Decentralized Many-to-One Matching With Random Search

I analyze a canonical many-to-one matching market within a decentralized search model with frictions, where a finite number of firms and workers meet randomly until the market clears. I compare the stable matchings of the underlying market and equilibrium outcomes when time is nearly costless. In contrast to the case where each firm has just a single vacancy, I show that stable matchings are not obtained as easily. In particular, there may be no Markovian equilibrium that uniformly implements either the worker- or the firm-optimal stable matching in every subgame. The challenge results from the firms’ ability to withhold capacity strategically. Yet, this is not the case for markets with vertical preferences on one side, and I construct the equilibrium strategy profile that leads to the unique stable matching almost surely. Moreover, multiple vacancies enable firms to implicitly collude and achieve unstable but firm-preferred matchings, even under Markovian equilibria. Finally, I identify one sufficient condition on preferences to rule out such opportunities.