  We develop a framework for designing simple and efficient policies for a
family of online allocation and pricing problems, that includes online packing,
budget-constrained probing, dynamic pricing, and online contextual bandits with
knapsacks. In each case, we evaluate the performance of our policies in terms
of their regret (i.e., additive gap) relative to an offline controller that is
endowed with more information than the online controller. Our framework is
based on Bellman Inequalities, which decompose the loss of an algorithm into
two distinct sources of error: (1) arising from computational tractability
issues, and (2) arising from estimation/prediction of random trajectories.
Balancing these errors guides the choice of benchmarks, and leads to policies
that are both tractable and have strong performance guarantees. In particular,
in all our examples, we demonstrate constant-regret policies that only require
re-solving an LP in each period, followed by a simple greedy action-selection
rule; thus, our policies are practical as well as provably near optimal.
