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.     

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.     

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     

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.     

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.     

Strategy-proof allocation mechanisms for economies with an indivisible good

We consider economies with a single indivisible good and money. We characterize the set of mechanisms that satisfy strategy-proofness, individual rationality, equal compensation, and demand monotonicity. There are three types of mechanisms which have the following properties: (i) they determine the allocation of monetary compensation depending on who receives the indivisible good; (ii) they allocate the indivisible good to one of the pre-specified (one or two) agent(s); and (iii) they disregard preferences of agents other than the pre-specified agent(s). This result implies that the presence of an indivisible good induces serious asymmetry in mechanisms. Received: 26 March 1996 / Accepted: 23 September 1997     

Asymmetrically fair rules for an indivisible good problem with a budget constraint

We study a particular restitution problem where there is an indivisible good (land or property) over which two agents have rights: the dispossessed agent and the owner. A third party, possibly the government, seeks to resolve the situation by assigning rights to one and compensate the other. There is also a maximum amount of money available for the compensation. We characterize a family of asymmetrically fair rules that are immune to strategic behavior, guarantee minimal welfare levels for the agents, and satisfy the budget constraint.     

On monotonicity in economies with indivisible goods

We consider the problem of fair allocation in economies with indivisible objects that may or may not be desirable (for instance, activities that may or may not be pleasurable but have to be carried out unless there are not enough agents for that). We search for efficient solutions satisfying two additional properties. First, each agent should find his bundle at least as desirable as the bundle that would be assigned to him in the hypothetical economy in which all agents have preferences identical to his, under equal treatment of equals and efficiency. In a preliminary step, we show that there is no logical relation between this requirement and no-envy, and between it and egalitarian-equivalence. We also establish the existence of efficient allocations satisfying it. The second property, object monotonicity, says that the availability of additional objects either has a negative impact on everyone's welfare, or it has a positive impact on everyone's welfare. We show that there is no object-monotonic selection from the correspondence that associates with each economy its set of efficient allocations meeting an even weaker version of the bound.I am grateful to Atila Abdulkadiroglu, Koichi Tadenuma, and a referee for their very helpful comments.     

The replacement principle in economies with indivisible goods

 We consider the problem of allocating a list of indivisible goods and some amount of an infinitely divisible good among agents with equal rights on these resources, and investigate the implications of the following requirement on allocation rules: when the preferences of some of the agents change, all agents whose preferences are fixed should (weakly) gain, or they should all (weakly) lose. This condition is an application of a general principle of solidarity discussed in Thomson (1990b) under the name “replacement principle”. We look for selections from the no-envy solution satisfying this property. We show that in the general case, when the number of objects is arbitrary, there is no such selection. However, in the one-object case (a single prize), up to Pareto-indifference, there is only one selection from the no-envy solution satisfying the property. Such a solution always selects an envy-free allocation at which the winner of the prize is indifferent between his bundle and the losers’ common bundle. Received: 15 May 1995 / Accepted: 5 June 1996     

Identical preferences lower bound solution and consistency in economies with indivisible goods

We consider the problem of allocating a finite set of indivisible goods and a single infinitely divisible good among a group of agents, and we study a solution, called the Identical Preferences Lower Bound solution, in the presence of consistency properties. This solution is not consistent. We prove that its maximal consistent subsolution is the No-envy solution. Our main result is that the minimal consistent extension of the intersection of the Identical Preferences Lower Bound solution with the Pareto solution is the Pareto solution. This result remains true in the restricted domain when all the indivisible goods are identical, but not when there is a unique indivisible good.This paper was developed during my stay at Rochester University in the summer of 1992. I would like to express my special thanks to Professor William Thomson for all his help and advice. Iam also grateful to my supervisor Luis Corchón, to Koichi Tadenuma and to the anonymous referees for their helpful comments. The remaining errors are my exclusive responsibility. Financial support from the DGCYT under project PB 91-0756 and the Instituto Valenciano de Investigaciones Económicas are gratefully acknowledged.     

