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.     

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.     

Representatives and districts

Why would any group want to have a decision-making body composed of representatives? The best answer is found in the “Anti-Federalist ideal” identified by Wood [1992]: if within-group benefits are highly correlated, a legislature composed of randomly chosen representatives that maximized its own well-being would maximize the well-being of the entire polity. This argument for representative government implies that the best districting schemes are the most segregated: within-district variance is minimized. Received: 9 June 1997/Accepted 28 December 1998     

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.     

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.     

On fairness and welfare analysis under uncertainty

This paper investigates the role of fairness, uncertainty, and a “veil of ignorance” in efficient resource allocation. It focuses on the choice of private and public goods, the method of financing, as well as the choice of information available for public decision-making. A fair-equivalent and Pareto efficient allocation is presented using a maximin criterion defined in terms of individual willingness-to-pay. The paper investigates the role of information in public decision making in terms of its implications for both efficiency and fairness. While better information typically generates improved efficiency, it can also contribute to unfair allocations. The effects of asymmetric information are discussed. Received: 4 May 1998/Accepted: 28 February 2002 We would like to thank two anonymous referees for very useful comments on an earlier draft of the paper. The research was supported in part by a Hatch grant from the University of Wisconsin, Madison.     

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).     

Extended Pareto rules and relative utilitarianism

This paper introduces the “Extended Pareto” axiom on Social Welfare Functions and gives a characterization of the axiom when it is assumed that the Social Welfare Functions that satisfy it in a framework of preferences over lotteries also satisfy the restrictions (on the domain and range of preferences) implied by the von-Neumann Morgenstern axioms. With the addition of 2 other axioms: “Anonymity” and a weak version of Arrow's Independence of Irrelevant Alternatives axiom: “Weak IIA” it is shown that there is a unique Social Welfare Function called “Relative Utilitarianism” that consists of normalising individual utilities between 0 and 1 and adding them. Received: 7 June 1994 / Accepted: 28 April 1997     

An algorithm for envy-free allocations in an economy with indivisible objects and money

This paper studies envy-free allocations for economies with indivisible objects, quasi-linear utility functions, and an amount of money. We give a polynomially bounded algorithm for finding envy-free allocations. Connectedness of envy-graphs, which are used in the algorithm, characterizes the extreme points of the polytopes of sidepayments corresponding with envy-free allocations. Received: 22 October 1997/Accepted: 19 January 1999     

Inequality, new directions: full ethical foundation and consequences

Deriving comparisons and measures of inequality from full ethical foundations was a main innovation of the 1960s and pursuing it is still a most fruitful direction. This implies using “equal equivalents” and some principles particularly rich in meanings. Multidimensional inequalities can be measured and compared thanks to the “equal-equivalent manifolds”. The “equal-equivalent utility function” defines individual “welfare” cleaned of differences in sui generis individual tastes and hedonic capacities deemed irrelevant for “macrojustice”. Then, equal allocation is a deeper end-value than equal welfare but has to be complemented by free choice for freedom, Pareto efficiency and a demanded partial self-ownership. The result is the richly multi-meaning “equal-labour income equalization”.     

Strategy-proofness and efficiency in a simple production economy

We consider an economy with two agents, “firm” and “worker.” The firm owns a technology which transforms a single input into a single output and the worker owns a limited amount of input good, for example, leisure. The firm is interested in profit measured in terms of output and the worker's preferences are defined over the input-output space. Manipulability comes not only from a lack of information about the (worker's) preferences but also about the technology. With a possibility for manipulation, can we still obtain efficient allocations? We show that there is no allocation mechanism which is Pareto efficient, strategy-proof, and non-dictatorial. Received: 30 March 1998/Accepted: 06 July 1999     

Objective measures of well-being and the cooperative production problem

 The concern for measuring well-being objectively (as opposed to subjectively, that is, relying only on preferences) is found in modern political philosophy, especially in J. Rawls’s, A. Sen’s and G. Cohen’s writings. This paper explains the implications of using an objective well-being index as equalisandum or, close to the so-called “safety net preoccupation”, to guarantee a well-being lower bound. In the simple production model studied here, five characterization results and two related theorems prove the convergence of both approaches. We come to the conclusion that the Proportional Solution is the dominant solution according to “objectivist” axioms. Received: 5 September 1994/Accepted: 29 June 1996     

