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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: 15 December 2026 | Viewed by 8689

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Special Issue Editor

Prof. Dr. Joanna Olbryś

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Guest Editor

Faculty of Computer Science, Bialystok University of Technology, Wiejska Street 45A, 15-351 Bialystok, Poland
Interests: information theory; econometrics; statistics; empirical finance; financial economics; operations research in finance; computational economics; stock market microstructure; computing in social science
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Special Issue Information

Dear Colleagues,

Following the success of the three volumes of this Special Issue, with this fourth volume, we aim to provide a forum for the presentation of entropy-based applications in economics, finance, and management studies. The concept of entropy originates from thermodynamics, but it is utilized in many research fields to characterize the complexity of systems and to investigate the information content of probability distributions. Entropy is a general measure, and therefore many definitions and applications have been proposed in the literature.

Areas of interest include, but are not limited to, the following topics:

Entropy-based applications in portfolio selection, asset pricing, and risk management;
Entropy measures as indicators for systematic risk and market informational efficiency;
Entropy optimization approaches in economics and finance;
Entropy-based applications in market microstructure research;
Shannon theory in multi-criteria decision-making methods with applications to economic and management problems;
Structural entropy in network-based applications in economics, finance, and management;
Entropy measures in econophysics.

Theoretical and empirical contributions addressing any of the aforementioned topics are especially welcome.

Prof. Dr. Joanna Olbryś
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

information entropy
fuzzy entropy
maximum entropy
copula entropy
structural entropy
transfer entropy
permutation entropy
slope entropy
approximate entropy
sample entropy

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Related Special Issues

Entropy-Based Applications in Economics, Finance, and Management in Entropy (14 articles)
Entropy-Based Applications in Economics, Finance, and Management II in Entropy (5 articles)
Entropy-Based Applications in Economics, Finance and Management, 3rd Edition in Entropy (7 articles)

Published Papers (10 papers)

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Research

24 pages, 10218 KB

Open AccessArticle

Bank Resolution Trade-Offs Under Coupled Liquidity and Credit Risks: An Agent-Based Network Analysis of Systemic Stability

by Qianqian Gao, Hongjie Pan, Yinglin Liu and Naixi Chen

Entropy 2026, 28(6), 618; https://doi.org/10.3390/e28060618 - 31 May 2026

Viewed by 221

Abstract

Prolonged downturns in the global economy have simultaneously increased banks’ credit risk exposures and intensified the need for effective liquidity management. This study develops a dynamic agent-based financial network comprising banks, depositors, firms, and the central bank to examine trade-offs in bank resolution under coupled liquidity and credit risks from the perspective of systemic stability. The simulation results show that, for liquidity risk management, when banks adopt the asset-sale strategy, both default probability and expected returns in the banking system exhibit a nonlinear pattern: they first decline and then rise as the asset depreciation ratio increases. Furthermore, at moderate levels of asset depreciation, the asset-sale strategy helps preserve heterogeneity within the banking system, thereby preventing excessive risk concentration, and performs better than the liability-expansion strategy. Regarding credit risk resolution, the debt-relief strategy significantly improves systemic stability, whereas the effectiveness of the debt-extension strategy depends critically on liquidity management conditions. Under liability-expansion scenarios, default risk initially declines but later rises as debt maturity is extended, whereas expected returns move in the opposite direction. Under asset-sale conditions, the debt-extension strategy enhances systemic stability only when the allowable number of debt extensions is sufficiently high. The analysis of strategic trade-offs indicates that combining the debt-relief strategy with the asset-sale strategy generates a positive synergistic effect and strengthens systemic resilience, whereas the interaction between the debt-extension and asset-sale strategies produces offsetting effects. These findings offer useful implications for banks and regulators in designing coordinated and adaptive frameworks for risk resolution and systemic stability. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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18 pages, 1091 KB

