  We consider contextual bandits with knapsacks, with an underlying structure
between rewards generated and cost vectors suffered. We do so motivated by
sales with commercial discounts. At each round, given the stochastic i.i.d.\
context $\mathbf{x}_t$ and the arm picked $a_t$ (corresponding, e.g., to a
discount level), a customer conversion may be obtained, in which case a reward
$r(a,\mathbf{x}_t)$ is gained and vector costs $c(a_t,\mathbf{x}_t)$ are
suffered (corresponding, e.g., to losses of earnings). Otherwise, in the
absence of a conversion, the reward and costs are null. The reward and costs
achieved are thus coupled through the binary variable measuring conversion or
the absence thereof. This underlying structure between rewards and costs is
different from the linear structures considered by Agrawal and Devanur [2016]
(but we show that the techniques introduced in the present article may also be
applied to the case of these linear structures). The adaptive policies
exhibited solve at each round a linear program based on upper-confidence
estimates of the probabilities of conversion given $a$ and $\mathbf{x}$. This
kind of policy is most natural and achieves a regret bound of the typical order
(OPT/$B$) $\sqrt{T}$, where $B$ is the total budget allowed, OPT is the optimal
expected reward achievable by a static policy, and $T$ is the number of rounds.
