{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T15:12:56Z","timestamp":1767193976211,"version":"3.48.0"},"reference-count":65,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,26]],"date-time":"2025-12-26T00:00:00Z","timestamp":1766707200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate the fundamental trade-off between entropy and the Gini index within income distributions, employing a stochastic framework to expose deficiencies in conventional inequality metrics. Anchored in the principle of maximum entropy (ME), we position entropy as a key marker of societal robustness, while the Gini index, identical to the (second-order) K-spread coefficient, captures spread but neglects dynamics in distribution tails. We recommend supplanting Lorenz profiles with simpler graphs such as the odds and probability density functions, and a core set of numerical indicators (K-spread K2\/\u03bc, standardized entropy \u03a6\u03bc, and upper and lower tail indices, \u03be,\u00a0\u03b6) for deeper diagnostics. This approach fuses ME into disparity evaluation, highlighting a path to harmonize fairness with structural endurance. Drawing from percentile records in the World Income Inequality Database from 1947 to 2023, we fit flexible models (Pareto\u2013Burr\u2013Feller, Dagum) and extract K-moments and tail indices. The results unveil a concave frontier: moderate Gini reductions have little effect on entropy, but aggressive equalization incurs steep stability costs. Country-level analyses (Argentina, Brazil, South Africa, Bulgaria) link entropy declines to political ruptures, positioning low entropy as a precursor to instability. On the other hand, analyses based on the core set of indicators for present-day geopolitical powers show that they are positioned in a high stability area.<\/jats:p>","DOI":"10.3390\/e28010035","type":"journal-article","created":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T06:25:51Z","timestamp":1766989551000},"page":"35","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Trade-Off Between Entropy and Gini Index in Income Distribution"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6226-0241","authenticated-orcid":false,"given":"Demetris","family":"Koutsoyiannis","sequence":"first","affiliation":[{"name":"Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical University of Athens, Heroon Polytechneiou 5, 15772 Zographou, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0555-4021","authenticated-orcid":false,"given":"G.-Fivos","family":"Sargentis","sequence":"additional","affiliation":[{"name":"Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical University of Athens, Heroon Polytechneiou 5, 15772 Zographou, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,26]]},"reference":[{"key":"ref_1","unstructured":"(2025, December 22). 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