Open AccessArticle

Informational Content of the VIX Index: Dynamic Entropy Approach

by Joanna Olbryś and Dawid Toczydłowski

Entropy 2026, 28(5), 528; https://doi.org/10.3390/e28050528 - 6 May 2026

Viewed by 347

Abstract

The aim of this study is to thoroughly assess the informational content of the CBOE Volatility Index^® (VIX^® Index) in the context of various turbulent periods. The VIX Index is especially important from an investor perspective. It is often referred to as the “investor fear gauge”, because its level tends to spike during periods of market turmoil and other extreme events. Therefore, this index significantly differs from other market indices and financial instruments. Information theory and normalized Shannon entropy, combined with a rolling-window dynamic approach, are used to explore the evolution of the VIX Index over time. The research hypothesis states that the informational content of the VIX Index varies substantially across periods affected by crucial events. To verify this hypothesis, three important periods of the twenty-first century are analyzed: (1) the Global Financial Crisis, (2) the COVID-19 pandemic outbreak, and (3) the period covering the sub-periods before and after the Donald Trump’s Presidential Inauguration. The results provide no reason to reject the research hypothesis. The empirical findings show that the entropy values appear to be quite sensitive to the choice of discretizaton procedure. However, this evidence is consistent with the existing literature. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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23 pages, 1745 KB

Open AccessArticle

Landauer-Based Economic Temperature in Blockspace Markets: Evidence from Bitcoin and Ethereum

by Michael Zouari, Ilan Alon and Zeev Shtudiner

Entropy 2026, 28(5), 508; https://doi.org/10.3390/e28050508 - 1 May 2026

Viewed by 667

Abstract

The Landauer principle motivates the definition of economic temperature as the monetary price of processing a bit irreversibly. No empirical test of this definition exists in transparent fee markets. This paper fills that gap using daily Bitcoin and Ethereum data, constructing canonical thermodynamic state variables and evaluating five diagnostic layers: state variable behavior, Maxwell-type integrability, Carnot-style efficiency bounds, nonlinear regime separation, and structural break sensitivity to protocol events. Bitcoin’s log-temperature behaves as a persistent mean-reverting process with an AR(1) coefficient of 0.97 and a half-life of 21 days; Ethereum is highly persistent, with weaker formal evidence of stationarity than Bitcoin. Maxwell integrability is frequency-dependent: Bitcoin passes all four relations at monthly frequency, whereas Ethereum passes two of four. Carnot-style evidence is the strongest: realized fee extraction efficiency stays well below the implied bound, with daily compliance exceeding 97% on both chains. Structural breaks around Bitcoin ordinals, EIP-1559, the merge, and Shanghai confirm that protocol changes reorganize the temperature relation. The thermodynamic framework provides structure that standard fee market analysis does not, including a first principles efficiency bound and a state space coherence test. The findings provide partial, frequency-dependent, and chain-specific empirical support for a Landauer-based thermodynamic description of blockspace markets. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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23 pages, 1624 KB

Open AccessArticle

Measurement of China’s “External Market Provider” Role: Trade-Margin Decomposition and Gravity Determinants

by Manru Zhao and Yujia Lu

Entropy 2026, 28(5), 504; https://doi.org/10.3390/e28050504 - 30 Apr 2026

Viewed by 467

Abstract

This paper measures China’s role as an “external market provider” by quantifying, for 168 source countries during 2001–2022, the share of each country’s exports absorbed by China and decomposing that share into extensive (product coverage), quantity, and price margins using the Hummels–Klenow framework. To characterize destination-market concentration, we construct an HHI-based network diversification indicator from export-destination shares and interpret it from a complementary information-theoretic perspective, where higher concentration corresponds to lower diversification and stronger dependence. We document the dynamics of China’s market-provision role and estimate an extended gravity-type model with country- and year-fixed effects. The results show that China’s external market-provider role expanded markedly after WTO accession, with growth driven mainly by the quantity margin and, after 2018, increasingly supported by the price margin. Economic proximity and similarity in global value-chain position are associated with stronger China-absorption shares, while greater destination concentration relative to China is associated with lower China-absorption shares. Free trade agreements are linked to stronger, more extensive, and larger margins. Robustness checks based on lagged covariates, additional controls, higher-dimensional fixed effects, Tobit estimation, and winsorization support the main findings. Overall, the paper provides a replicable framework for measuring destination-market pull and shows how China’s import-side role varies across products, regions, and development groups, while using the information-theoretic perspective as a supplementary interpretation of diversification patterns rather than as a separate empirical tool. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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24 pages, 1336 KB

