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.
@cite_9 defined a set of Pareto-efficient procedures generalizing the TTC algorithm when allowing for indifferences, which however include procedures that are not strategy-proof. @cite_20 and @cite_37 independently proposed general frameworks for efficient and strategy-proof generalization of the TTC procedure with indifferences.
@cite_9 @cite_37 @cite_20
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The (Shapley-Scarf) housing market is a well-studied and fundamental model of an exchange economy. Each agent owns a single house and the goal is to reallocate the houses to the agents in a mutually beneficial and stable manner. Recently, Alcalde-Unzu and Molis (2011) and Jaramillo and Manjunath (2011) independently examined housing markets in which agents can express indifferences among houses. They proposed two important families of mechanisms, known as TTAS and TCR respectively. We formulate a family of mechanisms which not only includes TTAS and TCR but also satisfies many desirable properties of both families. As a corollary, we show that TCR is strict core selecting (if the strict core is non-empty). Finally, we settle an open question regarding the computational complexity of the TTAS mechanism. Our study also raises a number of interesting research questions. 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.