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_1 explored another direction by restricting preferences to single-peaked domains. In this case she presents the that satisfies the same properties as TTC hence overcoming Ma's result on the single-peaked domain.
@cite_1
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We define a general notion of single-peaked preferences based on abstract betweenness relations. Special cases are the classical example of single-peaked preferences on a line, the separable preferences on the hypercube, the “multi-dimensionally single-peaked” preferences on the product of lines, but also the unrestricted preference domain. Generalizing and unifying the existing literature, we show that a social choice function is strategy-proof on a sufficiently rich domain of generalized single-peaked preferences if and only if it takes the form of voting by issues (“voting by committees”) satisfying a simple condition called the “Intersection Property.” Based on the Intersection Property, we show that the class of preference domains associated with “median spaces” gives rise to the strongest possibility results; in particular, we show that the existence of strategyproof social choice rules that are non-dictatorial and neutral requires an underlying median space. A space is a median space if, for every triple of elements, there is a fourth element that is between each pair of the triple; numerous examples are given (some well-known, some novel), and the structure of median spaces and the associated preference domains is analysed. Many NP-hard winner determination problems admit polynomial-time algorithms when restricting inputs to be single-peaked. Commonly, such algorithms employ dynamic programming along the underlying axis. We introduce a new technique: carefully chosen integer linear programming (IP) formulations for certain voting problems admit an LP relaxation which is totally unimodular if preferences are single-peaked, and which thus admits an integral optimal solution. This technique gives fast algorithms for finding optimal committees under the PAV and Chamberlin-Courant voting rules under single-peaked preferences, as well as for certain OWA-based rules. Under single-crossing preferences, Young scores can also be calculated. An advantage of this technique is that no special-purpose algorithm needs to be used to exploit structure in the input preferences: any standard IP solver will terminate in the first iteration if the input is single-peaked, and will continue to work otherwise. 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. Many text classification tasks are known to be highly domain-dependent. Unfortunately, the availability of training data can vary drastically across domains. Worse still, for some domains there may not be any annotated data at all. In this work, we propose a multinomial adversarial network (MAN) to tackle the text classification problem in this real-world multidomain setting (MDTC). We provide theoretical justifications for the MAN framework, proving that different instances of MANs are essentially minimizers of various f-divergence metrics (Ali and Silvey, 1966) among multiple probability distributions. MANs are thus a theoretically sound generalization of traditional adversarial networks that discriminate over two distributions. More specifically, for the MDTC task, MAN learns features that are invariant across multiple domains by resorting to its ability to reduce the divergence among the feature distributions of each domain. We present experimental results showing that MANs significantly outperform the prior art on the MDTC task. We also show that MANs achieve state-of-the-art performance for domains with no labeled data.