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. We prove a regret upper bound of
$\widetilde{\mathcal{O}}\big(nT^{3/4}\big)$ and a lower bound of
$\Omega\big((nT)^{3/4}\big)$ for monotonic demand curves, and a regret bound of
$\widetilde{\Theta}\big(nT^{2/3}\big)$ for demands curves that are monotonic in
price and concave in the ancillary variables.

该研究针对具有不同需求曲线的顺序盈利最大化问题，考虑价格和市场支出等附属变量的优化，提出了接近最佳的算法，并给出了不同类型需求曲线的后悔上限。