Negatively interdependent preferences

We develop a theory of representation of interdependent preferences that reflect the widely acknowledged phenomenon of keeping up with the Joneses (i.e. of those preferences which maintain that well-being depend on “relative standing” in the society as well as on material consumption). The principal ingredient of our analysis is the assumption that individuals desire to occupy a (subjectively) better position than their peers. This is quite a primitive starting point in that it does not give any reference to what is actually regarded as “status” in the society. We call this basic postulate negative interdependence, and study its implications. In particular, combining this assumption with some other basic postulates that are widely used in a number of other branches of the theory of individual choice, we axiomatize the relative income hypothesis, and obtain an operational representation of interdependent preferences. Received: 7 December 1998/Accepted: 24 August 1999     

The direct social preference relation in the theory of public expenditure

It is shown that if there is a finite number of private goods, a single public good, and the individual preferences satisfy certain monotonicity and separability conditions then there is a unique and complete social preference relation defined on the set of allocations by the requirement that the relation is individualistic in terms of the individual ordinal preferences. This relation, called the direct social preference relation, is derived without imposing interpersonal welfare comparisons and all its ordinal properties are inherited directly from those of the individual preferences. However, an allocation which is maximal in terms of the direct social preference relation on the set of feasible allocations is in general not Pareto optimal and the relation may therefore not be suitable as a guide to optimizing social choice.This version of the paper has benefitted from J. Weymark's insightful and detailed comments. In addition, valuable suggestions have been received from G. C. Archibald, C. Blackorby, D. Donaldson and D. Primont. All remaining errors are the sole responsibility of the author     

A multi-systemic approach to AIDS and adolescents

The AIDS epidemic poses a significant challenge for mental health professionals, educators and for society at large. Recent studies (Kennedy, 1988) have shown that AIDS has entered the adolescent population. Other studies (Hirschorn, 1987a, 1987b; Simpkins and Kushna, 1986; Weinstein, Atwood, and Rosen, in press) have shown that while college students are educated about AIDS, they inconsistently use safer sex practices. Most of the behavior change attempts have focused on the developmental psychodynamics of the adolescent stage. This paper discusses the notion that behavior change in the adolescent is not only based on the psychodynamics of the intrapersonal approaches; but, rether, that what is additionally needed is an examination of the interpersonal and social processes which support the psychodynamics of the adolescent. A multi-systemic approach to AIDS and adolescents is then presented utilizing intrapersonal, interpersonal, and social processes. This approach incorporates the school-family-community systems.     

Rational choice and revealed preference without binariness

 This paper attempts to provide a unified account of the rationalization of possibly non-binary choice-functions by “Extended Preference Relations” (relations between sets and elements). The analysis focuses on transitive EPRs for which three choice-functional characterizations are given, two of them based on novel axioms. Transitive EPRs are shown to be rationalizable by sets of orderings that are “closed under compromise”; this novel requirement is argued to be the key to establish a canonical relationship between sets of orderings and choice-functions. The traditional assumption of “binariness” on preference relations or choice functions is shown to be analytically unhelpful and normatively unfounded; non-binariness may arise from “unresolvedness of preference”, a previously unrecognized aspect of preference incompleteness. Received: 28 August 1995/Accepted: 14 February 1996     

Argumentation in Multi-issue Debates

An axiomatic modeling approach to multi-issue debates is proposed. A debate is viewed as a decision procedure consisting of two stages: (1) an “argumentation rule” determines what arguments are admissible for each party, given the “raw data”, depending on the issue or set of issues under discussion; (2) a “persuasion rule” determines the strength of the admissible arguments and selects the winning party. Persuasion rules are characterized for various alternative specifications of the argumentation rule. These characterizations capture rhetorical effects that we sometimes encounter in real-life multi-issue debates.     

Quaternary dichotomous voting rules

In this article, we provide a general model of “quaternary” dichotomous voting rules (QVRs), namely, voting rules for making collective dichotomous decisions (to accept or reject a proposal), based on vote profiles in which four options are available to each voter: voting (“yes”, “no”, or “abstaining”) or staying home and not turning out. The model covers most of actual real-world dichotomus rules, where quorums are often required, and some of the extensions considered in the literature. In particular, we address and solve the question of the representability of QVRs by means of weighted rules and extend the notion of “dimension” of a rule.     

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.