Bribe-proof rules in the division problem

The division problem consists of allocating an amount of a perfectly divisible good among a group of n agents with single-peaked preferences. A rule maps preference profiles into n shares of the amount to be allocated. A rule is bribe-proof if no group of agents can compensate one of its subgroups to misrepresent their preferences and, after an appropriate redistribution of their shares, each obtains a weakly preferred share and all agents in the misrepresenting subgroup obtain a strictly preferred share. We characterize all bribe-proof rules as the class of Pareto efficient, strategy-proof, and weakly replacement monotonic rules. This class is larger than the set of sequential allotment rules identified in Barberà et al. [Barberà, S., Jackson, M., Neme, A., 1997. Strategy-proof allotment rules. Games Econ. Behav. 18, 1–21].     

Bribe-proof rules in the division problem

The division problem consists of allocating an amount of a perfectly divisible good among a group of n agents with single-peaked preferences. A rule maps preference profiles into n shares of the amount to be allocated. A rule is bribe-proof if no group of agents can compensate one of its subgroups to misrepresent their preferences and, after an appropriate redistribution of their shares, each obtains a weakly preferred share and all agents in the misrepresenting subgroup obtain a strictly preferred share. We characterize all bribe-proof rules as the class of Pareto efficient, strategy-proof, and weakly replacement monotonic rules. This class is larger than the set of sequential allotment rules identified in Barberà et al. [Barberà, S., Jackson, M., Neme, A., 1997. Strategy-proof allotment rules. Games Econ. Behav. 18, 1–21].     

Priorities in the location of multiple public facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfy anonymity, avoid the no-show paradox, strategy-proofness and population-monotonicity. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.     

Arrow's Possibility Theorem for one-dimensional single-peaked preferences

In one-dimensional environments with single-peaked preferences we consider social welfare functions satisfying Arrow's requirements, i.e. weak Pareto and independence of irrelevant alternatives. When the policy space is a one-dimensional continuum such a welfare function is determined by a collection of 2^N strictly quasi-concave preferences and a tie-breaking rule. As a corollary we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voter's preference is strictly quasi-concave.     

Strategy-Proof Probabilistic Mechanisms in Economies with Pure Public Goods

Public good economies where agents are endowed with strictly convex continuous single-peaked preferences on a convex subset of Euclidean space are considered. Such an economy arises for instance in the classical problem of allocating a given budget to finance the provision of several public goods where the agents have monotonically increasing strictly convex continuous preferences. A probabilistic mechanism assigns a probability distribution over the feasible alternatives to any profile of reported preferences. The main result of the paper establishes that any strategy-proof (in the sense of A. Gibbard, Econometrica45 (1977), 665-681) and unanimous mechanism must be a random dictatorship. Journal of Economic Literature Classification Numbers: D70, D71, H40, C60.     

Solidarity and Probabilistic Target Rules

We consider a probabilistic approach to collective choice problems where a group of agents with single-peaked preferences have to decide on the level or location of a public good. We show that every probabilistic rule that satisfies Pareto efficiency and "solidarity" (population-monotonicity or replacement-domination) must equal a so-called target rule.     

Public decisions: Solidarity and the status quo

A public decision model specifies a set of alternatives, a variable population, and a common set of admissible preferences. We study the implications of the principle of solidarity, for social choice functions in all such models. The principle says that when the environment changes, all agents not responsible for the change should either all weakly win, or all weakly lose. Under weak additional requirements, but regardless the domain of preferences, each of two formulations of this principle, population-monotonicity and replacement-domination, imply coalition-strategy-proofness; that the choice only depends on the set of preferences that are present in the society, but not on the number, nor on the labels of agents having particular preferences; and that there exists an alternative always weakly Pareto-dominated by the alternative selected by the rule. Replacement-domination is generally at least as strong as population-monotonicity.     

Strategy-Proof Probabilistic Decision Schemes for One-Dimensional Single-Peaked Preferences

Collective decision problems are considered with a finite number of agents who have single-peaked preferences on the real line. A probabilistic decision scheme assigns a probability distribution over the real line to every profile of reported preferences. The main result of the paper is a characterization of the class of unanimous and strategy-proof probabilistic schemes with the aid of fixed probability distributions that play a role similar to that of the phantom voters in H. Moulin (Public Choice35 (1980), 437-455). Thereby, the work of Moulin (1980) is extended to the probabilistic framework. Journal of Economic Literature Classification Numbers: D71, D81.     

