Computing a subgame perfect equilibrium of a sequential matching game

We study a decentralized matching market in which each firm sequentially makes 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. Our aim is to establish the complexity of computing the SPE, or more precisely, deciding whether each offer is accepted in the SPE. We show that the tractability of this problem drastically changes according to the number of potential offers related to each firm and worker. If each firm makes offers to at most two workers (or each worker receives offers from at most two firms), then the problem is efficiently solved by a variant of the deferred acceptance algorithm. In contrast, the problem is PSPACE-hard even if both firms and workers are related to at most three offers.