  Matching markets are often organized in a multi-stage and decentralized
manner. Moreover, participants in real-world matching markets often have
uncertain preferences. This article develops a framework for learning optimal
strategies in such settings, based on a nonparametric statistical approach and
variational analysis. We propose an efficient algorithm, built upon concepts of
"lower uncertainty bound" and "calibrated decentralized matching," for
maximizing the participants' expected payoff. We show that there exists a
welfare-versus-fairness trade-off that is characterized by the uncertainty
level of acceptance. Participants will strategically act in favor of a low
uncertainty level to reduce competition and increase expected payoff. We prove
that participants can be better off with multi-stage matching compared to
single-stage matching. We demonstrate aspects of the theoretical predictions
through simulations and an experiment using real data from college admissions.
