{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:HBGENUHESSXRLUMFPMWTZOCUCB","short_pith_number":"pith:HBGENUHE","canonical_record":{"source":{"id":"2606.20013","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-06-18T09:47:00Z","cross_cats_sorted":[],"title_canon_sha256":"79c50c798987ea71be54c7ebe23892309a57c3e11e37d77ad729c0336055c4d0","abstract_canon_sha256":"08c64fb9899a6035a93a64683714b04e5d17f9512876907e07522b6394e6057e"},"schema_version":"1.0"},"canonical_sha256":"384c46d0e494af15d1857b2d3cb854106256758c49b73592202131535199586c","source":{"kind":"arxiv","id":"2606.20013","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.20013","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"arxiv_version","alias_value":"2606.20013v1","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.20013","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"pith_short_12","alias_value":"HBGENUHESSXR","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"pith_short_16","alias_value":"HBGENUHESSXRLUMF","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"pith_short_8","alias_value":"HBGENUHE","created_at":"2026-06-19T16:13:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:HBGENUHESSXRLUMFPMWTZOCUCB","target":"record","payload":{"canonical_record":{"source":{"id":"2606.20013","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-06-18T09:47:00Z","cross_cats_sorted":[],"title_canon_sha256":"79c50c798987ea71be54c7ebe23892309a57c3e11e37d77ad729c0336055c4d0","abstract_canon_sha256":"08c64fb9899a6035a93a64683714b04e5d17f9512876907e07522b6394e6057e"},"schema_version":"1.0"},"canonical_sha256":"384c46d0e494af15d1857b2d3cb854106256758c49b73592202131535199586c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:13:00.648009Z","signature_b64":"/PuRDJreJcx+TwnSepnx4M62ztLfn/kk/PiH/P1DNAQpO1isCo5duicMBBAZnXiLYMdVPnU5C/lEyZXXkf3vCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"384c46d0e494af15d1857b2d3cb854106256758c49b73592202131535199586c","last_reissued_at":"2026-06-19T16:13:00.647650Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:13:00.647650Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.20013","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":"2026-06-19T16:13:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eP7QZncRLuIpLBgKH6iiWnyEGuO4JG2nFPBKtMaajisFif+Fp40FpBXeoLDZfCYqlEoOBRRAb8C2UAqVi1D+BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:35:34.455107Z"},"content_sha256":"8244f0d08d313ba71d1660f0cad3bf4fb3c0d78d2d2a9f6b7086247208e4240a","schema_version":"1.0","event_id":"sha256:8244f0d08d313ba71d1660f0cad3bf4fb3c0d78d2d2a9f6b7086247208e4240a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:HBGENUHESSXRLUMFPMWTZOCUCB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Regularized Nikaido-Isoda Function Approach to Multi-Leader-Follower Games","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Atsushi Hori, Ellen H. Fukuda, Takayuki Okuno","submitted_at":"2026-06-18T09:47:00Z","abstract_excerpt":"A multi-leader--follower game (MLFG) is a hierarchical noncooperative game in which leaders compete at the upper level while taking into account the followers' best responses at the lower level. A typical approach to solving the MLFG reformulates it as an equilibrium problem with equilibrium constraints (EPECs) by replacing the lower-level game with its KKT conditions. Another approach, when each follower's response is unique, is to reformulate the MLFG as a Nash equilibrium problem by substituting these response functions into each leader's problem. However, both reformulations may lack scala"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.20013","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/2606.20013/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":"2026-06-19T16:13:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fFNQcYeuf/X5W/BFYy+lwdemF6SM7oyZv/BFhVdlkUde0c6SPWwFKZ9cst8Ipxls99Ve+yIp+17IG1GwT/xjBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:35:34.455726Z"},"content_sha256":"9b13f45827c47c841e795c996b3f4cef9419ec1caafa4ebcf56d182dc6b66654","schema_version":"1.0","event_id":"sha256:9b13f45827c47c841e795c996b3f4cef9419ec1caafa4ebcf56d182dc6b66654"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HBGENUHESSXRLUMFPMWTZOCUCB/bundle.json","state_url":"https://pith.science