Integrated Multi-system Prediction via Equilibrium State Evaluation

Authors: Beinan Xu, Andy Song, Jiti Gao
Primary area: learning on time series and dynamical systems
Keywords: Multi-system, Equilibrium, Prediction

Abstract:
This study presents a new paradigm of prediction, Equilibrium State Evaluation (ESE), which excels in multi-system prediction where systems interact with each other and every system needs its own prediction. Unlike mainstream prediction approaches, ESE views each system as an integral part under one structure and predicts all systems simultaneously in one go.  It evaluates these systems' equilibrium state by analyzing the dynamics of their attributes in a holistic manner, instead of treating each system as an individual time series. The effectiveness of ESE is verified in synthetic and real world scenarios, in particular COVID-19 transmission, where each geographic region can be viewed as a system.  So cases spreading across regions against the medical competency and demographic traits of these regions can be considered as an equilibrium problem rather than a time series problem.  Extensive analysis and experiments show that ESE is linear in complexity and can be 10+ times faster than SOTA methods, yet achieving comparable or better prediction accuracy.  More importantly, ESE can be integrated with these prediction methods to achieve both high accuracy and high speed, making it a powerful prediction mechanism, especially for scenarios that involve multiple systems. When the dimensionality of the multi-system increases, e.g. more systems joining, the advantages of ESE would become even more apparent.

Decision: reject
Reviewer scores: 1, 5, 1

Reviews and discussion:
<thread>
creation date:
1730519188366

thread_type:
official_review

<thread_object>
creator: reviewer_i1ob

<content>
summary:
This paper presents a new prediction approach called Equilibrium State Evaluation for handling multi-system prediction problems where multiple interacting systems must be predicted simultaneously. Where the authors treat each system as independent time series and consider multiple-interacting systems as components of a "super-system". The method maintains an Equilibrium Index (EI) that measures the distance between the current state and the equilibrium state.

strengths:
There aren't any significant strengths in the paper

weaknesses:
- The paper is confusing to follow. The introduction is poorly written. The mathematical notation is very dense without sufficient explanation. 
- Complex method workflow is not fully explained
- It is not apparent to me why proportions must follow zero-sum rule. Doesn't explain how to handles growing-shrinking total system values.
- It is unclear why is specific normalization is chosen λᵢ,ⱼ = (αᵢ,ⱼ - mean(atⱼ)) / (max(atⱼ) - min(atⱼ)), why is this normalization beneficial as compared to other ways to normalize? 
- Algorithm 1 is vaguely described, and testing and training process aren't well defined. It doesn't map succinctly with the figure 2 of the paper.

questions:
The authors could provide a more detailed algorithm description and better justification of experimental and methodological design
</content>
</thread_object>

</thread>
<thread>
creation date:
1730664362150

thread_type:
official_review

<thread_object>
creator: reviewer_ddnh

<content>
summary:
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.

questions:
1. Convergence Mechanism: Since Nash equilibrium and the zero-sum constraint do not inherently ensure convergence to equilibrium, does the model include any dynamic process to guide the system toward equilibrium from any initial state? Would the authors consider adding time-based dynamics or stability conditions to ensure convergence?
2. Exclusion of Oscillatory and Chaotic Behavior: In traditional multi-compartment models, stability conditions can be defined to ensure convergence to equilibrium. However, the equilibrium condition in this paper does not seem to exclude stable oscillatory or chaotic behavior. Could the authors clarify if such behaviors are possible in their model? If so, how might these be addressed to ensure the system reaches a steady-state equilibrium?
3. Role of Cointegration: Cointegration is typically used to identify statistical relationships between non-stationary variables, assuming they share a long-term equilibrium path. Is cointegration in this model intended merely as a statistical tool for parameter estimation, or is it expected to actively drive the system toward equilibrium? Could the authors clarify how cointegration contributes to achieving or maintaining equilibrium in the system?
4. Distinction from Multi-Compartment Models: The ESE model shows strong similarities to multi-compartment models, which also use conservation principles to maintain balance. Beyond the conceptual framing of Nash equilibrium and payoff functions, is there a fundamental structural difference that distinguishes ESE from these existing models? Specifically, does ESE offer any new insights or capabilities that go beyond what is possible with traditional multi-compartment approaches?
5. Computational Complexity: The paper claims linear computational complexity but does not provide a formal proof. Could the authors offer a theoretical complexity analysis to substantiate this claim? Additionally, how does the model handle scenarios with extensive interactions or feedback loops between subsystems, which might increase computational demands?
6. Applicability to Non-Stationary or Oscillatory Systems: If the system exhibits non-stationary or oscillatory behavior, how reliable is the equilibrium index (EI) as a measure of stability? Would the model need adjustments to handle such cases, or does it implicitly assume stationarity and convergence to a single steady state?
</content>
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</thread>
<thread>
creation date:
1730677731048

thread_type:
official_review

<thread_object>
creator: reviewer_wbf8

<content>
summary:
I believe this paper either has a high chance of being generated by a language model, or was rushed into submission. In both cases, I recommend a rejection.

strengths:
-

weaknesses:
**I stopped reviewing this paper, as there were multiple issues. The objective and the problem setting are not clear, assumptions are not clearly stated, related work are missing, and writing is very poor.**

The definition of a "system" in the first place is not clear. The authors casually drop "properties" of this said system/multi-system, which vaguely correspond to some concepts, such as Nash equilibria, but not clear at all if those are assumptions/setting the authors consider. 

