{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T14:04:46Z","timestamp":1778681086300,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T00:00:00Z","timestamp":1749513600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005323","name":"North Minzu University","doi-asserted-by":"publisher","award":["ZDZX201805"],"award-info":[{"award-number":["ZDZX201805"]}],"id":[{"id":"10.13039\/501100005323","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005323","name":"North Minzu University","doi-asserted-by":"publisher","award":["NXYLXK2017B09"],"award-info":[{"award-number":["NXYLXK2017B09"]}],"id":[{"id":"10.13039\/501100005323","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005323","name":"North Minzu University","doi-asserted-by":"publisher","award":["2022AAC03287"],"award-info":[{"award-number":["2022AAC03287"]}],"id":[{"id":"10.13039\/501100005323","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Governance and Social Management Research Center of Northwest Ethnic Regions, the First-Class Disciplines Foundation of Ningxia","award":["ZDZX201805"],"award-info":[{"award-number":["ZDZX201805"]}]},{"name":"Governance and Social Management Research Center of Northwest Ethnic Regions, the First-Class Disciplines Foundation of Ningxia","award":["NXYLXK2017B09"],"award-info":[{"award-number":["NXYLXK2017B09"]}]},{"name":"Governance and Social Management Research Center of Northwest Ethnic Regions, the First-Class Disciplines Foundation of Ningxia","award":["2022AAC03287"],"award-info":[{"award-number":["2022AAC03287"]}]},{"name":"Ningxia Youth Talent Support Program","award":["ZDZX201805"],"award-info":[{"award-number":["ZDZX201805"]}]},{"name":"Ningxia Youth Talent Support Program","award":["NXYLXK2017B09"],"award-info":[{"award-number":["NXYLXK2017B09"]}]},{"name":"Ningxia Youth Talent Support Program","award":["2022AAC03287"],"award-info":[{"award-number":["2022AAC03287"]}]},{"name":"Leading Talent Support Program of North Minzu University and the Natural Science Foundation of Ningxia","award":["ZDZX201805"],"award-info":[{"award-number":["ZDZX201805"]}]},{"name":"Leading Talent Support Program of North Minzu University and the Natural Science Foundation of Ningxia","award":["NXYLXK2017B09"],"award-info":[{"award-number":["NXYLXK2017B09"]}]},{"name":"Leading Talent Support Program of North Minzu University and the Natural Science Foundation of Ningxia","award":["2022AAC03287"],"award-info":[{"award-number":["2022AAC03287"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper proposes a two-stage stochastic non-cooperative game model to solve relief supplies procurement and distribution optimization of multiple rescue organizations in crisis rescue. Rescue organizations with limited budgets minimize rescue costs through relief supply procurement, storage, and transportation in an uncertain environment. Under a mild assumption, we establish the existence and uniqueness of the equilibrium point and derive the optimality conditions by using the duality theory, characterizing the saddle point in the Lagrange framework. The problem is further reformulated as a constraint system governed by Lagrange multipliers, and its optimality is characterized by the Karush\u2013Kuhn\u2013Tucker condition. The economic interpretation of the multipliers as shadow prices is elucidated. Numerical experiments verify the effectiveness of the model in cost optimization in crisis rescue scenarios.<\/jats:p>","DOI":"10.3390\/axioms14060456","type":"journal-article","created":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T06:48:56Z","timestamp":1749538136000},"page":"456","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Stochastic Nash Equilibrium Problem for Crisis Rescue"],"prefix":"10.3390","volume":"14","author":[{"given":"Cunlin","family":"Li","sequence":"first","affiliation":[{"name":"Ningxia Key Laboratory of Intelligent Information and Big Data Processing, North Minzu University, Wenchang Street, Yinchuan 750021, China"},{"name":"School of Mathematics and Information Science, North Minzu University, Wenchang Street, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiyan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, North Minzu University, Wenchang Street, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"106458","DOI":"10.1016\/j.cie.2020.106458","article-title":"A two-stage stochastic programming framework for evacuation planning in disaster responses","volume":"145","author":"Wang","year":"2020","journal-title":"Comput. 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