{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T15:17:36Z","timestamp":1767885456632,"version":"3.49.0"},"reference-count":52,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,9,30]],"date-time":"2015-09-30T00:00:00Z","timestamp":1443571200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>We consider indirect reciprocity with optional interactions and private information. A game is offered between two players and accepted unless it is known that the other person is a defector. Whenever a defector manages to exploit a cooperator, his or her reputation is revealed to others in the population with some probability. Therefore, people have different private information about the reputation of others, which is a setting that is difficult to analyze in the theory of indirect reciprocity. Since a defector loses a fraction of his social ties each time he exploits a cooperator, he is less efficient at exploiting cooperators in subsequent rounds. We analytically calculate the critical benefit-to-cost ratio above which cooperation is successful in various settings. We demonstrate quantitative agreement with simulation results of a corresponding Wright\u2013Fisher process with optional interactions and private information. We also deduce a simple necessary condition for the critical benefit-to-cost ratio.<\/jats:p>","DOI":"10.3390\/g6040438","type":"journal-article","created":{"date-parts":[[2015,10,1]],"date-time":"2015-10-01T02:15:03Z","timestamp":1443665703000},"page":"438-457","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Indirect Reciprocity with Optional Interactions and Private Information"],"prefix":"10.3390","volume":"6","author":[{"given":"Jason","family":"Olejarz","sequence":"first","affiliation":[{"name":"Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, MA 02138, USA"}]},{"given":"Whan","family":"Ghang","sequence":"additional","affiliation":[{"name":"Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, MA 02138, USA"},{"name":"Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA"},{"name":"National Institute for Mathematical Sciences, Daejeon 34047, Korea"}]},{"given":"Martin","family":"Nowak","sequence":"additional","affiliation":[{"name":"Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, MA 02138, USA"},{"name":"Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA"},{"name":"Department of Organismic and Evolutionary Biology, Harvard University, 26 Oxford Street, Cambridge, MA 02138, USA"}]}],"member":"1968","published-online":{"date-parts":[[2015,9,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1560","DOI":"10.1126\/science.1133755","article-title":"Five rules for the evolution of cooperation","volume":"314","author":"Nowak","year":"2006","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Nowak, M.A. (2006). 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