{"paper":{"title":"Positional Determinacy with Colored Vertices: a 1-to-2-Player Lift","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Rapha\\\"el Berthon, St\\'ephane Le Roux","submitted_at":"2026-07-08T13:46:35Z","abstract_excerpt":"Positional determinacy of vertex-colored parity games was proved in the 1990s, which directly implies positional determinacy of edge-colored parity games. In 2006, it was shown that if a prefix-independent color-based objective ensures that every edge-colored two-player turn-based game is positionally determined, this objective is equivalent to a parity objective. We prove a similar result for vertex-colored games, namely that the following are equivalent for any prefix-independent objective $W$ over a finite set of colors:\n  - $W$ is positionally determined on all vertex-colored one-player ga"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07415","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/2607.07415/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"}