Swap Dynamics in Single-Peaked House Markets

This paper focuses on the problem of fairly and efficiently allocating resources to agents. We consider a restricted framework in which all the resources are initially owned by the agents, with exactly one resource per agent (house market). In this framework, and with strict preferences, the Top Trading Cycle (TTC) algorithm is the only procedure satisfying Pareto-optimality, individual rationality and strategy-proofness. When preferences are single-peaked, the Crawler enjoys the same properties. These two centralized procedures might involve long trading cycles. In this paper we focus instead on a procedure involving the shortest cycles: bilateral swap deals. In such a swap dynamics, the agents perform pairwise mutually improving deals until reaching a swap-stable allocation (no improving swap-deal is possible). We prove that on the single-peaked domain every swap-stable allocation is Pareto-optimal, showing the efficiency of the swap dynamics. Besides, both the outcome of TTC and the Crawler can always be reached by sequences of swaps. However, some Pareto-optimal allocations are not reachable through improving swap-deals. We further analyze the swap-deal procedure through the study of the average or minimum rank of the resources obtained by agents in the final allocation. We start by providing the price of anarchy of these procedures. Finally, we present an extensive experimental study in which different versions of swap dynamics as well as other existing allocation procedures are compared. We show that swap-deal procedures exhibit good results on average in this domain, under different cultures for generating synthetic data.Comment: Replaces our previous submission: "House Markets and Single-Peaked Preferences: From Centralized to Decentralized Allocation Procedures". Following reviewers' comments, leaves out our contribution on a variant of the Crawler procedure (goes in a separate submission) to concentrate on swap dynamics (new results added

On Reachable Assignments in Cycles and Cliques

The efficient and fair distribution of indivisible resources among agents is a common problem in the field of \emph{Multi-Agent-Systems}. We consider a graph-based version of this problem called Reachable Assignments, introduced by Gourves, Lesca, and Wilczynski [AAAI, 2017]. The input for this problem consists of a set of agents, a set of objects, the agent's preferences over the objects, a graph with the agents as vertices and edges encoding which agents can trade resources with each other, and an initial and a target distribution of the objects, where each agent owns exactly one object in each distribution. The question is then whether the target distribution is reachable via a sequence of rational trades. A trade is rational when the two participating agents are neighbors in the graph and both obtain an object they prefer over the object they previously held. We show that Reachable Assignments is NP-hard even when restricting the input graph to be a clique and develop an $O(n^3)$-time algorithm for the case where the input graph is a cycle with $n$ vertices

Object Reachability via Swaps under Strict and Weak Preferences

The \textsc{Housing Market} problem is a widely studied resource allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We first consider whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie among objects is allowed). This problem is polynomially solvable in a star-network and NP-complete in a tree-network. It is left as a challenging open problem whether the problem is polynomially solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Then we show that when the preferences of the agents are weak (ties among objects are allowed), the problem becomes NP-hard even when the network is a path. In addition, we consider the computational complexity of finding different optimal assignments for the problem under the network being a path or a star.Comment: This version is to appear in Autonomous Agents and Multi-Agent System

Envy-Free Allocations Respecting Social Networks

Finding an envy-free allocation of indivisible resources to agents is a central task in many multiagent systems. Often, non-trivial envy-free allocations do not exist, and, when they do, finding them can be computationally hard. Classical envy-freeness requires that every agent likes the resources allocated to it at least as much as the resources allocated to any other agent. In many situations this assumption can be relaxed since agents often do not even know each other. We enrich the envy-freeness concept by taking into account (directed) social networks of the agents. Thus, we require that every agent likes its own allocation at least as much as those of all its (out)neighbors. This leads to a "more local" concept of envy-freeness. We also consider a "strong" variant where every agent must like its own allocation more than those of all its (out)neighbors. We analyze the classical and the parameterized complexity of finding allocations that are complete and, at the same time, envy-free with respect to one of the variants of our new concept. To this end, we study different restrictions of the agents' preferences and of the social network structure. We identify cases that become easier (from $\Sigma^\textrm{p}_2$-hard or NP-hard to polynomial-time solvability) and cases that become harder (from polynomial-time solvability to NP-hard) when comparing classical envy-freeness with our graph envy-freeness. Furthermore, we spot cases where graph envy-freeness is easier to decide than strong graph envy-freeness, and vice versa. On the route to one of our fixed-parameter tractability results, we also establish a connection to a directed and colored variant of the classical SUBGRAPH ISOMORPHISM problem, thereby extending a known fixed-parameter tractability result for the latter.Comment: 49 pages; 7 figures; A preliminary version of this article appeared in the Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems (AAMAS'18

Object Allocation Over a Network of Objects: Mobile Agents with Strict Preferences

In recent work, Gourv\`es, Lesca, and Wilczynski propose a variant of the classic housing markets model where the matching between agents and objects evolves through Pareto-improving swaps between pairs of adjacent agents in a social network. To explore the swap dynamics of their model, they pose several basic questions concerning the set of reachable matchings. In their work and other follow-up works, these questions have been studied for various classes of graphs: stars, paths, generalized stars (i.e., trees where at most one vertex has degree greater than two), trees, and cliques. For generalized stars and trees, it remains open whether a Pareto-efficient reachable matching can be found in polynomial time. In this paper, we pursue the same set of questions under a natural variant of their model. In our model, the social network is replaced by a network of objects, and a swap is allowed to take place between two agents if it is Pareto-improving and the associated objects are adjacent in the network. In those cases where the question of polynomial-time solvability versus NP-hardness has been resolved for the social network model, we are able to show that the same result holds for the network-of-objects model. In addition, for our model, we present a polynomial-time algorithm for computing a Pareto-efficient reachable matching in generalized star networks. Moreover, the object reachability algorithm that we present for path networks is significantly faster than the known polynomial-time algorithms for the same question in the social network model.Comment: List of all changes from v1: (1) publication month on title page corrected from February to March (original submission date was March 1, 2021); (2) page number on title page removed; (3) cleaned up some bibtex entrie