Recently, the problem of allocating one resource per agent with initial endowments () has seen a renewed interest: indeed, while in the general domain Top Trading Cycle is known to be the only procedure guaranteeing Pareto-optimality, individual rationality, and strategy proofness, the situation differs in single-peaked domains. Bade (2019) presented the Crawler, an alternative procedure enjoying the same properties (with the additional advantage of being implementable in obviously dominant strategies); while (2015) showed that allowing mutually beneficial swap-deals among the agents was already enough to guarantee Pareto-optimality. In this paper we significantly deepen our understanding of this decentralized procedures: we show in particular that the single-peaked domains happen to be maximal'' if one wishes to guarantee this convergence property. Interestingly, we also observe that the set of allocations reachable by swap-deals always contains the outcome of the Crawler. To further investigate how these different mechanisms compare, we pay special attention to the average and minimum rank of the resource obtained by the agents in the outcome allocation. We provide theoretical bounds on the loss potentially induced by these procedures with respect to these criteria, and complement these results with an extensive experimental study which shows how different variants of swap dynamics behave. In fact, even the simplest dynamics exhibit very good results, and it is possible to further guide the process towards our objectives, if one is ready to sacrifice a bit in terms of decentralization. On our way, we also show that a simple variant of the Crawler allows to check efficiently that an allocation is Pareto-optimal in single-peaked domains.
Following this line of work, we assume single-peaked preferences in this paper. This domain of preferences has been introduced by @cite_36 and @cite_16 . It has been more specifically studied in voting theory and now are a very common domain of preferences . Numerous works have used single-peaked preferences in the context of fair division. @cite_23 studied the fair division problem with single-peaked preferences and divisible objects. He defines and characterizes the , the unique strategy-proof, efficient and anonymous allocation rule in this setting. The fairness properties of this rule have been subsequently explored by @cite_17 and @cite_6 @cite_2 who showed that it is envy-free, one-sided resource monotonic and consistent. As already mentioned, @cite_15 extended this research area to indivisible resources and considered the problem of assigning objects to a line under single-peaked preferences. Subsequently, @cite_18 investigated the computational aspects of this assignment problem.
@cite_18 @cite_36 @cite_6 @cite_23 @cite_2 @cite_15 @cite_16 @cite_17
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We consider the problem of assigning agents to slots on a line, where only one agent can be served at a slot and each agent prefers to be served as close as possible to his target. We introduce a general approach to compute aggregate gap-minimizing assignments, as well as gap-egalitarian assignments. The approach relies on an algorithm which is shown to be faster than general purpose algorithms for the assignment problem. We also extend the approach to probabilistic assignments and explore the computational features of existing, as well as new, methods for this setting. When a decision is reached by voting or is arrived at by a group all of whose members are not in complete accord, there is no part of economic theory which applies. This paper is intended to help fill this gap; to provide a type of reasoning which will contribute to the development of the theory of tradeunions, the firm, and the cartel; and to provide the basis for a theory of the equilibrium distribution of taxation or of public expenditure. Still other uses of the theory might be not less important. For reasons of space we avoid discussion of many points that demand fuller treatment and only attempt to indicate the course of the argument. Abstract We consider the problem of fairly allocating an infinitely divisible commodity among agents with single-peaked preferences. We search for methods of performing this division, or solutions , that satisfy the following property of consistency : any recommendation made for any economy is in agreement with the recommendation made for any "reduced" economy obtained by imagining the departure of some of the agents with their allotted consumptions. Our main result is that essentially all efficient subsolutions of the no-envy solution satisfying consistency must contain a certain solution known as the uniform rule. We also characterize the uniform rule on the basis of a "converse" of consistency and the distributional requirement of individual rationality from equal division. Journal of Economic Literature Classification Numbers: D63, D71. We consider the problem of assigning agents to slots on a line, where only one agent can be served at a slot and each agent prefers to be served as close as possible to his target. Our focus is on aggregate gap minimizing methods, i.e., those that minimize the total gap between targets and assigned slots. We first consider deterministic assignment of agents to slots, and provide a direct method for testing if a given deterministic assignment is aggregate gap minimizing. We then consider probabilistic assignment of agents to slots, and make use of the previous method to propose an aggregate gap minimizing modification of the classic random priority method to solve this class of problems. We also provide some logical relations in our setting among standard axioms in the literature on assignment problems, and explore the robustness of our results to several extensions of our setting. Originally published in 1951, Social Choice and Individual Values introduced "Arrow's Impossibility Theorem" and founded the field of social choice theory in economics and political science. This new edition, including a new foreword by Nobel laureate Eric Maskin, reintroduces Arrow's seminal book to a new generation of students and researchers. "Far beyond a classic, this small book unleashed the ongoing explosion of interest in social choice and voting theory. A half-century later, the book remains full of profound insight: its central message, 'Arrow's Theorem,' has changed the way we think."-Donald G. Saari, author of Decisions and Elections: Explaining the Unexpected 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.