General allocation problems with single-peaked preferences: path-independence and related topics

We face the problem of allocationg a fixed amount of a perfectly divisible good among a group of agents with single-peaked preferences. We survey the three different cases studied in the literature: the pure distribution case, the redistribution case, and the gerneral case. The so called general case provide with a natural framework to analyze the idea of path-independence. In this framework, we explore the existence of rules fulfilling this property. Our first result is negative: a strong version of this property cannot be fulfilled together with efficiency. Nonetheless, some restricted versions of the path-independence property are compatible with interesting properties, in particular no manipulability, and no envy. We then identify two solutions satisfying this sort of property: the equal distance rule, and a new extension of the uniform rule.     

Strategy-proof Location on a Network

We consider rules that choose a location on a graph (e.g. a road network) based on agents' single-peaked preferences. First, we characterize the class of strategy-proof, onto rules when the graph is a tree. Such a rule is based on a collection of generalized median voter rules (Moulin, 1980) satisfying a consistency condition. Second, we characterize such rules for graphs containing cycles. We show that while such a rule is not necessarily dictatorial, the existence of a cycle grants some agent an amount of decisive power, unlike the case of trees. Rules for this case can be described in terms of a subclass of such rules for trees. Journal of Economic Literature Classification Numbers: C72, D78.     

Strategy-Proof Voting on Compact Ranges

Strategy-proof social choice functions are characterized for societies where the space of alternatives is any full dimensional compact subset of a Euclidean space and all voters have generalized single-peaked preferences. Our results build upon and extend those obtained for cartesian product ranges by Border and Jordan (1983). By admitting a large set of non-Cartesian ranges, we give a partial answer to the major open question left unresolved in this pioneering article. We prove that our class is composed by generalized median voter schemes which satisfy an additional condition, called the intersection property [Barberà, Massó, and Neme (1997)].Journal of Economic LiteratureClassification Number: D71.     

Resource-monotonic allocation when preferences are single-peaked

Summary. We consider the problem of allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. Thomson (1994a), S?nmez (1994), and Moulin (1999) introduce three different resource-monotonicity conditions. In each characterization they derive, the axioms are independent. Under Pareto-optimality, the three resource-monotonicity conditions are equivalent. We investigate whether the interchange of these conditions still yields a valid characterization, and when the characterization still holds, whether it is a tight result or not. We strengthen each of the results, that is either the used resource-monotonicity condition can be replaced by a weaker one, or by using another resource-monotonicity condition, the result is not tight. Our main result is that when at least three agents are present, the class of fixed-path rationing methods is characterized by weak one-sided resource-monotonicity, strategy-proofness, and consistency. Received: April 24, 2000; revised version: April 10, 2001     

Inefficiency of Strategy-Proof Rules for Pure Exchange Economies

In his pioneering article, (in “Decision and Organization” (C. B. McGuire and R. Radner, Eds.), pp. 297-336, North-Holland, Amsterdam, 1972) Hurwicz establishes that there is no strategy-proof, Pareto-efficient, and individually rational rule for pure exchange economies with two agents and two goods, provided that the domain includes a sufficiently wide class of classical preferences. In this article, we extend his result to pure exchange economies with any finite number of agents and goods. We establish that (i) there is no strategy-proof, Pareto-efficient, and individually rational rule on the class of classical, homothetic, and smooth preferences; and (ii) there is no strategy-proof, Pareto-efficient, and symmetric rule on the same class of preferences. Journal of Economic Literature Classification Numbers: D78, D71, C72.     

