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It turns out that the only additional requirement is that every non-unitary feasible coalition can be written in a unique way as a union of non-unitary supports. As a consequence, a fairness axiom for solutions for cooperative games on cycle-free accessible union stable network structures can be defined that, together with the well-known component efficiency, characterizes the Shapley value on the class of cycle-free accessible union stable network structures. Since this fairness axiom combines ideas behind the traditional fairness axiom and balanced contributions, we refer to it as <jats:italic>balanced fairness<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s10288-025-00594-y","type":"journal-article","created":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T15:22:12Z","timestamp":1757344932000},"page":"329-347","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On cycle-free accessible union stable network structures"],"prefix":"10.1007","volume":"23","author":[{"given":"Encarnaci\u00f3n","family":"Algaba","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8570-3706","authenticated-orcid":false,"given":"Ren\u00e9","family":"van den Brink","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chris","family":"Dietz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,8]]},"reference":[{"key":"594_CR1","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1080\/03081079.2019.1615908","volume":"48","author":"E Algaba","year":"2019","unstructured":"Algaba E, B\u00e9al S, R\u00e9mila E, Solal P (2019) Harsanyi power solutions for cooperative games on voting structures. 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