Entanglement: Balancing Punishment and Compensation, Repeated Dilemma Game-Theoretic Analysis of Maximum Compensation Problem for Bypass and Least Cost Paths in Fact-Checking, Case of Fake News with Weak Wallace's Law

This research note is organized with respect to a novel approach to solving problems related to the spread of fake news and effective fact-checking. Focusing on the least-cost routing problem, the discussion is organized with respect to the use of Metzler functions and Metzler matrices to model the dynamics of information propagation among news providers. With this approach, we designed a strategy to minimize the spread of fake news, which is detrimental to informational health, while at the same time maximizing the spread of credible information. In particular, through the punitive dominance problem and the maximum compensation problem, we developed and examined a path to reassess the incentives of news providers to act and to analyze their impact on the equilibrium of the information market. By applying the concept of entanglement to the context of information propagation, we shed light on the complexity of interactions among news providers and contribute to the formulation of more effective information management strategies. This study provides new theoretical and practical insights into issues related to fake news and fact-checking, and will be examined against improving informational health and public digital health.This paper is partially an attempt to utilize "Generative AI" and was written with educational intent. There are currently no plans for it to become a peer-reviewed paper.