{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:ND2FR34GVXZYR6OEQQFQX62JBK","short_pith_number":"pith:ND2FR34G","canonical_record":{"source":{"id":"1109.2152","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-09-09T20:42:30Z","cross_cats_sorted":[],"title_canon_sha256":"46cc558da6599a75e2632bb1f45b2594e229c26e867075e5e07e8b1f235e0d79","abstract_canon_sha256":"3a3a02c9daa2425b5370ea72c6ab6ab3f3a7ce69f3836333372cf668b3cad1ec"},"schema_version":"1.0"},"canonical_sha256":"68f458ef86adf388f9c4840b0bfb490ab76272df73ff19abf98337ebe6a108bb","source":{"kind":"arxiv","id":"1109.2152","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2152","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2152v1","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2152","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"pith_short_12","alias_value":"ND2FR34GVXZY","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"ND2FR34GVXZYR6OE","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"ND2FR34G","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:ND2FR34GVXZYR6OEQQFQX62JBK","target":"record","payload":{"canonical_record":{"source":{"id":"1109.2152","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-09-09T20:42:30Z","cross_cats_sorted":[],"title_canon_sha256":"46cc558da6599a75e2632bb1f45b2594e229c26e867075e5e07e8b1f235e0d79","abstract_canon_sha256":"3a3a02c9daa2425b5370ea72c6ab6ab3f3a7ce69f3836333372cf668b3cad1ec"},"schema_version":"1.0"},"canonical_sha256":"68f458ef86adf388f9c4840b0bfb490ab76272df73ff19abf98337ebe6a108bb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:08.430755Z","signature_b64":"CLts4eT9YFm0KLfqCkeDEwpwcDTm4tVGL3gq/5+KMVoPd6o/Y1HXkm87cTXFpIc5qVgsK0BtKxtT3yBrD2AiDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68f458ef86adf388f9c4840b0bfb490ab76272df73ff19abf98337ebe6a108bb","last_reissued_at":"2026-05-18T03:18:08.430130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:08.430130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.2152","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MUeHT+DdRgRevtPm1W12Fh2ijiam3rwVCZ1DuEeUhBI61X8WutxfceQ8sGH1xbzfcFRZn5rcoMv3k3mSKgQgDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T20:05:49.275681Z"},"content_sha256":"a77f902d588e1522ae8b18c06ea13e6e2e3ef37f3421fd68a292a4cb5be9287c","schema_version":"1.0","event_id":"sha256:a77f902d588e1522ae8b18c06ea13e6e2e3ef37f3421fd68a292a4cb5be9287c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:ND2FR34GVXZYR6OEQQFQX62JBK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pure Nash Equilibria: Hard and Easy Games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"F. Scarcello, G. Gottlob, G. Greco","submitted_at":"2011-09-09T20:42:30Z","abstract_excerpt":"We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete.  We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players.  We say that a game has small neighborhood if the utility function for each player depends only on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2152","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CIGzmeVeV7j7DunTLmJHiBvIzqk61oCJqETMriyWXERmOg8Ktb+8MrRYU6xTqp0oS1FJSbJxg9iW7mO7V9nACQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T20:05:49.276325Z"},"content_sha256":"ea9a451c7a6436e0aa0dee0512925f411078a72a41f2bdae96ab0fcb65dc706d","schema_version":"1.0","event_id":"sha256:ea9a451c7a6436e0aa0dee0512925f411078a72a41f2bdae96ab0fcb65dc706d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ND2FR34GVXZYR6OEQQFQX62JBK/bundle.json","state_url":"https://pith.science/pith/ND2FR34GVXZYR6OEQQFQX62JBK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ND2FR34GVXZYR6OEQQFQX62JBK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T20:05:49Z","links":{"resolver":"https://pith.science/pith/ND2FR34GVXZYR6OEQQFQX62JBK","bundle":"https://pith.science/pith/ND2FR34GVXZYR6OEQQFQX62JBK/bundle.json","state":"https://pith.science/pith/ND2FR34GVXZYR6OEQQFQX62JBK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ND2FR34GVXZYR6OEQQFQX62JBK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ND2FR34GVXZYR6OEQQFQX62JBK","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":"3a3a02c9daa2425b5370ea72c6ab6ab3f3a7ce69f3836333372cf668b3cad1ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-09-09T20:42:30Z","title_canon_sha256":"46cc558da6599a75e2632bb1f45b2594e229c26e867075e5e07e8b1f235e0d79"},"schema_version":"1.0","source":{"id":"1109.2152","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2152","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2152v1","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2152","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"pith_short_12","alias_value":"ND2FR34GVXZY","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"ND2FR34GVXZYR6OE","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"ND2FR34G","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:ea9a451c7a6436e0aa0dee0512925f411078a72a41f2bdae96ab0fcb65dc706d","target":"graph","created_at":"2026-05-18T03:18:08Z","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"},"paper":{"abstract_excerpt":"We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete.  We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players.  We say that a game has small neighborhood if the utility function for each player depends only on","authors_text":"F. Scarcello, G. Gottlob, G. Greco","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-09-09T20:42:30Z","title":"Pure Nash Equilibria: Hard and Easy Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2152","kind":"arxiv","version":1},"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:a77f902d588e1522ae8b18c06ea13e6e2e3ef37f3421fd68a292a4cb5be9287c","target":"record","created_at":"2026-05-18T03:18:08Z","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":"3a3a02c9daa2425b5370ea72c6ab6ab3f3a7ce69f3836333372cf668b3cad1ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2011-09-09T20:42:30Z","title_canon_sha256":"46cc558da6599a75e2632bb1f45b2594e229c26e867075e5e07e8b1f235e0d79"},"schema_version":"1.0","source":{"id":"1109.2152","kind":"arxiv","version":1}},"canonical_sha256":"68f458ef86adf388f9c4840b0bfb490ab76272df73ff19abf98337ebe6a108bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68f458ef86adf388f9c4840b0bfb490ab76272df73ff19abf98337ebe6a108bb","first_computed_at":"2026-05-18T03:18:08.430130Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:08.430130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CLts4eT9YFm0KLfqCkeDEwpwcDTm4tVGL3gq/5+KMVoPd6o/Y1HXkm87cTXFpIc5qVgsK0BtKxtT3yBrD2AiDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:08.430755Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.2152","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a77f902d588e1522ae8b18c06ea13e6e2e3ef37f3421fd68a292a4cb5be9287c","sha256:ea9a451c7a6436e0aa0dee0512925f411078a72a41f2bdae96ab0fcb65dc706d"],"state_sha256":"8c7c56a2953e7f01da17f875f8398cb956b9fa56382ac52283a0c9f5eda1e9bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8u5bxJFbjhfsgmjSM2zr4AU1do9Cfp2YTaU8SfPnzaUlJ4s0FA7cC6asTESn7qOVbaAJeTxeBvTQeWZgtdyrBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T20:05:49.279701Z","bundle_sha256":"fac0a324cb57ac1cea8ed87c91e1e3ee0e242f7665cfbc84a8d3427d93f8e1ca"}}