Coalitional strategy-proof and resource-monotonic solutions for multiple assignment problems

We consider the problem of allocating indivisible objects when agents may desire to consume more than one object and monetary transfers are not possible. Each agent receives a set of objects and free disposal is allowed. We are interested in allocation rules that satisfy appealing properties from an economic and social point of view. Our main result shows that sequential dictatorships are the only efficient and coalitional strategy-proof solutions to the multiple assignment problem. Adding resource-monotonicity narrows this class down to serial dictatorships.We thank Francois Maniquet, two anonymous referees, and the participants of the GREBE-FRANCQUI Summer School on Axiomatic Resource Allocation Theory, held in Namur, Belgium, for their 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     

An alternative characterization of the uniform rule

We consider the problem of allocating some amount of a commodity among a group of agents with single-peaked preferences. We show that the uniform rule is the only rule satisfying equal treatment of equals, Pareto efficiency, and strategy-proofness. This characterization strengthens two interesting results due to Sprumont (1991). Our method of proof involves only elementary arguments.I wish to thank Professor William Thomson for his enormous efforts in supervision. I am grateful to Professor Marcus Berliant and Hideo Konishi for their useful remarks and especially to an anonymous referee for insightful comments. All remaining errors are my own responsibility.     

Characterizing Vickrey allocation rule by anonymity

We consider the problem of allocating finitely many units of an indivisible good among a group of agents when each agent receives at most one unit of the good and pays a non-negative price. For example, imagine that a government allocates a fixed number of licenses to private firms, or that it distributes equally divided lands to households. Anonymity in welfare is a condition of impartiality in the sense that it requires allocation rules to treat agents equally in welfare terms from the viewpoint of agents who are ignorant of their own valuations or identities. We show that the Vickrey allocation rule is the unique allocation rule satisfying strategy-proofness, anonymity in welfare, and individual rationality.     

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.     

Balanced consistency and balanced cost reduction for sequencing problems

We investigate the implications of balanced consistency and balanced cost reduction in the context of sequencing problems. Balanced consistency requires that the effect on the payoff from the departure of one agent to another agent should be equal between any two agents. On the other hand, balanced cost reduction requires that if one agent leaves a problem, then the total payoffs of the remaining agents should be affected by the amount previously assigned to the leaving agent. We show that the minimal transfer rule is the only rule satisfying efficiency and Pareto indifference together with either one of our two main axioms, balanced consistency and balanced cost reduction.     

Maximal Domain for Strategy-proof Rules in Allotment Economies

We consider the problem of allocating an amount of a perfectly divisible good among a group of n agents. We study how large a preference domain can be to allow for the existence of strategy-proof, symmetric and efficient allocation rules when the amount of the good is a variable. This question is qualified by an additional requirement that a domain should include a minimally rich domain. We first characterize the uniform rule (Bennasy in The economics of market disequilibrium, Academic, New York, 1982) as the unique strategy-proof, symmetric, and efficient rule on a minimally rich domain when the amount of the good is fixed. Then, exploiting this characterization, we establish the following: there is a unique maximal domain that includes a minimally rich domain and allows for the existence of strategy-proof, symmetric, and efficient rules when the amount of good is a variable. It is the single-plateaueddomain.     

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.     

A Note on the Separability Principle in Economies with Single-Peaked Preferences

We consider the problem of allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. A rule that has played a central role in the analysis of the problem is the so-called uniform rule. Chun (2001) proves that the uniform rule is the only rule satisfying Pareto optimality, no-envy, separability, and Ω-continuity. We obtain an alternative characterization by using a weak replication-invariance condition, called duplication-invariance, instead of Ω-continuity. Furthermore, we prove that the equal division lower bound and separability imply no-envy. Using this result, we strengthen one of Chun’s (2001) characterizations of the uniform rule by showing that the uniform rule is the only rule satisfying Pareto optimality, the equal division lower bound, separability, and either Ω-continuity or duplication-invariance.     

Single-peaked preferences with several commodities

We consider the problem of allocating m commodities among n agents with single-peaked preferences. When m≥2 and n=2 any strategy-proof and efficient solution is dictatorial. We propose an extension of the Uniform Rule that (in the two-agents case) is the only one that satisfies strategy-proofness, envy-freeness, and a weak requirement related to efficiency. Alternatively, the envy-freeness property may be replaced by weak-anonymity. Received: 7 November 1997/Accepted: 1 August 2000     

Fair allocation of indivisible goods: the two-agent case

One must allocate a finite set of indivisible goods among two agents without monetary compensation. We impose Pareto-efficiency, anonymity, a weak notion of no-envy, a welfare lower bound based on each agent’s ranking of the subsets of goods, and a monotonicity property w.r.t. changes in preferences. We prove that there is a rule satisfying these axioms. If there are three goods, it is the only rule, together with one of its subcorrespondences, satisfying each fairness axiom and not discriminating between goods.     

