{"paper":{"title":"Polytopality of simple games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Filip D. Jevti\\'c, Marinko Timotijevi\\'c, Rade T. \\v{Z}ivaljevi\\'c","submitted_at":"2023-09-26T11:26:18Z","abstract_excerpt":"The Bier sphere $Bier(\\mathcal{G}) = Bier(K) = K\\ast_\\Delta K^\\circ$ and the canonical fan $Fan(\\Gamma) = Fan(K)$ are combinatorial/geometric companions of a simple game $\\mathcal{G} = (P,\\Gamma)$ (equivalently the associated simplicial complex $K$), where $P$ is the set of players, $\\Gamma\\subseteq 2^P$ is the set of wining coalitions, and $K = 2^P\\setminus \\Gamma$ is the simplicial complex of losing coalitions. We characterize roughly weighted majority games as the games $\\Gamma$ such that $Bier(\\mathcal{G})$ (respectively $Fan(\\Gamma)$) is canonically polytopal (canonically pseudo-polytopal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.14848","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2309.14848/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}