{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:T5OWEZ6NZVI5CDVPXUJ5UK2MGW","short_pith_number":"pith:T5OWEZ6N","canonical_record":{"source":{"id":"2108.00680","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-08-02T07:27:01Z","cross_cats_sorted":[],"title_canon_sha256":"a34276c41b62a1c4e9c77f194d419c05d57ce388e80dc1c5842acb23ae8912e9","abstract_canon_sha256":"d4e720d8e3773f509b1cae4bd74dc092926e49cf8f8b9cf2126d4c5e0defd4eb"},"schema_version":"1.0"},"canonical_sha256":"9f5d6267cdcd51d10eafbd13da2b4c35921f13bc235e16598e138d2bbcf1215d","source":{"kind":"arxiv","id":"2108.00680","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.00680","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"arxiv_version","alias_value":"2108.00680v2","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.00680","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"pith_short_12","alias_value":"T5OWEZ6NZVI5","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"pith_short_16","alias_value":"T5OWEZ6NZVI5CDVP","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"pith_short_8","alias_value":"T5OWEZ6N","created_at":"2026-07-05T04:15:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:T5OWEZ6NZVI5CDVPXUJ5UK2MGW","target":"record","payload":{"canonical_record":{"source":{"id":"2108.00680","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-08-02T07:27:01Z","cross_cats_sorted":[],"title_canon_sha256":"a34276c41b62a1c4e9c77f194d419c05d57ce388e80dc1c5842acb23ae8912e9","abstract_canon_sha256":"d4e720d8e3773f509b1cae4bd74dc092926e49cf8f8b9cf2126d4c5e0defd4eb"},"schema_version":"1.0"},"canonical_sha256":"9f5d6267cdcd51d10eafbd13da2b4c35921f13bc235e16598e138d2bbcf1215d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:15:27.795516Z","signature_b64":"VviRxVczVQaPWDz/wf3DL7n2wsV4sAKpC/yAk+SEflag/nE1zAv0ZWbBQ/lR9i6WEf4657GGSefvFK318N0qCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f5d6267cdcd51d10eafbd13da2b4c35921f13bc235e16598e138d2bbcf1215d","last_reissued_at":"2026-07-05T04:15:27.795005Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:15:27.795005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2108.00680","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":"2026-07-05T04:15:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TXjKcgprxQf56hxeDBvVwHb1u900LwFcpqAHWEUNzxss9DuKd9adpZYKEjW1m5iqIF0RZUuMcALHVLN0urQ4Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T10:09:05.881627Z"},"content_sha256":"eb0f867a6e62d2e45ed0a900f22f0dfeb526df5f3b0ec31063d4f56e4d467d6c","schema_version":"1.0","event_id":"sha256:eb0f867a6e62d2e45ed0a900f22f0dfeb526df5f3b0ec31063d4f56e4d467d6c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:T5OWEZ6NZVI5CDVPXUJ5UK2MGW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Game Theory Using Stochastic Tail Orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fabian Wirth, Jeremias Epperlein, Sandra K\\\"onig, Stefan Rass, Stefan Schauer, Vincent B\\\"urgin","submitted_at":"2021-08-02T07:27:01Z","abstract_excerpt":"We consider a family of distributions on which natural tail orders can be constructed upon a representation of a distribution by a (single) hyper-real number. Past research revealed that the ordering can herein strongly depend on the particular model of the hyperreals, specifically the underlying ultrafilter. Hence, our distribution family is constructed to order invariantly of an ultrafilter. Moreover, we prove that it lies dense in the set of all distributions with the (same) compact support, w.r.t. the supremum norm. Overall, this work resents a correction to [10, 12], in response to recent"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.00680","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/2108.00680/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":"2026-07-05T04:15:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"34QuIIcauGcbTd0nu5NAtmdBzi+5XIgufSCWtgBgcHMxljwOaNfS4k78UwgM0K+eltmnMh04nNNVxEINjuE3DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T10:09:05.882127Z"},"content_sha256":"8f2ca25a0f725f84442c32e9fe9f3a9d00abec5bf7c70b4b45afc53878fb9196","schema_version":"1.0","event_id":"sha256:8f2ca25a0f725f84442c32e9fe9f3a9d00abec5bf7c70b4b45afc53878fb9196"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T5OWEZ6NZVI5CDVPXUJ5UK2MGW/bundle.json","state_url":"https