Learning in Conjectural Stackelberg Games

2025-01-23

AI Summary

This paper addresses the strategic learning challenge faced by leaders in nonconvex multi-leader single-follower Stackelberg games, where leaders lack knowledge of the follower’s response. To handle response uncertainty, we propose a “conjecture-driven sequential decision-making” framework and introduce the **Conjectural Stackelberg Equilibrium (CSE)**—a novel equilibrium concept that refines the classical Stackelberg equilibrium. We design a two-stage algorithm with theoretical convergence guarantees: Stage I employs supervised learning to fit a conjectural model of the follower’s response; Stage II optimizes the leader’s strategy over this learned conjecture, thereby decoupling conjecture learning from strategy optimization. We establish global convergence of the algorithm under mild assumptions. Numerical experiments demonstrate that the proposed method significantly enhances the robustness and practicality of computed equilibrium strategies compared to baseline approaches.

Technology Category

• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

We extend the formalism of Conjectural Variations games to Stackelberg games involving multiple leaders and a single follower. To solve these nonconvex games, a common assumption is that the leaders compute their strategies having perfect knowledge of the follower's best response. However, in practice, the leaders may have little to no knowledge about the other players' reactions. To deal with this lack of knowledge, we assume that each leader can form conjectures about the other players' best responses, and update its strategy relying on these conjectures. Our contributions are twofold: (i) On the theoretical side, we introduce the concept of Conjectural Stackelberg Equilibrium -- keeping our formalism conjecture agnostic -- with Stackelberg Equilibrium being a refinement of it. (ii) On the algorithmic side, we introduce a two-stage algorithm with guarantees of convergence, which allows the leaders to first learn conjectures on a training data set, and then update their strategies. Theoretical results are illustrated numerically.

Problem

• Stackelberg Game
• Uncertainty
• Equilibrium State

Innovation

• Hypothetical Dominant Equilibrium
• Two-step Algorithm
• Predictive Strategy Adjustment