{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:O3XIMJTNMWZOB46AHELSYCWDJA","short_pith_number":"pith:O3XIMJTN","canonical_record":{"source":{"id":"2304.07187","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2023-04-14T15:00:22Z","cross_cats_sorted":[],"title_canon_sha256":"32c46efb62f6068374f6fd2245533e18482f811d26e9cc500436a30d061ae330","abstract_canon_sha256":"e4a9f02c51f0093e66762823a673aed5f266dc9d786fc97b645fd15c59747d55"},"schema_version":"1.0"},"canonical_sha256":"76ee86266d65b2e0f3c039172c0ac348198eb7b7904f64bf9d439695bd294a1d","source":{"kind":"arxiv","id":"2304.07187","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2304.07187","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"arxiv_version","alias_value":"2304.07187v2","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.07187","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"pith_short_12","alias_value":"O3XIMJTNMWZO","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"pith_short_16","alias_value":"O3XIMJTNMWZOB46A","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"pith_short_8","alias_value":"O3XIMJTN","created_at":"2026-07-05T07:01:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:O3XIMJTNMWZOB46AHELSYCWDJA","target":"record","payload":{"canonical_record":{"source":{"id":"2304.07187","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2023-04-14T15:00:22Z","cross_cats_sorted":[],"title_canon_sha256":"32c46efb62f6068374f6fd2245533e18482f811d26e9cc500436a30d061ae330","abstract_canon_sha256":"e4a9f02c51f0093e66762823a673aed5f266dc9d786fc97b645fd15c59747d55"},"schema_version":"1.0"},"canonical_sha256":"76ee86266d65b2e0f3c039172c0ac348198eb7b7904f64bf9d439695bd294a1d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:01:24.800702Z","signature_b64":"fy6EyeuJMiRgn9p39Mdjn1NPJvQHYAkbtiXnQpybKAZmukKv5NfX9Q1nnS4NaHKnWc9KmpfgSJSNdqdU/V1WCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76ee86266d65b2e0f3c039172c0ac348198eb7b7904f64bf9d439695bd294a1d","last_reissued_at":"2026-07-05T07:01:24.800225Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:01:24.800225Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2304.07187","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:01:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BDMIAighyuPf2PB6d4oavUtIdEtZTQGNJLTIJmp/xczUeQeZnfWbPQG2JFd3JGwOlW2ux+DEZu38yNx1VM3dAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-22T09:28:29.743172Z"},"content_sha256":"2277ae4d932969abddf01c3de7bf64b0f3573da451095ac72428ef114c3f78ce","schema_version":"1.0","event_id":"sha256:2277ae4d932969abddf01c3de7bf64b0f3573da451095ac72428ef114c3f78ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:O3XIMJTNMWZOB46AHELSYCWDJA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Proof that Coarse Correlated Equilibrium Implies Nash Equilibrium in Two-Player Zero-Sum Games","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Revan MacQueen","submitted_at":"2023-04-14T15:00:22Z","abstract_excerpt":"We give a simple proof of the well-known result that the marginal strategies of a coarse correlated equilibrium form a Nash equilibrium in two-player zero-sum games. A corollary of this fact is that no-external-regret learning algorithms that converge to the set of coarse correlated equilibria will also converge to Nash equilibria in two-player zero-sum games. We show an approximate version: that $\\epsilon$-coarse correlated equilibria imply $2\\epsilon$-Nash equilibria."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.07187","kind":"arxiv","version":2},"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/2304.07187/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":"2026-07-05T07:01:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nSvE/7uPrS9B06owMxFoPEyUM/df3PNl5kih1edikY8sqLEIPeB19G/Jat22TGQSffZacgxzDm+pdsuKqHe9DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-22T09:28:29.743556Z"},"content_sha256":"905a24826cfbdff1539c87eb019301bca145099fff7c5c84c96cb6be4f776de9","schema_version":"1.0","event_id":"sha256:905a24826cfbdff1539c87eb019301bca145099fff7c5c84c96cb6be4f776de9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O3XIMJTNMWZOB46AHELSYCWDJA/bundle.json","state_url":"https://pith