/pith/HBGENUHESSXRLUMFPMWTZOCUCB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HBGENUHESSXRLUMFPMWTZOCUCB/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":"2026-06-22T12:35:34Z","links":{"resolver":"https://pith.science/pith/HBGENUHESSXRLUMFPMWTZOCUCB","bundle":"https://pith.science/pith/HBGENUHESSXRLUMFPMWTZOCUCB/bundle.json","state":"https://pith.science/pith/HBGENUHESSXRLUMFPMWTZOCUCB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HBGENUHESSXRLUMFPMWTZOCUCB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:HBGENUHESSXRLUMFPMWTZOCUCB","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":"08c64fb9899a6035a93a64683714b04e5d17f9512876907e07522b6394e6057e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-06-18T09:47:00Z","title_canon_sha256":"79c50c798987ea71be54c7ebe23892309a57c3e11e37d77ad729c0336055c4d0"},"schema_version":"1.0","source":{"id":"2606.20013","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.20013","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"arxiv_version","alias_value":"2606.20013v1","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.20013","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"pith_short_12","alias_value":"HBGENUHESSXR","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"pith_short_16","alias_value":"HBGENUHESSXRLUMF","created_at":"2026-06-19T16:13:00Z"},{"alias_kind":"pith_short_8","alias_value":"HBGENUHE","created_at":"2026-06-19T16:13:00Z"}],"graph_snapshots":[{"event_id":"sha256:9b13f45827c47c841e795c996b3f4cef9419ec1caafa4ebcf56d182dc6b66654","target":"graph","created_at":"2026-06-19T16:13:00Z","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/2606.20013/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A multi-leader--follower game (MLFG) is a hierarchical noncooperative game in which leaders compete at the upper level while taking into account the followers' best responses at the lower level. A typical approach to solving the MLFG reformulates it as an equilibrium problem with equilibrium constraints (EPECs) by replacing the lower-level game with its KKT conditions. Another approach, when each follower's response is unique, is to reformulate the MLFG as a Nash equilibrium problem by substituting these response functions into each leader's problem. However, both reformulations may lack scala","authors_text":"Atsushi Hori, Ellen H. Fukuda, Takayuki Okuno","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-06-18T09:47:00Z","title":"A Regularized Nikaido-Isoda Function Approach to Multi-Leader-Follower Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.20013","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":"sha256:8244f0d08d313ba71d1660f0cad3bf4fb3c0d78d2d2a9f6b7086247208e4240a","target":"record","created_at":"2026-06-19T16:13:00Z","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":"08c64fb9899a6035a93a64683714b04e5d17f9512876907e07522b6394e6057e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-06-18T09:47:00Z","title_canon_sha256":"79c50c798987ea71be54c7ebe23892309a57c3e11e37d77ad729c0336055c4d0"},"schema_version":"1.0","source":{"id":"2606.20013","kind":"arxiv","version":1}},"canonical_sha256":"384c46d0e494af15d1857b2d3cb854106256758c49b73592202131535199586c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"384c46d0e494af15d1857b2d3cb854106256758c49b73592202131535199586c","first_computed_at":"2026-06-19T16:13:00.647650Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:13:00.647650Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/PuRDJreJcx+TwnSepnx4M62ztLfn/kk/PiH/P1DNAQpO1isCo5duicMBBAZnXiLYMdVPnU5C/lEyZXXkf3vCg==","signature_status":"signed_v1","signed_at":"2026-06-19T16:13:00.648009Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.20013","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8244f0d08d313ba71d1660f0cad3bf4fb3c0d78d2d2a9f6b7086247208e4240a","sha256:9b13f45827c47c841e795c996b3f4cef9419ec1caafa4ebcf56d182dc6b66654"],"state_sha256":"88af86ce6a1074c6da1144c71ad3daf8967c3092bb61c231bcd562dc5d2df2df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"npLwilqdwgRAA8f0Wy3yULvO5t/nuWA1i3Z+Js8KID0r06/Sic6ya6o/kcWY5x4haSlCXqthcczPUJ2ed1GLDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T12:35:34.459461Z","bundle_sha256":"43784b0ed06af3fb9de35e0e0a5ef674c3ee13684624e813310a4e89231d3fb1"}}