifically, the resulting allocations are Pareto-optimal and maximize the well-being of the worst-off individual. As a criterion of intrapersonal equitability, the property of envy-freeness is considered. To accommodate envy-freeness, a modification of the basic procedure is suggested. With two individuals, the modified procedure is shown to select the envy-free allocations that are balanced, i.e. the allocations that maximize the well-being of the worse-off individual among all envy-free allocations. Received: 3 March 2000/Accepted: 27 November 2000     

Trade-off between equity and efficiency in a general economy with indivisible goods

In most of the recent literature on fair allocation in economies with indivisible goods and a single infinitely divisible good, it is assumed that each agent can consume at most one indivisible good. In this paper, we show that if this assumption is dropped, there do not necessarily exist envy-free and Pareto efficient allocations. However, envy-free allocations still exist and so do Pareto efficient allocations. Hence, a trade-off between equity and efficiency arises.     

Money-egalitarian-equivalent and gain-maximin allocations of indivisible items with monetary compensation

Suppose that a certain quantity M of money and a finite number of indivisible items are to be distributed among n people, all of whom have equal claims on the whole. Different allocations are presented using various criteria of fairness in the special case where each player's utility function is additively separable. An allocation is “money-egalitarian-equivalent” (MEE) if each player's monetary valuation of his or her bundle is a fixed constant. We show that there is an essentially unique allocation that is MEE and Pareto-optimal; it is also envy-free. Alternatively, the “gain” of a player may be defined as the difference between how the player evaluates his bundle and an exact nth part of the whole according to his numerical evaluation of the whole. A “gain-maximin” criterion would maximize the minimum gain obtained by any player. We show that Knaster's procedure finds an allocation which is optimal under the gain-maximin criterion. That allocation is not necessarily envy-free, so we also find the envy-free allocation that is optimal under the gain-maximin criterion among all envy-free allocations. It turns out that, even though there exist allocations that are simultaneously envy-free and Pareto-optimal, this optimal allocation may fail to be Pareto-optimal, and it may also violate monotonicity criteria. Received: 30 September 1996/Accepted: 6 March 2002 The author would like to thank Professor William Thomson for a discussion on this subject; and he would like to thank the anonymous referees, who made many substantive suggestions for improving this paper – shortening it, streamlining the arguments, improving the terminology, making further ties with the literature, and improving the exposition.     

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.I am grateful to William Thomson, Marcus Berliant, Robert Gilles, Hervé Moulin, Thomas Palfrey, Tomoichi Shinotsuka, Simon Wilkie, and two anonymous referees for their comments. This paper is a revised version of Chap. 1 of my doctoral dissertation submitted to the University of Rochester in July 1988. The financial support of the University of Rochester in the form of the Raymond N. Ball Dissertation Year Fellowship is gratefully acknowledged. All shortcomings of this paper are solely of my own creation.     

Dividing resources by flexible majority rules

We examine the division of resources among individuals by flexible majority rules where the majority necessary to adopt a proposal depends on the proposal itself. For instance, the size of the majority may increase with the maximal difference between the shares individuals receive. For large discount factors such rules imply an efficient and even distribution of resources. For low discount factors flexible majority rules supplemented by specific agenda-planning rules such as agenda rights for the opposition guarantee envy-free distribution. Uncertainty about discount rates can make it easier to achieve efficient and envy-free allocations.I would like to thank Clive Bell, Peter Bernholz, Theresa Fahrenberger, Amihai Glazer, Ulrich Erlenmaier, Volker Hahn, Hans Haller, Klaus Schmidt, Eva Terberger-Stoy, seminar participants in Heidelberg and Munich and conference participants at the meeting of the Public Choice Society for their helpful comments.     

