{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:MKS5HS3WKUPPDHQQMZBR6FR2QZ","short_pith_number":"pith:MKS5HS3W","canonical_record":{"source":{"id":"2107.03898","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2021-07-08T15:30:52Z","cross_cats_sorted":[],"title_canon_sha256":"67dc4f6c424b492e41df5d4b256b23d8600d73de54e395781a43f4dbfa6ec886","abstract_canon_sha256":"00fa01d542fe3d46fb7f851cbb44119ef09754aa49aba865e7271aee60ea3cd5"},"schema_version":"1.0"},"canonical_sha256":"62a5d3cb76551ef19e1066431f163a8647941a976d9974abe14dfcf82cfc9e3f","source":{"kind":"arxiv","id":"2107.03898","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.03898","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"arxiv_version","alias_value":"2107.03898v1","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.03898","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"pith_short_12","alias_value":"MKS5HS3WKUPP","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"pith_short_16","alias_value":"MKS5HS3WKUPPDHQQ","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"pith_short_8","alias_value":"MKS5HS3W","created_at":"2026-07-05T02:56:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:MKS5HS3WKUPPDHQQMZBR6FR2QZ","target":"record","payload":{"canonical_record":{"source":{"id":"2107.03898","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2021-07-08T15:30:52Z","cross_cats_sorted":[],"title_canon_sha256":"67dc4f6c424b492e41df5d4b256b23d8600d73de54e395781a43f4dbfa6ec886","abstract_canon_sha256":"00fa01d542fe3d46fb7f851cbb44119ef09754aa49aba865e7271aee60ea3cd5"},"schema_version":"1.0"},"canonical_sha256":"62a5d3cb76551ef19e1066431f163a8647941a976d9974abe14dfcf82cfc9e3f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:56:20.573719Z","signature_b64":"TDjBW5CRDY/C76/E4QnvXRClTfOtv+Y2c0sGCh0hY+PlTnydgO//QX0fmF3upFdiz6Oa3psfGaB+r0gq7n+4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62a5d3cb76551ef19e1066431f163a8647941a976d9974abe14dfcf82cfc9e3f","last_reissued_at":"2026-07-05T02:56:20.573305Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:56:20.573305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2107.03898","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":"2026-07-05T02:56:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SRLd7XdNZD74laYzDxyXkIJhSUyU0A/ftxmUWLYNptwZSNY397u0Ev/ZsB1lOaGw6r9rhcP2s2BEfZ7cjphVAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T18:45:04.307085Z"},"content_sha256":"18c30a379235244bc54bd45a2323583c25cee0e6571097fc2b5509a46788c106","schema_version":"1.0","event_id":"sha256:18c30a379235244bc54bd45a2323583c25cee0e6571097fc2b5509a46788c106"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:MKS5HS3WKUPPDHQQMZBR6FR2QZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Matthew J. Katzman, Paul W. Goldberg","submitted_at":"2021-07-08T15:30:52Z","abstract_excerpt":"Nearly a decade ago, Azrieli and Shmaya introduced the class of $\\lambda$-Lipschitz games in which every player's payoff function is $\\lambda$-Lipschitz with respect to the actions of the other players. They showed that such games admit $\\epsilon$-approximate pure Nash equilibria for certain settings of $\\epsilon$ and $\\lambda$. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.03898","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/2107.03898/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:56:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XTPvP1S+HxhqB7lZ2+Xtd8UbtOMpBP/tjKsqnY4ugOr8WolH9Uj1MmCtQYvEjux2Uo4WYPrr0dY/7nww09sJBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T18:45:04.308735Z"},"content_sha256":"efff577042380897ffb0615405a8b2d0653a12762429c051f004b57977a46843","schema_version":"1.0","event_id":"sha256:efff577042380897ffb0615405a8b2d0653a12762429c051f004b57977a46843"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MKS5HS3WKUPPDHQQMZBR6FR2QZ/bundle.json","state_url":"https://pith.science/pith/MKS5HS3WKUPPDHQQMZBR6FR2QZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