Marketing Hosting: From Fixed to Endogenous Budgets

The canonical model for multi-channel marketing, Bandits with Knapsacks (BwK), optimizes cumulative rewards subject to resource constraints but typically assumes a fixed, exogenous budget. This assumption is tenuous in real-world systems requiring performance-adaptive investment, where pre-committing to a budget is challenging and suboptimal. We introduce Marketing Hosting, a paradigm modeling the budget as an endogenous, performance-dependent variable. This yields a new problem class, Bandits with Endogenous Knapsacks (BwEK), characterized by a challenging feedback loop coupling rewards with constraints. We develop a specialized primal-dual algorithm to manage this coupling. For settings with hard, per-round constraints, we design a novel risk-aware algorithm that mitigates the path-dependent risk of ruin, providing the first high-probability safety guarantee for such problems. Finally, we solve the strategic bi-level problem of learning the optimal reinvestment rate. We validate our theoretical results through extensive simulations, real-world data experiments, and a live A/B test.