Open AccessArticle

Haken-Entropy-Based Analysis of the Synergy Among Financial Support, Technological Innovation, and Industrial Upgrading

by Yue Zhang, Jinchuan Ke and Jingqi He

Entropy 2026, 28(4), 465; https://doi.org/10.3390/e28040465 - 17 Apr 2026

Viewed by 518

Abstract

This study reveals the internal mechanism of the synergetic evolution of financial support, technological innovation, and industrial upgrading from the perspective of system synergy. It aims to provide a theoretical basis and reference for promoting benign interactions among these elements, thereby driving high-quality economic development. During the research process, an evaluation indicator system was constructed based on China’s industrial development data, utilizing the entropy method to determine indicator weights and the Haken model to analyze synergy effects. In a methodological innovation, this study identifies the system’s order parameters to derive the potential function. Through this approach, it systematically analyzes the dynamic evolution characteristics and synergetic mechanisms of the composite system. The research results indicate that the three systems have formed a mutually promoting and closely coupled compound synergetic mechanism, rather than following a single linear transmission path. The overall synergy level presents a medium-to-low development trend, following an asymmetric U-shaped evolution trajectory that first decreases and then slowly recovers. Furthermore, the degree of synergy exhibits an inverse relationship with the volatility of the subsystems, suggesting that the stability of synergy is highly susceptible to external forces and remains in a state of dynamic flux. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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82 pages, 6808 KB

Open AccessArticle

Agentic Finance: An Adaptive Inference Framework for Bounded-Rational Investing Agents

by Samuel Montañez Jacquez, John H. Clippinger and Matthew Moroney

Entropy 2026, 28(3), 321; https://doi.org/10.3390/e28030321 - 12 Mar 2026

Cited by 2 | Viewed by 1650

Abstract

We propose Adaptive Inference, a portfolio management framework extending Active Inference to non-stationary financial environments. The framework integrates inference, control, and execution under endogenous uncertainty, modeling investment decisions as coupled dynamics of belief updating, preference encoding, and action selection rather than optimization over fixed objectives. In this approach, portfolio behavior is governed by the expected free energy (EFE) minimization, showing that classical valuation models emerge as limiting cases when epistemic components vanish. Using train–test evaluation on the ARKK Innovation ETF (2015–2025), we identify a Passivity Paradox: frozen belief transfer outperforms naive adaptive learning. A Professional Agent achieves a Sharpe ratio of 0.39 while its adaptive counterpart degrades to

- 0.28

, reflecting belief contamination when learning from policy-dependent signals. Crucially, the architecture is not designed to generate alpha but to perform endogenous risk management that mitigates overtrading under regime ambiguity and distributional shift. Adaptive Inference Agents maintain long exposure most of the time while tactically reducing positions during high-entropy periods, implementing uncertainty-aware passive investing. All agents reduce realized volatility relative to ARKK Buy-and-Hold (43.0% annualized). Cross-asset validation on the S&P 500 ETF (SPY) shows that inference-guided risk shaping achieves a positive Entropic Sharpe Ratio (ESR), defined as excess return per unit of informational work, thereby quantifying the economic value of information under thermodynamic constraints on inference. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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19 pages, 325 KB

Open AccessArticle

The Impact of Green FinTech Promote Corporate Carbon Neutrality: Evidence from the Perspective of Financing Incentives and Scale Quality

by Lei Zhuang and Chuang Wu

Entropy 2026, 28(1), 6; https://doi.org/10.3390/e28010006 - 20 Dec 2025

Cited by 1 | Viewed by 824

Abstract

As an in-depth integration of green capital chains and technological innovation chains, green fintech provides strong support for enterprises in promoting green and low-carbon development and achieving carbon neutrality. Based on relevant data from Chinese listed companies between 2014 and 2023, this study constructs indices for green fintech development and corporate carbon neutrality to empirically examine the impact of green fintech on corporate carbon neutrality. Benchmark regression results show that green fintech exerts a significantly positive effect on corporate carbon neutrality. A mediation analysis of financing incentives indicates that alleviating corporate financing constraints and reducing financial distress are effective pathways through which green fintech facilitates carbon neutrality. Furthermore, a moderating effect analysis reveals that green fintech plays a more pronounced role in enhancing carbon neutrality for enterprises with higher audit quality and larger operational scales. Accordingly, policy recommendations are proposed, focusing on establishing a green fintech service-sharing platform, providing targeted policy support, and improving carbon information disclosure mechanisms. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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19 pages, 1866 KB