Coalitionally strategy-proof rules in allotment economies with homogeneous indivisible goods

We consider the allotment problem of homogeneous indivisible goods among agents with single-peaked and risk-averse von Neumann–Morgenstern expected utility functions. We establish that a rule satisfies coalitional strategy-proofness, same-sideness, and strong symmetry if and only if it is the uniform probabilistic rule. By constructing an example, we show that if same-sideness is replaced by respect for unanimity, this statement does not hold even with the additional requirements of no-envy, anonymity, at most binary, peaks-onlyness and continuity. We would like to thank the associate editor and two anonymous referees whose comments and suggestions significantly improve this paper. We are also grateful to Masaki Aoyagi, Kazuhiko Hashimoto, Hervé Moulin, Hiroo Sasaki, Koji Takamiya, William Thomson, Takuma Wakayama as well as other participants at the Eighth International Meeting of the Society for Social Choice and Welfare in Istanbul, the 2006 Autumn Annual Meeting of the Japanese Economic Association in Osaka, and Yokohama National University Seminar of Economic Theory for their helpful comments. We acknowledge financial supports from the Japan Society for the Promotion of Science via the Research Fellowship for Young Scientist (Hatsumi) and the Grant-in-Aid for Scientific Research (Serizawa).     

Inequality in an OLG economy with heterogeneous cohorts and pension systems

We analyze the consumption and wealth inequality in an OLG model with mandatory pension systems. Our framework features within-cohort heterogeneity of endowments and heterogeneity of preferences. We allow for population aging and gradual decline in TFP growth. We show four main results. First, increasing longevity translates to substantial increases in aggregate consumption inequality and wealth inequality. Second, a pension system reform from a defined benefit to a defined contribution works to reinforce consumption inequality and reduce wealth inequality. Third, minimum pension benefits are able to partially counteract an increase in inequality introduced by the defined contribution system, at a fiscal cost. Fourth the minimum pension benefit guarantee mostly addresses the sources of inequality which stem from differentiated endowments rather than those which stem from heterogeneous preferences.     

Circular economy and the city: an urban political economy agenda

ABSTRACT

The circular economy (CE) has become a matter of urban development. A literature review shows that the CE debate is biased toward technology-driven industrial change, while bracketing broader socio-political interests. We address this gap by exploring the political economy of scale of the CE. Looking into the case of Brussels (Belgium), a city that has recently adopted the CE as part of its socio-economic strategy, we explore how the anticipated transition to a ‘circular city’ chimes with long-standing urban development agendas. While there is little evidence of stable growth coalitions between corporate and political elites, we argue that the CE provides an ‘urban sustainability fix’ by selectively incorporating ecological goals in urban governance strategies. We further scrutinise the landscape of diverse and heterogenous CE practices in food and transport, highlighting how they are regulated and organised, what labour conditions they offer, and how they are anchored in urban space.     

Social loss with respect to the core of an economy

The dual notion of Pareto-efficiency (i.e., individual utilities cannot be reached with fewer resources than those of the allocation under consideration) is used in order to define the resources-core. A measure of social loss with respect to the core is then introduced, more or less as already done with respect to the locus of efficient allocations. Received: 4 December 2000/Accepted: 14 June 2001     

Fair and group strategy-proof good allocation with money

We completely characterize the class of fair and group strategy-proof mechanisms. We consider two notions of fairness, anonymity in welfare and no-envy. Both fairness axioms, when applied with strategy-proofness, imply decision efficiency, and lead to the same class of group strategy-proof mechanisms (where the group size is restricted to two). We find that the only feasible mechanism satisfying a mild zero transfer axiom, in this class, is the Pivotal mechanism.     

Perceived risk and the precautionary demand for money

Keynes contended that individuals hold money for fear of being unable to meet unforeseen future cash requirements. This ‘precautionary demand’ for money has long been an accepted part of monetary theory, but has played a subservient role because of our inability to measure an individual's degree of aversion to risk. This study, however, employs a risk taking scale, similar to that developed by Zuckerman, to empirically investigate Keynes's precautionary demand for money. The results are sufficiently encouraging to suggest that this scale might successfully be applied to other economics subfields in which risk plays a role.