  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 study
signaling mechanisms that help to clear the congestion in such decentralized
markets and find that the effects of signaling are heterogeneous, showing a
dependence on the participants and matching stages. We prove that participants
can be better off with multi-stage matching compared to single-stage matching.
The deferred acceptance procedure assumes no limit on the number of stages and
attains efficiency and fairness but may make some participants worse off than
multi-stage matching. We demonstrate aspects of the theoretical predictions
through simulations and an experiment using real data from college admissions.