Resolving an Insurance Allocation Problem: A Procedural Approach

In this paper we study the problem of the determination of a fair allocation in a co-insurance problem, i.e., how some insurance companies have to share the risk and the premium. We develop two procedures that produce a proportional and an equitable allocation, respectively. The procedures are applied to a real situation arising from environmental risk and the resulting allocations are compared with the classical quota share allocation and with an envy-free allocation resulting from a procedure presented in Fragnelli and Marina (Insur. Math Econ 33:75–85, 2003).     

A derivation of the money Rawlsian solution

We study the set of envy-free allocations for economies with indivisible objects and quasi-linear utility functions. We characterize the minimal amount of money necessary for its nonemptiness when negative distributions of money are not allowed. We also find that, when this is precisely the available amount of money, there is a unique way to combine objects and money such that these bundles may form an envy-free allocation. Based on this property, we describe a solution to the envy-free selection problem following a pseudo-egalitarian criterion. This solution coincides with the Money Rawlsian Solution proposed by Alkan et al. (1991).I am indebted to I. Gilboa for his valuable suggestions and his patience during the elaboration of the final version. I also wish to thank S. Barbera for his guidance in an earlier version and M. Boldrin, H. Moulin, Z. Neeman, W. Thomson and the referees for their comments. Financial support from FPU-MEC (Spain) is gratefully acknowledged.Northwestern University.     

Equity and efficiency in non-convex economies

The objective of this paper is to consider the following question. Does the presence of increasing returns introduce a fundamental trade-off between equity and efficiency objectives? We show that if the no-envy notion of Foley (1967) is taken as the equity criterion and Pareto optimality as the efficiency criterion, then the answer is yes; there exist economies with increasing returns and well-behaved preferences (and no agent-specific inputs) in which there do not exist any envy-free and Pareto optimal allocations. We also propose a weakening of the no-envy criterion and prove that this weaker equity notion is compatible with Pareto optimality in general non-convex economies.     

Resource-monotonic solutions to the problem of fair division when preferences are single-peaked

We consider the problem of fairly allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. We search for solutions satisfying resource-monotonicity, the requirement that all agents be affected in the same direction when the amount to divide changes. Although there are resource-monotonic selections from the Pareto solution, there are none satisfying the distributional requirements of no-envy or individual rationality from equal division. We then consider the weakening of resource-monotonicity obtained by allowing only changes in the amount to divide that do not reverse the direction of the inequality between the amount to divide and the sum of the preferred amounts. We show that there is essentially a unique selection from the solution that associates with each economy its set of envy-free and efficient allocations satisfying this property of one-sided resource-monotonicity: it is the uniform rule, a solution that has played a central role in previous analyses of the problem.     

The undercut procedure: an algorithm for the envy-free division of indivisible items

We propose a procedure for dividing a set of indivisible items between two players. We assume that each player’s preference over subsets of items is consistent with a strict ranking of the items, and that neither player has information about the other’s preferences. Our procedure ensures an envy-free division—each player receives a subset of items that it values more than the other player’s complementary subset—given that an envy-free division of “contested items,” which the players would choose at the same time, is possible. We show that the possibility of one player’s undercutting the other’s proposal, and implementing the reduced subset for himself or herself, makes the proposer “reasonable,” and generally leads to an envy-free division, even when the players rank items exactly the same. Although the undercut procedure is manipulable and its envy-free allocation may be Pareto-inferior, each player’s maximin strategy is to be truthful. Applications of the procedure are discussed briefly.     

Existence and Nash implementation of efficient sharing rules for a commonly owned technology

Suppose that a group of individuals owns collectively a technology which produces a consumption good by means of a (possibly heterogeneous) input. A sharing rule associates input contributions with a vector of consumptions that are technologically feasible. We show that the set of allocations obtained by any continuous sharing rule contains Pareto efficient allocations. We also present a mechanism that implements in Nash equilibrium the Pareto efficient allocations contained in an arbitrary sharing rule. Received: 29 June 1998/Accepted: 15 November 2000     

