{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:TDO4O45SQDKAI4K4LP6GTATSFR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"55891f843853b927b1a67ef1ce5fc6be2e92b24be93e5b08cdb826de4a2d3c46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2021-07-22T05:03:06Z","title_canon_sha256":"d9eeef2dca41e9699bd41e60d8da414388ff1841f6c3ce907bc4a5ff24e5a4ea"},"schema_version":"1.0","source":{"id":"2107.10458","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.10458","created_at":"2026-07-05T04:35:56Z"},{"alias_kind":"arxiv_version","alias_value":"2107.10458v2","created_at":"2026-07-05T04:35:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.10458","created_at":"2026-07-05T04:35:56Z"},{"alias_kind":"pith_short_12","alias_value":"TDO4O45SQDKA","created_at":"2026-07-05T04:35:56Z"},{"alias_kind":"pith_short_16","alias_value":"TDO4O45SQDKAI4K4","created_at":"2026-07-05T04:35:56Z"},{"alias_kind":"pith_short_8","alias_value":"TDO4O45S","created_at":"2026-07-05T04:35:56Z"}],"graph_snapshots":[{"event_id":"sha256:73f8c9de29a4bf7e14055b59b659bf5e0084a6bda817303f4e993f881d92618b","target":"graph","created_at":"2026-07-05T04:35:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2107.10458/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Public Good index is a power index for simple games introduced by Holler and later axiomatized by Holler and Packel, so that some authors also speak of the Holler--Packel index. A generalization to the class of games with transferable utility was given by Holler and Li. Here we generalize the underlying ideas to games with several levels of approval in the input and output -- so-called $(j,k)$ simple games. Corresponding axiomatizations are also provided.","authors_text":"Sascha Kurz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2021-07-22T05:03:06Z","title":"The Public Good index for games with several levels of approval in the input and output"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.10458","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:091108ac2931fe196512504cb2dd8e1478e226ffe2fa45265844dd0773b0c58b","target":"record","created_at":"2026-07-05T04:35:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"55891f843853b927b1a67ef1ce5fc6be2e92b24be93e5b08cdb826de4a2d3c46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2021-07-22T05:03:06Z","title_canon_sha256":"d9eeef2dca41e9699bd41e60d8da414388ff1841f6c3ce907bc4a5ff24e5a4ea"},"schema_version":"1.0","source":{"id":"2107.10458","kind":"arxiv","version":2}},"canonical_sha256":"98ddc773b280d404715c5bfc6982722c6cb61a84b62023b5b1044d592d7d2900","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98ddc773b280d404715c5bfc6982722c6cb61a84b62023b5b1044d592d7d2900","first_computed_at":"2026-07-05T04:35:56.672864Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:35:56.672864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7lLIQVw6vi+MPMX8t4gTbidKOLQ3g7m6MPhCr+f89U6srw+n1RH11f9qvonGrncFLkZa/+CbGtzFyI7Sgug8Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T04:35:56.673226Z","signed_message":"canonical_sha256_bytes"},"source_id":"2107.10458","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:091108ac2931fe196512504cb2dd8e1478e226ffe2fa45265844dd0773b0c58b","sha256:73f8c9de29a4bf7e14055b59b659bf5e0084a6bda817303f4e993f881d92618b"],"state_sha256":"9e695f1868d3ae8d45aa47614f4a5d8c1d78a6448eeab836a17a194b4df8fdb7"}