  We study generalizations of online bipartite matching in which each arriving
vertex (customer) views a ranked list of offline vertices (products) and
matches to (purchases) the first one they deem acceptable. The number of
products that the customer has patience to view can be stochastic and dependent
on the products seen. We develop a framework that views the interaction with
each customer as an abstract resource consumption process, and derive new
results for these online matching problems under the adversarial,
non-stationary, and IID arrival models, assuming we can (approximately) solve
the product ranking problem for each single customer. To that end, we show new
results for product ranking under two cascade-click models: an optimal
algorithm when each item has its own hazard rate for making the customer
depart, and a 1/2-approximate algorithm when the customer has a general
item-independent patience distribution. We also present a constant-factor
0.027-approximate algorithm in a new model where items are not initially
available and arrive over time. We complement these positive results by
presenting three additional negative results relating to these problems.