Solidarity in choosing a location on a cycle

We study the implications of two solidarity conditions on the efficient location of a public good on a cycle, when agents have single-peaked, symmetric preferences. Both conditions require that when circumstances change, the agents not responsible for the change should all be affected in the same direction: either they all gain or they all loose. The first condition, population-monotonicity, applies to arrival or departure of one agent. The second, replacement-domination, applies to changes in the preferences of one agent. Unfortunately, no Pareto-efficient solution satisfies any of these properties. However, if agents’ preferred points are restricted to the vertices of a small regular polygon inscribed in the circle, solutions exist. We characterize them as a class of efficient priority rules.     

Social threshold aggregations

A problem of axiomatic construction of a social decision function is studied for the case when individual opinions of agents are given as m-graded preferences with arbitrary integer m ≥ 3. It is shown that the only rule satisfying the introduced axioms of Pairwise Compensation, Pareto Domination and Noncompensatory Threshold and Contraction is the threshold rule.     

A characterization of the uniform rule with several commodities and agents

We consider a problem of allocating infinitely divisible commodities among a group of agents. More specifically, there are several commodities to be allocated and agents have continuous, strictly convex, and separable preferences. We establish that a rule satisfies strategy-proofness, unanimity, weak symmetry, and nonbossiness if and only if it is the uniform rule. This result extends to the class of continuous, strictly convex, and multidimensional single-peaked preferences.     

Cooperative production under diminishing marginal returns: interpreting fixed-path methods

Fixed-path methods (FPMs) were introduced to manage situations where several individuals jointly operate a single technology (see Math Soc Sci 44:145–154 (2002)). In the production context, they consist in allocating marginal increments of output according to a proportions vector which changes along an arbitrary path. While very appealing from an incentives viewpoint under diminishing marginal returns, the asymmetry of these methods lacks solid economic interpretation. We provide such an interpretation by considering a situation where the technology to be shared results from the aggregation of private production processes. We propose a group-strategyproof mechanism under which no single agent wishes to secede from the partnership: the inverse marginal product proportions mechanism. It is the only FPM satisfying autarkic individual rationality; its path is uniquely determined by the technological contributions of the agents.     

A characterization of the nucleolus without homogeneity in airport problems

We consider the problem of sharing the cost of a public facility among agents who have different needs for the facility. We show that the nucleolus is the only rule satisfying equal treatment of equals, last-agent cost additivity, and consistency.     

Priority,solidarity and egalitarianism

We provide alternative axiomatic characterizations of the extended egalitarian rules (Moreno-Ternero and Roemer, Econometrica 74:1419–1427, 2006) in a fixed-population setting of the canonical resource allocation model based on individual capabilities (output functions). Our main axioms are disability monotonicity (no reduction in the amount of resources allocated to an agent after she becomes more disabled) and agreement (when there is a change in agents’ capabilities or total resources, all agents who remain unchanged should be influenced in the same direction: all unchanged agents get more or all get less or all get the same amount as before).     

An axiomatization of the Kalai-Smorodinsky solution when the feasible sets can be finite

We axiomatize the Kalai-Smorodinsky solution (1975) in the Nash bargaining problems if the feasible sets can be finite. We show that the Kalai-Smorodinsky solution is the unique solution satisfying Continuity (in the Hausdorff topology endowed with payoffs space), Independence (which is weaker than Nash's one and essentially equivalent to Roth (1977)'s one), Symmetry, Invariance (both of which are the same as in Kalai and Smorodinsky), and Monotonicity (which reduces to a little bit weaker version of the original if the feasible sets are convex). Received: 4 November 1999/Accepted: 6 June 2001     

The Separability Principle in Economies with Single-Peaked Preferences

We investigate the implications of the “separability principle” for the class of problems allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. The separability principle requires that for two problems with the same population, but possibly different social endowments, in which the preferences of agents may change, if there is a subgroup of agents whose preferences are the same and the total amounts awarded to them are the same, then the amount awarded to each agent in the subgroup should be the same. First, we investigate the logical relations between separability and other axioms. As it turns out, consistency implies separability. Then, we present characterizations of the uniform rule on the basis of separability and also on the basis of other axioms.     

Probabilistic allocation rules and single-dipped preferences

We consider the problem of allocating an infinitely divisible endowment among a group of agents with single-dipped preferences. A probabilistic allocation rule assigns a probability distribution over the set of possible allocations to every preference profile. We discuss characterizations of the classes of Pareto-optimal and strategy-proof probabilistic rules which satisfy in addition replacement-domination or no-envy. Interestingly, these results also apply to problems of allocating finitely many identical indivisible objects – to probabilistic and to deterministic allocation. Received: 23 November 1998/Accepted: 20 October 2000