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.
In the present paper we focus on the model defined by @cite_22 , called or , in which there are exactly as many indivisible resources as agents and no money. @cite_22 defined the (TTC) procedure which has been extensively studied and shown to be unique when preferences are strict .
@cite_22
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Matching a set of agents to a set of objects has many real applications. One well-studied framework is that of priority-based matching, in which each object is assumed to have a priority order over the agents. The Deferred Acceptance (DA) and Top-Trading-Cycle (TTC) mechanisms are the best-known strategy-proof mechanisms. However, in highly anonymous environments, the set of agents is not known a priori, and it is more natural for objects to instead have priorities over characteristics (e.g., the student's GPA or home address). In this paper, we extend the model so that each agent reports not only its preferences over objects, but also its characteristic. We derive results for various notions of strategy-proofness and false-name-proofness, corresponding to whether agents can only report weaker characteristics or also incomparable or stronger ones, and whether agents can only claim objects allocated to their true accounts or also those allocated to their fake accounts. Among other results, we show that DA and TTC satisfy a weak version of false-name-proofness. Furthermore, DA also satisfies a strong version of false-name-proofness, while TTC fails to satisfy it without an acyclicity assumption on priorities. We consider the problem of reallocating indivisible objects amongst a set of agents when the preference ordering of each agent may contain indifferences. The same model, but with strict preferences, goes back to the seminal work of Shapley and Scarf in 1974. When preferences are strict, we now know that the Top-Trading Cycles (TTC) mechanism invented by Gale is Pareto efficient, strategy-proof, and finds a core allocation, and that it is the only mechanism satisfying these properties. In the extensive literature on this problem since then, the TTC mechanism has been characterized in multiple ways, establishing its central role within the class of all allocation mechanisms. The question motivating our work is the extent to which these results can be generalized to the setting with indifferences. Our main contribution is a general framework to design strategyproof mechanisms that find a Pareto optimal allocation in the weak-core. Along the way, we establish a sufficient condition for a mechanism (within a broad class of mechanisms) to be strategyproof and use this condition to design fast algorithms for finding a "good" reallocation. Our results generalize and unify two (different) mechanisms for the reallocation problem derived, independently of each other, by Manjunath and Jaramillo, and Alcalde-Unzu and Molis. We initiate the work on fair and strategyproof allocation of indivisible chores. The fairness concept we consider in this paper is maxmin share (MMS) fairness. We consider three previously studied models of information elicited from the agents: the ordinal model, the cardinal model, and the public ranking model in which the ordinal preferences are publicly known. We present both positive and negative results on the level of MMS approximation that can be guaranteed if we require the algorithm to be strategyproof. Our results uncover some interesting contrasts between the approximation ratios achieved for chores versus goods. The paper presents a model of an exchange economy with indivisible goods and money. There are a finite number of agents, each one initially endowed with a certain amount of money and at most one indivisible good. Each agent is assumed to have no use for more than one indivisible good. It is proved that the core of the economy is nonempty. If utility functions are increasing in money, and if the initial resources in money are in some sense “sufficient” the core allocations coincide with the competitive equilibrium allocations.