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Swap dynamics in single-peaked housing markets

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This paper focuses on the problem of fairly and efficiently allocating resources to agents. We consider a specific setting, usually referred to as a housing market, where each agent must receive exactly one resource (and initially owns one). In this framework, in the domain of linear preferences, the Top Trading Cycle (TTC) algorithm is the only procedure satisfying Pareto-optimality, individual rationality and strategy-proofness. Under the restriction of single-peaked preferences, Crawler enjoys the same properties. These two centralized procedures might however involve long trading cycles. In this paper we focus instead on procedures involving the shortest cycles: bilateral swap-deals. In such swap dynamics, the agents perform pairwise mutually improving deals until reaching a swap-stable allocation (no improving swap-deal is possible). We prove that in the single-peaked domain every swap-stable allocation is Pareto-optimal, showing the efficiency of the swap dynamics. In fact, this domain turns out to be maximal when it comes to guaranteeing this property. Besides, both the outcome of TTC and 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 outcome of swap dynamics through social welfare notions, in our context the average or minimum rank of the resources obtained by agents in the final allocation. We start by providing a worst-case analysis of these procedures. Finally, we present an extensive experimental study in which different versions of swap dynamics are compared to other existing allocation procedures. We show that they exhibit good results on average in this domain, under different cultures for generating synthetic data.

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  1. In principle, it is possible to distribute the execution of central procedures, by letting all agents broadcast, compute locally their own version of the central algorithm, and coordinate. This might however induce a significant cost.

  2. Note that the algorithm can equivalently be executed from the last agent to the first one.

  3. The underlined allocation will only be relevant in forthcoming proofs.

  4. Again, the underlined allocation will only be relevant in a later proof.

  5. The Kendall’s tau distance between two preference order \(\succ _i\) and \(\succ _{i'}\) over \(\mathcal {R}\) is the number of pairs \((r, r') \in \mathcal {R}^2\) such that \(\succ _i\) and \(\succ _{i'}\) do not rank r and \(r'\) in the same order. The Spearman distance between two preference order \(\succ _i\) and \(\succ _{i'}\) over \(\mathcal {R}\) is defined as \(\sum _{r \in \mathcal {R}} |rank_{a_i}(r) - rank_{a_{i'}}(r)|\). In both cases, the diversity index of a profile L is the sum of the distance between every pair of preference orders \((\succ _i, \succ _{i'})\) of L.


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We thank Sophie Bade, Yann Chevaleyre, Anastasia Damamme, and Julien Lesca, for discussions related to this topic as well as the anonymous reviewers for their comments and suggestions which significantly improved the paper.

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Correspondence to Nicolas Maudet.

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Beynier, A., Maudet, N., Rey, S. et al. Swap dynamics in single-peaked housing markets. Auton Agent Multi-Agent Syst 35, 20 (2021).

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