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However, to make such a selection fair for every player, we have to rely on an unrealistic assumption, that is, the availability of a trusted center that does not induce any bias among players. In this paper, to ensure fairness for every player even in the process of equilibrium selection, we propose a new equilibrium selection problem, named the upper-level v-GNEP. The proposed upper-level v-GNEP is formulated as a v-GNEP defined over the solution set of the lower-level v-GNEP. We also present an iterative algorithm, of guaranteed convergence to a solution of the upper-level v-GNEP, as an application of the hybrid steepest descent method to a fixed point set characterization of the solution of the lower-level v-GNEP. Numerical experiments illustrate the proposed equilibrium selection and algorithm.<\/jats:p>","DOI":"10.1007\/s10957-025-02887-y","type":"journal-article","created":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T05:06:42Z","timestamp":1766552802000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hierarchical Variational Inequality Problem for Noncooperative Game-Theoretic Selection of Generalized Nash Equilibrium"],"prefix":"10.1007","volume":"208","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-3603-7661","authenticated-orcid":false,"given":"Shota","family":"Matsuo","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0006-7022-6146","authenticated-orcid":false,"given":"Keita","family":"Kume","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6563-7526","authenticated-orcid":false,"given":"Isao","family":"Yamada","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,12,24]]},"reference":[{"key":"2887_CR1","doi-asserted-by":"publisher","first-page":"5","DOI":"10.5802\/ojmo.7","volume":"2","author":"S Alwadani","year":"2021","unstructured":"Alwadani, S., Bauschke, H.H., Revalski, J.P., Wang, X.: The difference vectors for convex sets and a resolution of the geometry conjecture. 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