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There are three games, the fair, spatially independent game $A$, the spatially dependent game $B$, and game $C$, which is a random mixture or nonrandom pattern of games $A$ and $B$. Of interest is $\\mu_B$ (or $\\mu_C$), the mean profit per turn at equilibrium to the set of $MN$ players playing game $B$ (or game $C$). Game $A$ is fair, so if $\\mu_B\\le0$ and $\\mu_C>0$, then we say the Parrondo"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1510.06947","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-10-23T14:39:40Z","cross_cats_sorted":[],"title_canon_sha256":"896812359ff8b2d60449b350a43e7d4cee6b6a2884cb1ba5117e3d6a78b65e18","abstract_canon_sha256":"719ee214175ac9eadd79fa1a314ed00692f32176edade3b53736d588cd2148a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:22.617218Z","signature_b64":"zk/kz7v29s4gJzZgxna2mz2ztScP57oSzJKAe9pztq2UKLlQ2kxLvsY55mJzAeObfpdKswwMSZDzXpihgSmBDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e14518a2a38b9fecd3a04493dd858424b32b3efe6b6ed39e515b0e3319c2be3","last_reissued_at":"2026-05-18T01:29:22.616605Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:22.616605Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parrondo games with two-dimensional spatial dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jiyeon Lee, S. 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