Majority cycles in a multi-dimensional setting

Summary. We consider a set of alternatives (electoral platforms, bills, etc. ...) defined as a Cartesian product of k finite discrete sets. We assume that the preferences of the individuals (voters) are marginally single-peaked and separable. The main result of this paper states that the pairwise majority relation satisfies these two properties but that it might exhibit several cycles. This result is important when related to classical problems of multi-dimensional decisions such as logrolling and vote trading. We relate our result with a continuous version of it (McKelvey, 1976). Received: March 21, 2000; revised version: April 12, 2001     

Second-best efficiency of allocation rules: strategy-proofness and single-peaked preferences with multiple commodities

We study strategy-proof allocation rules in economies with perfectly divisible multiple commodities and single-peaked preferences. In this setup, it is known that the incompatibility among strategy-proofness, Pareto efficiency and non-dictatorship arises in contrast with the Sprumont (Econometrica 59:509–519, 1991) one commodity model. We first investigate the existence problem of strategy-proof and second-best efficient rules, where a strategy-proof rule is second-best efficient if it is not Pareto-dominated by any other strategy-proof rules. We show that there exists an egalitarian rational (consequently, non-dictatorial) strategy-proof rule satisfying second-best efficiency. Second, we give a new characterization of the generalized uniform rule with the second-best efficiency in two-agent case.     

Reallocation of an infinitely divisible good

Summary. We consider the problem of reallocating the total initial endowments of an infinitely divisible commodity among agents with single-peaked preferences. With the uniform reallocation rule we propose a solution which satisfies many appealing properties, describing the effect of population and endowment variations on the outcome. The central properties which are studied in this context are population monotonicity, bilateral consistency, (endowment) monotonicity and (endowment) strategy-proofness. Furthermore, the uniform reallocation rule is Pareto optimal and satisfies several equity conditions, e.g., equal-treatment and envy-freeness. We study the trade-off between properties concerning variation and properties concerning equity. Furthermore, we provide several characterizations of the uniform reallocation rule based on these properties. Received: August 29, 1995; revised version June 26, 1996     

Water allocation by social choice rules: The case of sequential rules

This paper considers the problem of allocating shares of irrigation water to different agricultural agents with single-peaked preferences with respect to their own shares. We define two different sequential allocation rules that respect the asymmetry between the agents and maintain the properties of Pareto efficiency and strategy-proofness, and we design a specific algorithm to apply these rules. The results of the empirical application of these rules for the case of an irrigated area located in the Ebro Basin (Spain) show that the designed sequential rules are able to substantially improve the efficiency of the currently applied proportional rule in context of severe scarcity of water and/or high administrative water prices.     

Individually rational rules for the division problem when the number of units to be allotted is endogenous

We study individually rational rules to be used to allot, among a group of agents, a perfectly divisible good that is freely available only in whole units. A rule is individually rational if, at each preference profile, each agent finds that her allotment is at least as good as any whole unit of the good. We study and characterize two individually rational and efficient families of rules, whenever agents' preferences are symmetric single-peaked on the set of possible allotments. Rules in the two families are in addition envy-free, but they differ on whether envy-freeness is considered on losses or on awards. Our main result states that (a) the family of constrained equal losses rules coincides with the class of all individually rational and efficient rules that satisfy justified envy-freeness on losses and (b) the family of constrained equal awards rules coincides with the class of all individually rational and efficient rules that satisfy envy-freeness on awards.     

Two characterizations of the uniform rule for division problems with single-peaked preferences

Summary The uniform rule is considered to be the most important rule for the problem of allocating an amount of a perfectly divisible good between agents who have single-peaked preferences. The uniform rule was studied extensively in the literature and several characterizations were provided. The aim of this paper is to provide two different formulations and corresponding axiomatizations of the uniform rule. These formulations resemble the Nash and the lexicographic egalitarian bargaining solutions; the corresponding axiomatizations are based on axioms of independence of irrelevant alternatives and restricted monotonicity.The authors are grateful to Jeffrey Banks, Steve Ching, Vincent Feltkamp, Bezalel Peleg, Sang-Young Sonn, William Thomson and an anonymous referee for helpful comments.     

On strategy-proofness and symmetric single-peakedness

We characterize the class of strategy-proof social choice functions on the domain of symmetric single-peaked preferences. This class is strictly larger than the set of generalized median voter schemes (the class of strategy-proof and tops-only social choice functions on the domain of single-peaked preferences characterized by Moulin, 1980) since, under the domain of symmetric single-peaked preferences, generalized median voter schemes can be disturbed by discontinuity points and remain strategy-proof on the smaller domain. Our result identifies the specific nature of these discontinuities which allow to design non-onto social choice functions to deal with feasibility constraints.