The paper is almost impossible to read, with abuse of notation used absolutely for no reason. It is not really clear what problem the paper is dealing with and it is very poorly formulated throughout.

It is not clear how the COVID example fits into the Constraints 1 & 2. Why Eq. (4) needs to hold for the COVID case for instance?

Related work section mentions mostly datasets and equilibrium examples from different domains, but I could not spot any ML related work that consider a similar problem. 

Why do you define ${\cal M} {\cal S}$ to be the sum of the target variables in systems, whereas it was first defined as the set of systems? The notation should be improved across the board, but this is just one example.

Why does Eq. (1) hold *in general*? Is that a setting you consider?

You cannot refer to Constraint 1 when defining Constraint 1 itself?

Line 158 - Please refer to where did you describe your ESE method as claimed.

What is the takeaway from Figure 1 exactly?

The connection to the Nash equilibrium is never formally introduced or motivated. It is not clear where  Lemma 1 comes from. There are no proper citations as well.

questions:
-
</content>
</thread_object>

</thread>
<thread>
creation date:
1732149252202

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Paper updated according to the comments

comment:
Dear Reviewers, thank you for your time and effort, especially for these very insightful comments regarding convergence, cointegration and oscillation.  

***Please examine our responses and the updated paper, and reconsider your view on this study***.  Your concerns have been carefully addressed, with detailed clarifications and further explanations in reply to your comments.  The corresponding changes in the updated PDF file are ***highlighted in red***.  

Please note, this study is NOT incremental research, but aims to open a new paradigm for prediction, leveraging game theory and equilibrium which are not typical predictive models.  There is no direct prior work that we are aware of. Hence it may appear to lack references in ML.  However, this study does have many related works in game theory, econometrics and applied math (see our long list of bibliography). 

ESE's novelty and significance first lie in the new mathematical formulation of ESE, which turns equilibrium estimation of zero-sum games into predictive modelling.  As a result, there is no need to repeat the prediction for multiple systems of the same type, if they can be viewed together as a ``**multi-system**''.  

Note this approach does not work for every kind of time series prediction, as discussed at the end of the main paper.  Because most time series tasks/benchmarks cannot be viewed as a zero-sum game, so equilibrium does not apply.  Also, they do not have attribute data other than the target variables, making them unsuitable for estimating the equilibrium states, even if equilibrium can apply.  However the target problem studied in this paper, ``**multi-system prediction**'' can often be observed in the real world, for example, the spreading of viruses like COVID over multiple regions, the business activities of multiple companies under one market, and the economic development of multiple countries of one continent.  Hence the impact and the value of this research are foreseeable and significant. 

ESE's novelty and significance can be verified by its effectiveness, demonstrated through extensive experiments on a wide range of scenarios, presented in 17 tables (Tables 1-9, 12-19).  ESE can be at least comparable in prediction accuracy, in comparison with 12 SOTA time series prediction models.  The highlight is that ESE can be combined with these models to achieve the best accuracy in most cases.  

Another key advantage of ESE is its **simplicity, high speed and low complexity**.  It does not need to be repeated, no matter if there are 79 systems or 320 systems or more.  The complexity is as low as linear.  The prediction for all systems can be obtained in one run.  ***The speed increase can be as large as 70+ times (bold in Table 3).***  

In addition, we have studied other methods before ESE, e.g. differential equations, non-linear regression and Markov models, on the targeted multi-system prediction task.  None is suitable or as effective and efficient as ESE.  We would love to hear suggestions on alternative methods, that may be better than ESE for multisystem prediction tasks.  Our data are openly accessible.  We welcome separate studies.

In short, we view this work as a novel and significant advancement in predictive modelling, as well as an expansion of game theory applications, deserving of decent recognition in the field.

Thank you in advance 
(**Authors of 12376**)
</content>
</thread_object>

</thread>
<thread>
creation date:
1732546333878

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Please comment on our responses as early as possible.  Thank you in advance.

comment:
Dear Reviewers, could you please comment on our responses at your earliest convenience?  Our responses have been uploaded since last Thursday, 6 days ago.  We are eager to address any further questions or concerns before the discussion period deadline, which is fast approaching.

Looking forward to hearing from you.

Thank you in advance (**Authors of 12376**)
</content>
</thread_object>

</thread>


Metareview:
metareview:
The authors propose a Equilibrium State Evaluation (ESE) to model interconnected subsystems. The model uses econometrics techniques to model long term equilibrium trends. Reviewers have noted several methodological and foundational issues particularly in terms of assumptions made of existence of such an equilibrium and the ability of the method to achieve the equilibrium. Considering also the presentation issues, the paper would require significant revisions and another round of peer review before being ready for publication. As such despite the authors partially addressing the concerns raised, I am recommending a reject.

Please note that one of the reviewers made the claim that the paper is generated by a generative AI. I have not assumed it as such while going over the paper and reviewer comments.