Playing "hard to get"

This paper incorporates frictions derived from strategic interaction into a frictional matching model. In each period, unmarried agents are randomly matched with a partner. Agents observe noisy signals of their match partner’s quality and have two “rounds” in which they can accept or reject a match. In each round, the match is consummated only if there is mutual acceptance. Any unmarried agents after the second round return to the pool in order to be randomly matched again next period. Acceptance decisions in the second round must therefore incorporate not only information obtained via the noisy signal, but also information obtained through inference after observing the actions of one’s partner in the first round. I study a particular strategy of low-quality agents called “playing hard to get”. A low-quality agent playing hard to get rejects high signals in the first round in order to affect their match partner’s beliefs in a beneficial way. These benefits come at a cost, however, because a rejection may drive away an initially accepting partner.

I show the existence of an equilibrium in which a positive fraction of low-quality agents plays hard to get. In this equilibrium, high-quality agents who initially rejected a match, and were themselves rejected, update their beliefs and optimally accept a match in the second round. The benchmark of positive assortative matching (i.e. likes marry likes) was first established by Becker [J. Political Economy 1973] and more recently proved in a similar environment with noisy signals by Chade [J. Econ. Theory, 2006]. I show that matching, when allowing for intra-match strategic interaction, is not assortative. Low-type agents who send low signals match with high-type agents who send high signals with positive probability in equilibrium.