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.

翻译：本文研究一个序列利润最大化问题，同时优化价格及营销支出等辅助变量。具体而言，我们旨在通过任意序列的多个需求曲线实现利润最大化，每条需求曲线依赖于不同的辅助变量，但共享相同的价格。一个典型示例是目标营销：企业（卖方）希望在多个市场销售产品。企业可为不同市场投入差异化营销支出以优化客户获取，但必须在所有市场保持统一定价。此外，不同市场可能具有异质性需求曲线，每条曲线对价格和营销支出的响应方式各不相同。企业的目标是最大化其毛利润，即总收入减去营销成本。针对对抗性强盗设定下的此类问题——其中需求曲线为非自适应任意序列，企业仅能观测到需求曲线上选定点的噪声评估——我们提出了近似最优的算法。对于n条需求曲线（市场），我们证明单调需求曲线的遗憾上界为$\tilde{O}(nT^{3/4})$、下界为$\Omega((nT)^{3/4})$；对于价格单调且辅助变量凹的需求曲线，其遗憾边界为$\tilde{\Theta}(nT^{2/3})$。