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.     

On the equivalence between progressive taxation and inequality reduction

We establish the precise connections between progressive taxation and inequality reduction, in a setting where the level of tax revenue to be raised is exogenously fixed and tax schemes are balanced. We show that, in contrast with the traditional literature on taxation, the equivalence between inequality reduction and the combination of progressivity and income order preservation does not always hold in this setting. However, we show that, among rules satisfying consistency and, either revenue continuity, or revenue monotonicity, the equivalence remains intact.     

Allocating multiple estates among agents with single-peaked preferences

We consider the problem of allocating multiple social endowments (estates) of a perfectly divisible commodity among a group of agents with single-peaked preferences when each agent’s share can come from at most one estate. We inquire if well-known single-estate rules, such as the Uniform rule, the Proportional rule or the fixed-path rules can be coupled with a matching rule so as to achieve efficiency in the multi-estate level. On the class of problems where all agents have symmetric preferences, any efficient single-estate rule can be extended to an efficient multi-estate rule. If we allow asymmetric preferences however, this is no more the case. For nondictatorial single-estate rules that satisfy efficiency, strategy proofness, consistency, and resource monotonicity, an efficient extension to multiple estates is impossible. A similar impossibility also holds for single-estate rules that satisfy efficiency, peak-only, and a weak fairness property. We would like to express our gratitude to Bhaskar Dutta, Semih Koray, Hervé Moulin, and Yuntong Wang as well as an associate editor and two anonymous referees of this journal for detailed comments and suggestions. We also thank the seminar participants at Bilkent University, Indian Statistical Institute, Bilgi University, University of Warwick, ASSET 2003, and BWED XXVI.     

Agreement, separability, and other axioms for quasi-linear social choice problems

A quasi-linear social choice problem is concerned with choosing one among a finite set of public projects and determining side payments among agents to cover the cost of the project, assuming each agent has quasi-linear preferences. We first investigate the logical relations between various axioms in this context. They are: agreement, separability, population solidarity, consistency, converse consistency, and population-and-cost solidarity. Also, on the basis of these axioms, we present alternative characterizations of egalitarian solutions; each solution assigns to each agent an equal share of the surplus derived from the public project over some reference utility level, but uses a different method to compute the reference utility level. Received: 18 May 1998/Accepted: 1 July 1999     

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.     

Consistency in one-sided assignment problems

One-sided assignment problems combine important features of two well-known matching models. First, as in roommate problems, any two agents can be matched and second, as in two-sided assignment problems, the division of payoffs to agents is flexible as part of the solution. We take a similar approach to one-sided assignment problems as Sasaki (Int J Game Theory 24:373–397, 1995) for two-sided assignment problems, and we analyze various desirable properties of solutions including consistency and weak pairwise-monotonicity. We show that for the class of solvable one-sided assignment problems (i.e., the subset of one-sided assignment problems with a non-empty core), if a subsolution of the core satisfies [Pareto indifference and consistency] or [invariance with respect to unmatching dummy pairs, continuity, and consistency], then it coincides with the core (Theorems 1 and 2). However, we also prove that on the class of all one-sided assignment problems (solvable or not), no solution satisfies consistency and coincides with the core whenever the core is non-empty (Theorem 4). Finally, we comment on the difficulty in obtaining further positive results for the class of solvable one-sided assignment problems in line with Sasaki’s (1995) characterizations of the core for two-sided assignment problems.     

Equal factor equivalence in economies with multiple public goods

We reconsider the problem of provision and cost-sharing of multiple public goods. The efficient equal factor equivalent allocation rule makes every agent indifferent between what he receives and the opportunity of choosing the bundle of public goods subject to the constraint of paying r times its cost, where r is set as low as possible. We show that this rule is characterized in economies with a continuum of agents by efficiency, a natural upper bound on everyone's welfare, and a property of solidarity with respect to changes in population and preferences. Received: 3 August 1995 / Accepted : 29 April 1997     

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.     

Markovian assignment rules

We analyze dynamic assignment problems where agents successively receive different objects (positions, offices, etc.). A finite set of n vertically differentiated indivisible objects are assigned to n agents who live n periods. At each period, a new agent enters society, and the oldest agent retires, leaving his object to be reassigned. We define independent assignment rules (where the assignment of an object to an agent is independent of the way other objects are allocated to other agents), efficient assignment rules (where there does not exist another assignment rule with larger expected surplus), and fair assignment rules (where agents experiencing the same circumstances have identical histories in the long run). When agents are homogenous, we characterize efficient, independent and fair rules as generalizations of the seniority rule. When agents draw their types at random, we prove that independence and efficiency are incompatible, and that efficient and fair rules only exist when there are two types of agents. We characterize two simple rules (type-rank and type-seniority) which satisfy both efficiency and fairness criteria in dichotomous settings.     