MKS5HS3WKUPPDHQQMZBR6FR2QZ/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":"2026-08-17T18:45:04Z","links":{"resolver":"https://pith.science/pith/MKS5HS3WKUPPDHQQMZBR6FR2QZ","bundle":"https://pith.science/pith/MKS5HS3WKUPPDHQQMZBR6FR2QZ/bundle.json","state":"https://pith.science/pith/MKS5HS3WKUPPDHQQMZBR6FR2QZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MKS5HS3WKUPPDHQQMZBR6FR2QZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:MKS5HS3WKUPPDHQQMZBR6FR2QZ","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":"00fa01d542fe3d46fb7f851cbb44119ef09754aa49aba865e7271aee60ea3cd5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2021-07-08T15:30:52Z","title_canon_sha256":"67dc4f6c424b492e41df5d4b256b23d8600d73de54e395781a43f4dbfa6ec886"},"schema_version":"1.0","source":{"id":"2107.03898","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.03898","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"arxiv_version","alias_value":"2107.03898v1","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.03898","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"pith_short_12","alias_value":"MKS5HS3WKUPP","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"pith_short_16","alias_value":"MKS5HS3WKUPPDHQQ","created_at":"2026-07-05T02:56:20Z"},{"alias_kind":"pith_short_8","alias_value":"MKS5HS3W","created_at":"2026-07-05T02:56:20Z"}],"graph_snapshots":[{"event_id":"sha256:efff577042380897ffb0615405a8b2d0653a12762429c051f004b57977a46843","target":"graph","created_at":"2026-07-05T02:56:20Z","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/2107.03898/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Nearly a decade ago, Azrieli and Shmaya introduced the class of $\\lambda$-Lipschitz games in which every player's payoff function is $\\lambda$-Lipschitz with respect to the actions of the other players. They showed that such games admit $\\epsilon$-approximate pure Nash equilibria for certain settings of $\\epsilon$ and $\\lambda$. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding ","authors_text":"Matthew J. Katzman, Paul W. Goldberg","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2021-07-08T15:30:52Z","title":"Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.03898","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":"sha256:18c30a379235244bc54bd45a2323583c25cee0e6571097fc2b5509a46788c106","target":"record","created_at":"2026-07-05T02:56:20Z","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":"00fa01d542fe3d46fb7f851cbb44119ef09754aa49aba865e7271aee60ea3cd5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2021-07-08T15:30:52Z","title_canon_sha256":"67dc4f6c424b492e41df5d4b256b23d8600d73de54e395781a43f4dbfa6ec886"},"schema_version":"1.0","source":{"id":"2107.03898","kind":"arxiv","version":1}},"canonical_sha256":"62a5d3cb76551ef19e1066431f163a8647941a976d9974abe14dfcf82cfc9e3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62a5d3cb76551ef19e1066431f163a8647941a976d9974abe14dfcf82cfc9e3f","first_computed_at":"2026-07-05T02:56:20.573305Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:56:20.573305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TDjBW5CRDY/C76/E4QnvXRClTfOtv+Y2c0sGCh0hY+PlTnydgO//QX0fmF3upFdiz6Oa3psfGaB+r0gq7n+4Dw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:56:20.573719Z","signed_message":"canonical_sha256_bytes"},"source_id":"2107.03898","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:18c30a379235244bc54bd45a2323583c25cee0e6571097fc2b5509a46788c106","sha256:efff577042380897ffb0615405a8b2d0653a12762429c051f004b57977a46843"],"state_sha256":"afbc10c3075abee9499d1b222d5accb35f7476b38030dad60120176f2a8bfb12"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LBjtrpLkB8QBEC3eyuECd5rgUwbD8UpaKI5iROYRwZPrFbMDaExfY3fQM6Gsntfmq0gL11uZCk3QJfNFVTLHAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T18:45:04.323333Z","bundle_sha256":"f46aa6138cfa2b4e496672b2006b96040dbb1b3133ca9d52a1afba0fabd8c3a8"}}