Stackelberg Evolutionary Games With Modulated Leadership: A Three-Agent Framework for Tumor–Immune Dynamics

In many real-world systems, agents interact in complex ways that challenge traditional control and game-theoretic approaches. Building on Stackelberg evolutionary game theory, which models the interplay between a rational leader and an adaptive follower, we introduce a three-agent framework with modulated leadership, where a passive modulator dynamically aligns with either the leader or the follower, thereby influencing the leader’s strategic choices. The main contribution of this work is the application of the proposed framework to a critical biomedical challenge: tumor growth control. Particularly, we develop a novel tumor-immune model capturing evolution, interactions, and environmental factors. We then frame tumor control as a strategic problem involving the physician (leader), the evolving tumor (follower), and the immune system, which can alternatively support treatment and promote tumor survival. Leveraging this framework, we design combination therapies that dynamically adjust chemotherapy and anti-PD1 immunotherapy. Numerical simulations demonstrate the potential of the proposed approach in oncology and, more broadly, in other fields involving dynamic, multi-agent interactions.