.science/pith/O3XIMJTNMWZOB46AHELSYCWDJA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O3XIMJTNMWZOB46AHELSYCWDJA/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":"2026-07-22T09:28:29Z","links":{"resolver":"https://pith.science/pith/O3XIMJTNMWZOB46AHELSYCWDJA","bundle":"https://pith.science/pith/O3XIMJTNMWZOB46AHELSYCWDJA/bundle.json","state":"https://pith.science/pith/O3XIMJTNMWZOB46AHELSYCWDJA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O3XIMJTNMWZOB46AHELSYCWDJA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:O3XIMJTNMWZOB46AHELSYCWDJA","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":"e4a9f02c51f0093e66762823a673aed5f266dc9d786fc97b645fd15c59747d55","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2023-04-14T15:00:22Z","title_canon_sha256":"32c46efb62f6068374f6fd2245533e18482f811d26e9cc500436a30d061ae330"},"schema_version":"1.0","source":{"id":"2304.07187","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2304.07187","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"arxiv_version","alias_value":"2304.07187v2","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.07187","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"pith_short_12","alias_value":"O3XIMJTNMWZO","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"pith_short_16","alias_value":"O3XIMJTNMWZOB46A","created_at":"2026-07-05T07:01:24Z"},{"alias_kind":"pith_short_8","alias_value":"O3XIMJTN","created_at":"2026-07-05T07:01:24Z"}],"graph_snapshots":[{"event_id":"sha256:905a24826cfbdff1539c87eb019301bca145099fff7c5c84c96cb6be4f776de9","target":"graph","created_at":"2026-07-05T07:01:24Z","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/2304.07187/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a simple proof of the well-known result that the marginal strategies of a coarse correlated equilibrium form a Nash equilibrium in two-player zero-sum games. A corollary of this fact is that no-external-regret learning algorithms that converge to the set of coarse correlated equilibria will also converge to Nash equilibria in two-player zero-sum games. We show an approximate version: that $\\epsilon$-coarse correlated equilibria imply $2\\epsilon$-Nash equilibria.","authors_text":"Revan MacQueen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2023-04-14T15:00:22Z","title":"A Proof that Coarse Correlated Equilibrium Implies Nash Equilibrium in Two-Player Zero-Sum Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.07187","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":"sha256:2277ae4d932969abddf01c3de7bf64b0f3573da451095ac72428ef114c3f78ce","target":"record","created_at":"2026-07-05T07:01:24Z","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":"e4a9f02c51f0093e66762823a673aed5f266dc9d786fc97b645fd15c59747d55","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2023-04-14T15:00:22Z","title_canon_sha256":"32c46efb62f6068374f6fd2245533e18482f811d26e9cc500436a30d061ae330"},"schema_version":"1.0","source":{"id":"2304.07187","kind":"arxiv","version":2}},"canonical_sha256":"76ee86266d65b2e0f3c039172c0ac348198eb7b7904f64bf9d439695bd294a1d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76ee86266d65b2e0f3c039172c0ac348198eb7b7904f64bf9d439695bd294a1d","first_computed_at":"2026-07-05T07:01:24.800225Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:01:24.800225Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fy6EyeuJMiRgn9p39Mdjn1NPJvQHYAkbtiXnQpybKAZmukKv5NfX9Q1nnS4NaHKnWc9KmpfgSJSNdqdU/V1WCw==","signature_status":"signed_v1","signed_at":"2026-07-05T07:01:24.800702Z","signed_message":"canonical_sha256_bytes"},"source_id":"2304.07187","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2277ae4d932969abddf01c3de7bf64b0f3573da451095ac72428ef114c3f78ce","sha256:905a24826cfbdff1539c87eb019301bca145099fff7c5c84c96cb6be4f776de9"],"state_sha256":"63649b394690f3c2ca24a813f5a34824062e7669fdb2ab56f88f9d6b4463bc5c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2At2aiefHGs24uInllnV5jKwGw8CQS2fYnJWHvfv0cWKxZtG+foCSwcqgVef9nOyWWd2ltOlIdn1RflIkThABw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-22T09:28:29.745675Z","bundle_sha256":"a20a2ed19846c19725f4bead4513b5b7fcbcf3f9d3dca71ae3cd5eed04b3f19f"}}