{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:44:17Z","timestamp":1760237057325,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,2,20]],"date-time":"2020-02-20T00:00:00Z","timestamp":1582156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["17KK0055, 17H02044, 18H03498, 18K21570, 19H02376"],"award-info":[{"award-number":["17KK0055, 17H02044, 18H03498, 18K21570, 19H02376"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>Intensive studies on indirect reciprocity have explored rational assessment rules for maintaining cooperation and several have demonstrated the effects of the stern-judging rule. Uchida and Sasaki demonstrated that the stern-judging rule is not suitable for maintaining cooperative regimes in private assessment conditions while a public assessment system has been assumed in most studies. Although both assessment systems are oversimplified and society is most accurately represented by a mixture of these systems, little analysis has been reported on their mixture. Here, we investigated how much weight on the use of information originating from a public source is needed to maintain cooperative regimes for players adopting the stern-judging rule when players get information from both public and private sources. We did this by considering a hybrid-assessment scheme in which players use both assessment systems and by using evolutionary game theory. We calculated replicator equations using the expected payoffs of three strategies: unconditional cooperation, unconditional defection, and stern-judging rule adoption. Our analysis shows that the use of the rule helps to maintain cooperation if reputation information from a unique public notice board is used with more than a threshold probability. This hybrid-assessment scheme can be applied to other rules, including the simple-standing rule and the staying rule.<\/jats:p>","DOI":"10.3390\/g11010013","type":"journal-article","created":{"date-parts":[[2020,2,26]],"date-time":"2020-02-26T04:18:29Z","timestamp":1582690709000},"page":"13","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Hybrid Assessment Scheme Based on the Stern- Judging Rule for Maintaining Cooperation under Indirect Reciprocity"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9943-2668","authenticated-orcid":false,"given":"Isamu","family":"Okada","sequence":"first","affiliation":[{"name":"Faculty of Business Administration, Soka University, Tangi 1-236, Hachioji City, Tokyo 192-8577, Japan"},{"name":"Department of Information Systems and Operations, Vienna University of Economics, 1020 Wien, Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7966-2629","authenticated-orcid":false,"given":"Hitoshi","family":"Yamamoto","sequence":"additional","affiliation":[{"name":"Department of Information Systems and Operations, Vienna University of Economics, 1020 Wien, Austria"},{"name":"Faculty of Business Administration, Rissho University, Shinagawa City, Tokyo 141-8602, Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6209-9638","authenticated-orcid":false,"given":"Satoshi","family":"Uchida","sequence":"additional","affiliation":[{"name":"Research Center for Ethi-Culture Studies, RINRI Institute, Chiyoda City, Tokyo 102-8561, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1038\/31225","article-title":"Evolution of indirect reciprocity by image scoring","volume":"393","author":"Nowak","year":"1998","journal-title":"Nature"},{"key":"ref_2","first-page":"745","article-title":"Evolution of cooperation through indirect reciprocity","volume":"268","author":"Leimar","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1038\/nature04131","article-title":"Evolution of indirect reciprocity","volume":"437","author":"Nowak","year":"2005","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1038\/nature05229","article-title":"The efficient interaction of indirect reciprocity and costly punishment","volume":"444","author":"Rockenbach","year":"2006","journal-title":"Nature"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"20160341","DOI":"10.1098\/rsbl.2016.0341","article-title":"Indirect reciprocity can overcome free-rider problems on costly moral assessment","volume":"12","author":"Sasaki","year":"2016","journal-title":"Biol. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1038\/415424a","article-title":"Reputation helps solve the \u2018tragedy of the commons\u2019","volume":"415","author":"Milinski","year":"2002","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"17435","DOI":"10.1073\/pnas.0704598104","article-title":"Gossip as an alternative for direct observation in games of indirect reciprocity","volume":"44","author":"Sommerfeld","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6134","DOI":"10.1038\/ncomms7134","article-title":"Reputation can enhance or suppress cooperation through positive feedback","volume":"6","author":"McNamara","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.evolhumbehav.2015.12.001","article-title":"Reputation-based cooperation: Empirical evidence for behavioral strategies","volume":"37","author":"Swakman","year":"2016","journal-title":"Evol. Hum. Behav."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"14813","DOI":"10.1038\/s41598-018-33147-x","article-title":"Experimental evidence of selective inattention in reputation-based cooperation","volume":"8","author":"Okada","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1126\/science.288.5467.850","article-title":"Cooperation through image scoring in humans","volume":"288","author":"Wedekind","year":"2000","journal-title":"Science"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Sigmund, K. (2010). The Calculus of Selfishness, Princeton University Press.","DOI":"10.1515\/9781400832255"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1634","DOI":"10.1371\/journal.pcbi.0020178","article-title":"Stern-judging: A simple, successful norm which promotes cooperation under indirect reciprocity","volume":"2","author":"Pacheco","year":"2006","journal-title":"PLoS Comput. Biol"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"63","DOI":"10.2307\/2297925","article-title":"Social norms and community enforcement","volume":"59","author":"Kandori","year":"1992","journal-title":"Rev. Econ. Stud."