Supply chain coordination under unknown demand distribution: Online learning and contracting

Abstract

Multi-echelon stochastic inventory models with known demand distributions have long underpinned supply chain coordination, yielding first-best policies in centralized systems and contract mechanisms that induce decentralized agents to implement these policies. We revisit the classic two-echelon inventory model in an online learning setting with unknown demand, which necessitates rethinking both inventory control and coordination strategies. This setting poses three key challenges: (i) the overall loss function may be non-convex, limiting the applicability of standard online convex optimization methods; (ii) the multi-echelon structure creates information asymmetry, as the upstream agent observes only order quantities—potentially distorted by downstream learning—rather than true consumer demand; and (iii) realized inventory levels may exceed desired targets, further complicating learning dynamics. To address these challenges, we develop algorithms that combine online optimization with low-switching mechanisms and augmented loss functions, enabling effective learning despite these complexities. In the centralized setting, our algorithm converges to the first-best policy with low regret. In the decentralized setting, we design an adaptive coordination mechanism that yields favorable individual regret guarantees while learning the optimal contract, thereby incentivizing agents to implement the first-best policy and minimizing overall system regret. Numerical experiments demonstrate that our approach consistently outperforms standard benchmarks such as explore-then-exploit and vanilla online gradient descent, highlighting its robustness and practical relevance for supply chain coordination under demand uncertainty.