Locally Stable Marriage with Strict Preferences

TL;DR

This paper explores how local information and memory influence the convergence to stable matchings in networked agents, revealing computational hardness and conditions for guaranteed convergence.

Contribution

It introduces a model of locally stable marriage with dynamic learning, analyzes the complexity of reaching stability, and characterizes scenarios ensuring convergence.

Findings

Deciding the existence of a path to stability is NP-hard.

Memory and network structure significantly affect convergence.

Certain memory types guarantee convergence, others do not.

Abstract

We study stable matching problems with locality of information and control. In our model, each agent is a node in a fixed network and strives to be matched to another agent. An agent has a complete preference list over all other agents it can be matched with. Agents can match arbitrarily, and they learn about possible partners dynamically based on their current neighborhood. We consider convergence of dynamics to locally stable matchings -- states that are stable with respect to their imposed information structure in the network. In the two-sided case of stable marriage in which existence is guaranteed, we show that the existence of a path to stability becomes NP-hard to decide. This holds even when the network exists only among one partition of agents. In contrast, if one partition has no network and agents remember a previous match every round, a path to stability is guaranteed and random dynamics converge with probability 1. We characterize this positive result in various ways. For instance, it holds for random memory and for cache memory with the most recent partner, but not for cache memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we present results for centralized computation of locally stable matchings, i.e., computing maximum locally stable matchings in the two-sided case and deciding existence in the roommates case.