reviews: 
Comments: The paper presents a new model called Equilibrium State Evaluation (ESE) for predicting the collective behavior of interconnected multi-system structures. ESE is designed to manage systems with multiple interdependent subsystems, each affecting and being affected by the others. The goal of the model is to determine an equilibrium state for the entire multi-system by applying concepts from Nash equilibrium along with a zero-sum constraint.
In ESE, each subsystem is given a payoff function that assesses its "benefit" based on the states of all the subsystems. The equilibrium state is defined as the configuration in which each subsystem achieves its optimal state (maximum payoff), given the conditions of the others, consistent with Nash equilibrium principles. The zero-sum constraint enforces a balance across subsystems, ensuring that any gain in one subsystem is countered by a loss in another, thereby maintaining overall stability.

To estimate the long-term equilibrium trends of the system, the model utilizes cointegration, an econometric technique that identifies shared equilibrium paths among interdependent variables. Additionally, an equilibrium index (EI) is introduced to quantify how closely the system's current state aligns with equilibrium, providing a measure of system stability.

The paper validates the ESE model using multi-regional COVID-19 transmission data, demonstrating that it achieves competitive prediction accuracy while maintaining linear computational complexity. The authors contend that ESE's equilibrium-based approach enables scalable and efficient prediction of complex multi-system dynamics, making it potentially valuable in high-dimensional environments where subsystems are closely interconnected.
Strengths: The primary strength of this paper lies in its effort to apply a novel conceptual framework for modeling interconnected systems. It draws on ideas from Nash equilibrium, zero-sum constraints, and cointegration. This approach offers an interesting perspective by reinterpreting multi-system predictions through game-theoretic concepts, particularly utilizing Nash equilibrium principles and payoff functions to illustrate the mutual influences among subsystems. While these concepts are mainly interpretative, they provide a fresh way to think about multi-system interactions, which could have significant implications for interdisciplinary applications across fields such as economics, epidemiology, and regional forecasting. The paper demonstrates the practical feasibility of the proposed model, ESE, by applying it to a real-world COVID-19 dataset. This highlights the model's potential to capture complex interdependencies and offers insights into multi-region transmission dynamics. The presentation is generally clear, with structured explanations of equilibrium concepts and the role of each component, such as the equilibrium index. Furthermore, if the model’s empirical performance in terms of accuracy and computational efficiency is further substantiated, it suggests that ESE could serve as a competitive alternative for high-dimensional applications where subsystems are highly interconnected.
Weaknesses: A significant weakness of the paper lies in its insufficient support for the claim that the system will reach a true equilibrium state. While the model employs Nash equilibrium and zero-sum constraints to define static equilibrium conditions, these concepts alone do not provide a mechanism to ensure that the system will naturally progress toward equilibrium over time. Without a dynamic framework or time-dependent interactions—such as differential equations or explicit stability conditions—there is no mathematical basis for assuming that the system will move from an arbitrary initial state toward equilibrium. Furthermore, although the paper incorporates cointegration in its training process, this is a statistical technique that assumes the existence of a long-term equilibrium relationship rather than actively guiding the system toward it. This distinction weakens the model’s theoretical foundation, as it does not establish how equilibrium is achieved dynamically. Another critical limitation is that the equilibrium definition based on Nash equilibrium and payoff functions does not exclude the possibility of stable oscillatory behavior. This means the system could theoretically settle into a stable oscillation rather than converging to a true steady state. Additionally, the ESE model shows strong mathematical similarities to traditional multi-compartment models, which use conservation principles to maintain balance across subsystems. While the introduction of game-theoretic concepts like Nash equilibrium and payoff functions offers a fresh interpretive layer, it does not constitute a substantial mathematical advancement over established multi-compartment models. To improve the model’s robustness, the authors would need to provide a formal analysis of convergence conditions, explicitly address the exclusion of oscillatory states, and better differentiate their approach from conventional multi-compartment modeling frameworks.
Recommendation: 5
Reviewer confidence: 4
Soundness/correctness: 2
Clarity: 3