Counting steps for re-stabilization in a labor matching market

TL;DR

This paper introduces an algorithm to measure the time it takes for a worker to re-stabilize in a new job after resignation, linking the process to the structure of stable matchings in a labor market.

Contribution

It presents a novel algorithm modeling re-stabilization as a vacancy chain, connecting chain length to the lattice of stable matchings and preference cycles.

Findings

Chain length correlates with the lattice structure of stable matchings.

The algorithm computes the waiting time based on preference cycles.

Re-stabilization time can be explicitly calculated from initial and final matchings.

Abstract

We study a one-to-one labor matching market. If a worker considers resigning from her current job to obtain a better one, how long does it take for this worker to actually get it? We present an algorithm that models this situation as a re-stabilization process involving a vacancy chain. Each step of the algorithm is a link of such a chain. We show that the length of this vacancy chain, which can be interpreted as the time the worker has to wait for her new job, is intimately connected with the lattice structure of the set of stable matchings of the market. Namely, this length can be computed by considering the cardinalities of cycles in preferences derived from the initial and final stable matchings involved.