# Swap Dynamics in Single-Peaked Housing Markets

## TL;DR

This paper investigates swap-based dynamics in single-peaked housing markets, proving their efficiency and exploring their reachability and social welfare outcomes through theoretical analysis and extensive experiments.

## Contribution

It demonstrates that swap-stable allocations are Pareto-optimal in single-peaked preferences and shows how swap dynamics can approximate known procedures like TTC and Crawler.

## Key findings

- Swap-stable allocations are Pareto-optimal in the single-peaked domain.
- Sequences of swaps can reach TTC and Crawler outcomes.
- Swap dynamics perform well on average in synthetic experiments.

## Abstract

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.