Open AccessArticle

A Cognitive Perspective on Information Frictions in Labor Markets

by Zeqiang Zhang and Ruxin Chen

Entropy 2025, 27(12), 1182; https://doi.org/10.3390/e27121182 - 21 Nov 2025

Viewed by 781

Abstract

During the Great Recession, labor markets often exhibit a slow unemployment recovery and persistent outward shifts in the Beveridge curve, which suggests a decline in the efficiency of the job-matching process. While it is often explained by worker search intensity, we argue that the direction of search behavior also matters by proposing a stylized theoretical model based on the Free Energy Principle. Through modeling agents who actively divide their effort between applying for jobs and learning about the market’s new state, our framework shows that agents endogenously shift effort from applications to learning when their uncertainty is high. Building on this micro-foundation, we design a macroeconomic model where matching efficiency is no longer an external parameter but is instead governed by two cognitive factors: the share of unemployed workers with misaligned beliefs and the average learning effort of the informed. Simulation results show that a structural shock will divert effort to learning and depress matching by creating widespread uncertainty, and the subsequent slow recovery is governed by the realignment of collective beliefs. Our work provides a cognitive explanation for this observed persistence of unemployment and the shift of the Beveridge curve. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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28 pages, 986 KB

Open AccessArticle

Unlocking Carbon Emissions and Total Factor Productivity Nexus: Causal Moderation of Ownership Structures via Entropy Methods in Chinese Enterprises

by Ruize Cai, Jie You and Minho Kim

Entropy 2025, 27(10), 1048; https://doi.org/10.3390/e27101048 - 9 Oct 2025

Cited by 2 | Viewed by 1155

Abstract

Amidst global imperatives for environmental sustainability, this study investigates the nexus between carbon emissions reduction (CER), ownership structures, and total factor productivity (TFP) in Chinese enterprises—recognized as vital economic drivers facing carbon emissions pressures. Based on the theoretical frameworks of innovation offsets, agency cost theory, and upper echelons theory, with data from CSMAR (2009–2023), we proposed a positive effect of CER on TFP while examining the moderating roles of ownership structure metrics: chairman shareholding ratio, manager shareholding ratio, and ownership–control separation ratio. TFP estimation employed dual approaches: mean consolidation (TFP-Mean) and entropy weighting (TFP-Entropy) methods. The results confirmed CER exerts significantly positive impacts on TFP, with ownership structures demonstrating statistically significant yet directionally heterogeneous moderation effects. Heterogeneity analysis reveals heightened TFP sensitivity to carbon emission initiatives among private enterprises, foreign-owned enterprises, and small enterprises. Notably, the entropy weighting method exhibits substantial comparative advantages in TFP measurement. These findings underscore that advancing TFP necessitates simultaneously optimizing carbon emissions efficiency and ownership governance. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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34 pages, 9259 KB

Open AccessArticle

Dynamic Evolution and Convergence of the Coupled and Coordinated Development of Urban–Rural Basic Education in China

by Fangyu Ju, Qijin Li and Zhiyong Chen

Entropy 2025, 27(10), 1021; https://doi.org/10.3390/e27101021 - 28 Sep 2025

Viewed by 1006

Abstract

Understanding the coupled and coordinated development of China’s urban and rural basic education systems is crucial for fostering their interaction and synergistic growth. Using China’s provincial panel data from 2011 to 2023, this study measures the coupled and coordinated development level of urban–rural basic education (CCD-URBE) via the entropy weight method, G1-method and coupling coordination degree model. On this basis, the Dagum Gini coefficient decomposition method, traditional and spatial Markov chain models, as well as convergence test models are employed for empirical research. The results show that: (1) During the study period, the CCD-URBE across the nation and the four major regions improves significantly. Both intra-regional and inter-regional disparities show a consistent downward trend. Inter-regional disparities are the main source of the overall disparities, and the contribution rate of transvariation density to the overall disparities exhibits the most significant increase. (2) The CCD-URBE demonstrates strong stability, as most regions tend to maintain their original CCD-URBE grades. Meanwhile, neighborhood grades moderate the local transition probability significantly. Neighborhoods with high CCD-URBE promote the upward improvement of the local CCD-URBE, while those with low CCD-URBE inhibit it. (3) The CCD-URBE across the nation and the four major regions shows obvious trends of σ-convergence, absolute β-convergence, and conditional β-convergence. The central region, which has lower CCD-URBE, exhibits higher convergence speed. Based on these findings, targeted policy implications are derived. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Economics, Finance, and Management, 4th Edition)

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