The replacement principle for the provision of multiple public goods on tree networks

This article considers the provision of two public goods on tree networks where each agent has a single-peaked preference. We show that if there are at least four agents, then no social choice rule exists that satisfies efficiency and replacement-domination. In fact, these properties are incompatible, even if agents’ preferences are restricted to a smaller domain of symmetric single-peaked preferences. However, for rules on an interval, we prove that Miyagawa’s (Soc Choice Welf 18:527–541, 2001) characterization that only the left-peaks rule and the right-peaks rule satisfy both of these properties also holds on the domain of symmetric single-peaked preferences. Moreover, if agents’ peak locations are restricted to either the nodes or the endpoints of trees, rules exist on a subclass of trees. We provide a characterization of a family of such rules for this tree subclass.     

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.     

Pairwise Strategy-Proofness and Self-Enforcing Manipulation

“Strategy-proofness” is one of the axioms that are most frequently used in the recent literature on social choice theory. It requires that by misrepresenting his preferences, no agent can manipulate the outcome of the social choice rule in his favor. The stronger requirement of “group strategy-proofness” is also often employed to obtain clear characterization results of social choice rules. Group strategy-proofness requires that no group of agents can manipulate the outcome in their favors. In this paper, we advocate “effective pairwise strategy-proofness.” It is the requirement that the social choice rule should be immune to unilateral manipulation and “self-enforcing” pairwise manipulation in the sense that no agent of a pair has the incentive to betray his partner. We apply the axiom of effective pairwise strategy-proofness to three types of economies: public good economy, pure exchange economy, and allotment economy. Although effective pairwise strategy-proofness is seemingly a much weaker axiom than group strategy-proofness, effective pairwise strategy-proofness characterizes social choice rules that are analyzed by using different axioms in the literature.     

Equity and the Vickrey allocation rule on general preference domains

We consider the problem of allocating multiple units of an indivisible good among a group of agents in which each agent demands at most one unit of the good and money payment or receipt is required. Under general preference domains that may contain non quasi-linear preferences, the Vickrey allocation rule is characterized by axioms for equity and continuity without use of efficiency: namely, the Vickrey rule is the only rule that satisfies strategy-proofness, weak envy-freeness for equals, non-imposition, and continuity of welfare.     

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.     

Shapley–Shubik methods in cost sharing problems with technological cooperation

In the discrete cost sharing model with technological cooperation (Bahel and Trudeau in Int J Game Theory 42:439–460, 2013a), we study the implications of a number of properties that strengthen the well-known dummy axiom. Our main axiom, which requires that costless units of demands do not affect the cost shares, is used to characterize two classes of rules. Combined with anonymity and a specific stability property, this requirement picks up sharing methods that allow the full compensation of at most one technological contribution. If instead we strengthen the well-known dummy property to include agents whose technological contribution is offset by the cost of their demand, we are left with an adaptation of the Shapley–Shubik method that treats technologies as private and rewards their contributions. Our results provide two interesting axiomatizations for the adaptations of the Shapley–Shubik rule to our framework.     

Characterizations of the cycle-complete and folk solutions for minimum cost spanning tree problems

Minimum cost spanning tree problems connect agents efficiently to a source when agents are located at different points and the cost of using an edge is fixed. The folk and cycle-complete cost sharing solutions always offer core allocations. We provide similar characterizations for both. A new property is based on the following observation: when all agents have the same cost to connect to the source, we can connect one of them to the source then connect all other agents to him, as if he was the source. Cost sharing should also be done in these two steps. We also use some common properties: core selection, piecewise linearity and an independence property. The solutions are differentiated by properties that apply when the cheapest edge to the source gets cheaper. Either the savings are equally distributed among all agents (folk) or the agent on that edge gets all of the savings (cycle-complete).     

Linear cost sharing in economies with non-Samuelsonian public goods: Core equivalence

 We consider an economy with non-Samuelsonian public goods and we focus on linear cost sharing. In a linear cost sharing equilibrium all agents in the economy optimize given a certain fixed cost share to be contributed towards the provision of public goods in the economy. Hence, each agent pays a certain fraction of the total establishment costs of public goods and these cost shares are common knowledge. We show that for a certain fixed contribution scheme the resulting linear cost share equilibria are equivalent to corresponding core allocations, in which the core is based on the integral of the individual cost shares. We also show that there is no equivalence of the Foley core with cost share equilibria, even in well-behaved large economies. Received: 16 August 1995/Accepted: 29 July 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.     

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.     

Rationality,external norms,and the epistemic value of menus

Ever since Sen’s criticism of internal consistency properties of choice, there exists a perception that the standard rational choice approach has difficulties in coping with the existence of external norms, or the information a menu of choice might convey to a decision maker. This article provides a brief survey of possible responses to these criticisms of traditional rational choice theory. We review approaches to the internalization of external norms as well as ways to formulate norm-conditional rationalizability. Furthermore, we illustrate how the epistemic value of an opportunity set can be incorporated into rational choice theory.