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.jtbi.2005.08.008","article-title":"The leading eight: Social norms that can maintain cooperation by indirect reciprocity","volume":"239","author":"Ohtsuki","year":"2006","journal-title":"J. Theor. Biol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"37517","DOI":"10.1038\/srep37517","article-title":"Evolution of cooperation under indirect reciprocity and arbitrary exploration rates","volume":"6","author":"Santos","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.chaos.2013.08.006","article-title":"Effect of assessment error and private information on stern-judging in indirect reciprocity","volume":"56","author":"Uchida","year":"2013","journal-title":"Chaos Solitons Fract."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1038\/nature02978","article-title":"Indirect reciprocity can stabilize cooperation without the second-order free rider problem","volume":"432","author":"Panchanathan","year":"2004","journal-title":"Nature"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.jtbi.2004.06.032","article-title":"The logic of reprobation: Action and assessment rules in indirect reciprocity","volume":"231","author":"Brandt","year":"2004","journal-title":"J. Theor. Biol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.jtbi.2005.08.045","article-title":"The good, the bad and the discriminator ? Errors in direct and indirect reciprocity","volume":"239","author":"Brandt","year":"2006","journal-title":"J. Theor. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.jtbi.2005.11.028","article-title":"The evolution of norms","volume":"241","author":"Chalub","year":"2006","journal-title":"J. Theor. Biol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1098\/rspb.2006.3759","article-title":"Tag-based indirect reciprocity by incomplete social information","volume":"274","author":"Masuda","year":"2007","journal-title":"Proc. R. Soc. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.jtbi.2011.02.009","article-title":"Two wrongs don\u2019t make a right: The initial viability of different assessment rules in the evolution of indirect reciprocity","volume":"277","author":"Panchanathan","year":"2011","journal-title":"J. Theor. Biol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"41870","DOI":"10.1038\/srep41870","article-title":"The evolution of conditional moral assessment in indirect reciprocity","volume":"7","author":"Sasaki","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.jtbi.2009.11.013","article-title":"The competition of assessment rules for indirect reciprocity","volume":"263","author":"Uchida","year":"2010","journal-title":"J. Theor. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"036111","DOI":"10.1103\/PhysRevE.82.036111","article-title":"Effect of private information on indirect reciprocity","volume":"82","author":"Uchida","year":"2010","journal-title":"Phy. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.jtbi.2011.03.024","article-title":"Moral assessment in indirect reciprocity","volume":"299","author":"Sigmund","year":"2012","journal-title":"J. Theor. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"052810","DOI":"10.1103\/PhysRevE.87.052810","article-title":"Evolutionary stability and resistance to cheating in an indirect reciprocity model based on reputation","volume":"87","author":"Cuesta","year":"2013","journal-title":"Phys. Review. E"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ohtsuki, H., Iwasa, Y., and Nowak, M.A. (2015). Reputation effects in public and private interactions. PLoS Comput. Biol., 11.","DOI":"10.1371\/journal.pcbi.1004527"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"438","DOI":"10.3390\/g6040438","article-title":"Indirect reciprocity with optional interactions and private information","volume":"6","author":"Olejarz","year":"2015","journal-title":"Games"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"9737","DOI":"10.1038\/s41598-017-09935-2","article-title":"Tolerant indirect reciprocity can boost social welfare through solidarity with unconditional cooperators in private monitoring","volume":"7","author":"Okada","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.jtbi.2018.06.018","article-title":"A solution of private assessment in indirect reciprocity using solitary observation","volume":"455","author":"Okada","year":"2018","journal-title":"J. Theor. Biol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1016\/j.jtbi.2006.05.014","article-title":"The importance of subjectivity in perceptual errors on the emergence of indirect reciprocity","volume":"243","author":"Takahashi","year":"2006","journal-title":"J. Theor. Biol."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Inaba, M., and Takahashi, N. (2019). Linkage Based on the Kandori Norm Successfully Sustains Cooperation in Social Dilemmas. Games, 10.","DOI":"10.3390\/g10010010"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"12241","DOI":"10.1073\/pnas.1810565115","article-title":"Indirect reciprocity with private, noisy, and incomplete information","volume":"115","author":"Hilbe","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_36","unstructured":"Sugden, R. (1986). The Economics of Rights, Cooperation and Welfare, Basil Blackwell."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2495","DOI":"10.1098\/rspb.2001.1809","article-title":"Cooperation through indirect reciprocity: Image scoring or standing strategy?","volume":"268","author":"Milinski","year":"2001","journal-title":"Pr. R. Soc. B"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jeth.2001.2853","article-title":"Introduction to repeated games with private monitoring","volume":"102","author":"Kandori","year":"2002","journal-title":"J. Econ. Theo."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/BF02409751","article-title":"The dynamical theory of coevolution: A derivation from stochastic ecological processes","volume":"34","author":"Dieckmann","year":"1996","journal-title":"J. Math. Biol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"44146","DOI":"10.1038\/srep44146","article-title":"A norm knockout method on indirect reciprocity to reveal indispensable norms","volume":"7","author":"Yamamoto","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"14","DOI":"10.3389\/fphy.2018.00014","article-title":"A theoretical approach to norm ecosystems: Two adaptive architectures of indirect reciprocity show different paths to the evolution of cooperation","volume":"6","author":"Uchida","year":"2018","journal-title":"Front. Phys."}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/11\/1\/13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:59:32Z","timestamp":1760173172000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/11\/1\/13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,20]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["g11010013"],"URL":"https:\/\/doi.org\/10.3390\/g11010013","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2020,2,20]]}}}