Double implementation of Lindahl allocations by a pure mechanism

This paper considers the issue of designing mechanisms whose Nash allocations and strong Nash allocations coincide with Lindahl allocations for public goods economies when coalition patterns, preferences, and endowments are unknown to the designer. It will be noted that the mechanism presented here is feasible and continuous, and the implementation result is obtained without defining an artificial preference profile on prices announced by individuals. In addition, unlike most existing Nash-implementing mechanisms which need to distinguish the case of two agents from that of three or more agents, this paper provides a unified mechanism which is irrespective of the number of agents. Received: 19 August 1997/Accepted: 16 November 1998     

Generic efficiency and collusion-proofness in exchange economies

The purpose of this paper is to study the kind of efficient allocations that can be achieved in exchange economies with asymmetric information, by means of a decentralized mechanism robust to coalitional, strategic deviations. To this end, we define a new strategic equilibrium concept – called strong collusion-proof contract – designed to characterize stable communication agreements in games with differential information against non-binding, self-enforcing and incentive compatible deviations by coalitions. We then construct a strategic market mechanism which, for quasi-linear economies, is such that its strong collusion-proof contracts generically induce the incentive compatible interim efficient allocations. Moreover, we show that these allocations can be achieved by strong collusion-proof contracts. We show that the internally consistent extension of the strong collusion-proof contracts generically yields the same set of efficient allocations. Received: 22 January 2001/Accepted: 15 April 2002 RID="*" ID="*"  This author was working at CORE when this paper was written. We wish to thank Claude d'Aspremont, David Martimort, Jean-Fran?ois Mertens and Heracles Polemarchakis for helpful comments on an earlier version. The usual disclaimer applies.     

Fair division of indivisible items between two people with identical preferences: Envy-freeness, Pareto-optimality, and equity

This paper focuses on the fair division of a set of indivisible items between two people when both have the same linear preference order on the items but may have different preferences over subsets of items. Surprisingly, divisions that are envy-free, Pareto-optimal, and ensure that the less well-off person does as well as possible (i.e., are equitable) can often be achieved. Preferences between subsets are assumed to satisfy axioms of qualitative probability without implying the existence of additive utilities, which is treated as a special case. Algorithms that render fair division practicable are proposed, and their vulnerability to strategic manipulation is investigated. Received: 18 May 1998/Accepted: 2 March 1999     

No-envy, efficiency, and collective rationality

We consider fair collective choice functions (hereafter fair CCFs) which associate with each profile of extended preference orderings and each set of feasible social states a subset of the set of Pareto efficient and envy-free states for the preference profile. Our main objective is to examine compatibility of fair social choices with collective rationality. We formulate desirable properties of collective rationality, and look for CCFs satisfying them. Next, we show that any fair CCF violates most of collective rationality properties. Moreover, one of our results implies that no fair CCF can be rationalized by a social preference relation.     

p-Weakly constrained Pareto efficiency and aggregation in incomplete markets

This paper studies aggregation and a weak form of optimality referring to the p-weakly constrained Pareto efficiency (p-WE) in stochastic finance economies with incomplete markets. We derive a representative agent utility function maximized at equilibrium and a characterization of the set of p-WE allocations. Moreover, we establish the correspondence between the set of competitive equilibrium allocations and the set of p-WE allocations. It is noted that these are accomplished for general time-state utility functions and without restrictions on the initial resource allocations.     

Money-metric utilitarianism

We discuss a method of ranking allocations in economic environments which applies when we do not know the names or preferences of individual agents. We require that two allocations can be ranked with the knowledge only of agents present, their aggregate bundles, and community indifference sets—a condition we refer to as aggregate independence. We also postulate a basic Pareto and continuity property, and a property stating that when two disjoint economies and allocations are put together, the ranking in the large economy should be consistent with the rankings in the two smaller economies (reinforcement). We show that a ranking method satisfies these axioms if and only if there is a probability measure over the strictly positive prices for which the rule ranks allocations on the basis of the random-price money-metric utilitarian rule. This is a rule which computes the money-metric utility for each agent at each price, sums these, and then takes an expectation according to the probability measure.