://pith.science/pith/T5OWEZ6NZVI5CDVPXUJ5UK2MGW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T5OWEZ6NZVI5CDVPXUJ5UK2MGW/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":"2026-08-06T10:09:05Z","links":{"resolver":"https://pith.science/pith/T5OWEZ6NZVI5CDVPXUJ5UK2MGW","bundle":"https://pith.science/pith/T5OWEZ6NZVI5CDVPXUJ5UK2MGW/bundle.json","state":"https://pith.science/pith/T5OWEZ6NZVI5CDVPXUJ5UK2MGW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T5OWEZ6NZVI5CDVPXUJ5UK2MGW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:T5OWEZ6NZVI5CDVPXUJ5UK2MGW","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":"d4e720d8e3773f509b1cae4bd74dc092926e49cf8f8b9cf2126d4c5e0defd4eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-08-02T07:27:01Z","title_canon_sha256":"a34276c41b62a1c4e9c77f194d419c05d57ce388e80dc1c5842acb23ae8912e9"},"schema_version":"1.0","source":{"id":"2108.00680","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.00680","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"arxiv_version","alias_value":"2108.00680v2","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.00680","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"pith_short_12","alias_value":"T5OWEZ6NZVI5","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"pith_short_16","alias_value":"T5OWEZ6NZVI5CDVP","created_at":"2026-07-05T04:15:27Z"},{"alias_kind":"pith_short_8","alias_value":"T5OWEZ6N","created_at":"2026-07-05T04:15:27Z"}],"graph_snapshots":[{"event_id":"sha256:8f2ca25a0f725f84442c32e9fe9f3a9d00abec5bf7c70b4b45afc53878fb9196","target":"graph","created_at":"2026-07-05T04:15:27Z","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/2108.00680/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider a family of distributions on which natural tail orders can be constructed upon a representation of a distribution by a (single) hyper-real number. Past research revealed that the ordering can herein strongly depend on the particular model of the hyperreals, specifically the underlying ultrafilter. Hence, our distribution family is constructed to order invariantly of an ultrafilter. Moreover, we prove that it lies dense in the set of all distributions with the (same) compact support, w.r.t. the supremum norm. Overall, this work resents a correction to [10, 12], in response to recent","authors_text":"Fabian Wirth, Jeremias Epperlein, Sandra K\\\"onig, Stefan Rass, Stefan Schauer, Vincent B\\\"urgin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-08-02T07:27:01Z","title":"On Game Theory Using Stochastic Tail Orders"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.00680","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:eb0f867a6e62d2e45ed0a900f22f0dfeb526df5f3b0ec31063d4f56e4d467d6c","target":"record","created_at":"2026-07-05T04:15:27Z","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":"d4e720d8e3773f509b1cae4bd74dc092926e49cf8f8b9cf2126d4c5e0defd4eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-08-02T07:27:01Z","title_canon_sha256":"a34276c41b62a1c4e9c77f194d419c05d57ce388e80dc1c5842acb23ae8912e9"},"schema_version":"1.0","source":{"id":"2108.00680","kind":"arxiv","version":2}},"canonical_sha256":"9f5d6267cdcd51d10eafbd13da2b4c35921f13bc235e16598e138d2bbcf1215d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9f5d6267cdcd51d10eafbd13da2b4c35921f13bc235e16598e138d2bbcf1215d","first_computed_at":"2026-07-05T04:15:27.795005Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:15:27.795005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VviRxVczVQaPWDz/wf3DL7n2wsV4sAKpC/yAk+SEflag/nE1zAv0ZWbBQ/lR9i6WEf4657GGSefvFK318N0qCA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:15:27.795516Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.00680","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eb0f867a6e62d2e45ed0a900f22f0dfeb526df5f3b0ec31063d4f56e4d467d6c","sha256:8f2ca25a0f725f84442c32e9fe9f3a9d00abec5bf7c70b4b45afc53878fb9196"],"state_sha256":"2f4ca95e44302672ad32e15a7b311878d0b75ecdd88ab90d51a40c82c0cf831c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4kgg3z6R1JJWXY7hMQsJeXTv/JsyPP5UdNYUafGEzXBdx02BGWsJMbDSgrcrlBv+iD67Ol/2y8MWZIAU0IUaCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T10:09:05.887586Z","bundle_sha256":"022b189b6f5e586e82f3e21bc7fb00a5a62c540a1270c2f57d1b284c86430e4d"}}