  We study a sequential profit-maximization problem, optimizing for both price
and ancillary variables like marketing expenditures. Specifically, we aim to
maximize profit over an arbitrary sequence of multiple demand curves, each
dependent on a distinct ancillary variable, but sharing the same price. A
prototypical example is targeted marketing, where a firm (seller) wishes to
sell a product over multiple markets. The firm may invest different marketing
expenditures for different markets to optimize customer acquisition, but must
maintain the same price across all markets. Moreover, markets may have
heterogeneous demand curves, each responding to prices and marketing
expenditures differently. The firm's objective is to maximize its gross profit,
the total revenue minus marketing costs.
  Our results are near-optimal algorithms for this class of problems in an
adversarial bandit setting, where demand curves are arbitrary non-adaptive
sequences, and the firm observes only noisy evaluations of chosen points on the
demand curves. For $n$ demand curves (markets), we prove a regret upper bound
of $\tilde{O}(nT^{3/4})$ and a lower bound of $\Omega((nT)^{3/4})$ for
monotonic demand curves, and a regret bound of $\tilde{\Theta}(nT^{2/3})$ for
demands curves that are monotonic in price and concave in the ancillary
variables.
