Subgame Perfect Equilibria of Sequential Matching Games

TL;DR

This paper analyzes a sequential matching market modeled as a perfect-information game, characterizing the complexity of computing subgame perfect equilibria and designing schedules to achieve stable matchings.

Contribution

It introduces a formal game-theoretic model of sequential matching, characterizes the complexity of computing SPE, and proposes methods to realize stable matchings in the equilibrium.

Findings

Computing SPE is tractable with at most two offers per firm or worker.

Computing SPE is PSPACE-hard with at most three offers per firm and worker.

Offering schedules can be designed to realize the worker-optimal stable matching in SPE.

Abstract

We study a decentralized matching market in which firms sequentially make offers to potential workers. For each offer, the worker can choose "accept" or "reject," but the decision is irrevocable. The acceptance of an offer guarantees her job at the firm, but it may also eliminate chances of better offers from other firms in the future. We formulate this market as a perfect-information extensive-form game played by the workers. Each instance of this game has a unique subgame perfect equilibrium (SPE), which does not necessarily lead to a stable matching and has some perplexing properties. We show a dichotomy result that characterizes the complexity of computing the SPE. The computation is tractable if each firm makes offers to at most two workers or each worker receives offers from at most two firms. In contrast, it is PSPACE-hard even if both firms and workers are related to at most three offers. We also study engineering aspects of this matching market. It is shown that, for any preference profile, we can design an offering schedule of firms so that the worker-optimal stable matching is realized in the SPE.