  We study the problem of designing revenue-maximizing auctions for allocating
multiple goods to flexible consumers. In our model, each consumer is interested
in a subset of goods known as its flexibility set and wants to consume one good
from this set. A consumer's flexibility set and its utility from consuming a
good from its flexibility set are its private information. We focus on the case
of nested flexibility sets --- each consumer's flexibility set can be one of
$k$ nested sets. We provide several examples where such nested flexibility sets
may arise. We characterize the allocation rule for an incentive compatible,
individually rational and revenue-maximizing auction as the solution to an
integer program. The corresponding payment rule is described by an integral
equation. We then leverage the nestedness of flexibility sets to simplify the
optimal auction and provide a complete characterization of allocations and
payments in terms of simple thresholds.
