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Efficient Online Learning for Stackelberg Pricing in a Newsvendor Supply Chain

Core Concepts
The core message of this paper is to devise an efficient online learning algorithm for a Stackelberg pricing game between a supplier (leader) and a retailer (follower) in a Newsvendor supply chain setting, where the demand parameters are initially unknown.
The paper introduces a Stackelberg game framework for modeling the economic interaction between a supplier (leader) and a retailer (follower) in a Newsvendor supply chain setting. The key highlights are: Proof of the existence of a unique Stackelberg equilibrium under perfect information for the Newsvendor pricing game. Development of an online learning algorithm that leverages stochastic linear contextual bandits to learn the demand parameters, while integrating established economic theory. Derivation of convergence properties of the online learning algorithm to an approximate Stackelberg equilibrium, and theoretical guarantees for bounds on finite-time regret. Demonstration of the theoretical results through economic simulations, showing the learning algorithm outperforming baseline algorithms in terms of finite-time cumulative regret. The authors address the challenge of optimizing for both environmental and strategic regret, particularly when facing stochastic environmental parameters and uncertainty in agent strategies. They propose innovative approaches for optimization and bounding functions to derive theoretical worst-case bounds on regret.
The paper does not contain any explicit numerical data or statistics. It focuses on the theoretical analysis and development of the online learning algorithm.
"We introduce the application of online learning in a Stackelberg game pertaining to a system with two learning agents in a dyadic exchange network, consisting of a supplier and retailer, specifically where the parameters of the demand function are unknown." "We prove the existence of a unique Stackelberg equilibrium when extending this to a two-player pricing game." "A novel algorithm based on contextual linear bandits with a measurable uncertainty set is used to provide a confidence bound on the parameters of the stochastic demand. Consequently, optimal finite time regret bounds on the Stackelberg regret, along with convergence guarantees to an approximate Stackelberg equilibrium, are provided."

Deeper Inquiries

How can the proposed online learning algorithm be extended to handle more complex demand functions or supply chain structures beyond the Newsvendor setting

The proposed online learning algorithm can be extended to handle more complex demand functions or supply chain structures beyond the Newsvendor setting by incorporating additional features and constraints into the model. One way to achieve this is by introducing more parameters to capture the intricacies of the demand function, such as seasonality, trends, or external factors that may influence demand. This can be done by expanding the feature space in the contextual bandit algorithm to include these variables and allowing the algorithm to learn the relationships between them and the optimal pricing and ordering decisions. Furthermore, the algorithm can be adapted to handle different supply chain structures by modifying the reward functions and best response strategies to accommodate the specific dynamics of the supply chain. For example, in a multi-tier supply chain with multiple suppliers and retailers, the algorithm can be extended to optimize decisions at each level while considering the interdependencies between the different entities. This can involve incorporating hierarchical decision-making processes and coordination mechanisms to ensure alignment and efficiency across the supply chain. Overall, by enhancing the flexibility and adaptability of the online learning algorithm, it can be tailored to address a wide range of demand functions and supply chain structures, making it applicable to diverse real-world scenarios.

What are the potential limitations or drawbacks of the optimistic best response approach used by the follower, and how could alternative strategies be incorporated

The optimistic best response approach used by the follower may have potential limitations or drawbacks in certain situations. One limitation is that the optimistic approach assumes that the follower always makes decisions that maximize their immediate reward, which may not always align with long-term strategic goals or overall system optimization. This can lead to suboptimal outcomes in the long run, especially in complex and dynamic supply chain environments where decisions are interconnected and have ripple effects. To address this limitation, alternative strategies could be incorporated into the algorithm to provide a more balanced approach to decision-making. For example, a risk-averse strategy could be implemented to account for uncertainty and variability in demand, ensuring more stable and consistent performance over time. Additionally, a strategic planning component could be integrated to enable the follower to consider the impact of their decisions on the overall supply chain performance and make choices that benefit the system as a whole. By incorporating a mix of optimistic, risk-averse, and strategic decision-making approaches, the algorithm can achieve a more robust and adaptive behavior that balances short-term gains with long-term sustainability and efficiency in the supply chain.

Given the focus on regret minimization, how could the framework be adapted to also consider other performance metrics, such as social welfare or fairness, in the supply chain competition

To adapt the framework to consider other performance metrics such as social welfare or fairness in the supply chain competition, additional objectives and constraints can be incorporated into the algorithm. This can involve modifying the reward functions to include terms that capture social welfare considerations, such as the impact on stakeholders, communities, or the environment. By optimizing for a combination of profit, social welfare, and fairness, the algorithm can promote more sustainable and ethical decision-making in the supply chain. Furthermore, fairness constraints can be introduced to ensure equitable outcomes for all parties involved in the supply chain competition. This can involve incorporating fairness metrics, such as distributional fairness or equality of opportunity, into the optimization process to prevent biases or disparities in decision-making. By expanding the framework to consider a broader set of performance metrics and incorporating fairness and social welfare considerations, the algorithm can promote more responsible and inclusive supply chain practices that benefit not only the individual firms but also society as a whole.