{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:33:04Z","timestamp":1777703584684,"version":"3.51.4"},"reference-count":33,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2017,2,24]],"date-time":"2017-02-24T00:00:00Z","timestamp":1487894400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2017,2,24]]},"abstract":"<jats:p>Considering the fact that the players may have different risk attitudes for the approximate values of strategies, this paper uses the possibility and necessity measures to denote the risk attitudes of the players, which are useful tools to address this issue. To do this, the concepts of possibility and necessity expectations of fuzzy variables are introduced, and several desirable properties are studied. Then, two vectors to denote the players\u2019 risk attitudes for their strategies are presented. Furthermore, the concept of possibility and necessity (PN) equilibrium strategy is defined, and the associated models are constructed, by which the optimal possibility and necessity equilibrium strategy is obtained. Meanwhile, a special case is briefly considered. Finally, a numerical example is offered to show the equilibrium strategy of the players.<\/jats:p>","DOI":"10.3233\/jifs-16229","type":"journal-article","created":{"date-parts":[[2017,2,28]],"date-time":"2017-02-28T11:33:29Z","timestamp":1488281609000},"page":"2195-2206","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":12,"title":["PN equilibrium strategy for matrix games with fuzzy payoffs"],"prefix":"10.1177","volume":"32","author":[{"given":"Chao","family":"Xu","sequence":"first","affiliation":[{"name":"School of International Audit, Nanjing Audit University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fanyong","family":"Meng","sequence":"additional","affiliation":[{"name":"School of International Audit, Nanjing Audit University, Nanjing, China"},{"name":"School of Business, Central South University, Changsha, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Management and Economics, Beijing Institute of Technology, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2017,2,24]]},"reference":[{"issue":"3","key":"e_1_3_2_2_2","first-page":"399","article-title":"Solutions for fuzzy matrix games","volume":"60","author":"Adem C.C.","year":"2010","unstructured":"AdemC.C. and MehmetA., Solutions for fuzzy matrix games, Applied Mathematics and Computation 60(3) (2010), 399\u2013410.","journal-title":"Applied Mathematics and Computation"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(89)90260-1"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2005.06.004"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(03)00178-7"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(87)90028-5"},{"key":"e_1_3_2_7_2","volume-title":"Possibility theory","author":"Duboisv D.","year":"1988","unstructured":"DuboisvD. and PradeH., Possibility theory, Academic Press, New York, London, 1988."},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(91)90050-Z"},{"key":"e_1_3_2_9_2","volume-title":"Fuzzy interval analysisThe handbook of fuzzy sets: Volume I. Fundamentals of fuzzy sets","author":"Dubois D.","year":"2000","unstructured":"DuboisD., KerreE., MesiarR. and PradeH., Fuzzy interval analysis, in: DuboisD., PradeH. (Eds.), The handbook of fuzzy sets: Volume I. Fundamentals of fuzzy sets. Kluwer Academic Publishers, Dordrecht, 2000."},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-5287-7"},{"key":"e_1_3_2_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2005.10.002"},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(02)00438-4"},{"key":"e_1_3_2_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2007.05.018"},{"key":"e_1_3_2_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2008.06.010"},{"key":"e_1_3_2_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2011.03.015"},{"key":"e_1_3_2_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.omega.2011.01.007"},{"key":"e_1_3_2_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/TFUZZ.2014.2333065"},{"key":"e_1_3_2_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-48476-0"},{"key":"e_1_3_2_19_2","doi-asserted-by":"publisher","DOI":"10.1109\/TFUZZ.2002.800692"},{"key":"e_1_3_2_20_2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218488503002016"},{"key":"e_1_3_2_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(02)00509-2"},{"key":"e_1_3_2_22_2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.2000.7142"},{"key":"e_1_3_2_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2005.06.021"},{"key":"e_1_3_2_24_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2010.09.002"},{"issue":"2","key":"e_1_3_2_25_2","first-page":"289","article-title":"Two-person zero-sum games with multiple fuzzy goals","volume":"4","author":"Nishizaki I.","year":"1992","unstructured":"NishizakiI. and SakawaM., Two-person zero-sum games with multiple fuzzy goals, Japanese Journal of Fuzzy Theory and Systems 4(2) (1992), 289\u2013300.","journal-title":"Japanese Journal of Fuzzy Theory and Systems"},{"key":"e_1_3_2_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(98)00455-2"},{"key":"e_1_3_2_27_2","doi-asserted-by":"publisher","DOI":"10.1504\/IJOR.2015.069182"},{"key":"e_1_3_2_28_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(94)90208-9"},{"key":"e_1_3_2_29_2","doi-asserted-by":"publisher","DOI":"10.1080\/02522667.2014.961785"},{"key":"e_1_3_2_30_2","doi-asserted-by":"publisher","DOI":"10.26708\/IJMSC.2015.1.5.10"},{"key":"e_1_3_2_31_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-007-9015-9"},{"key":"e_1_3_2_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.omega.2004.07.007"},{"key":"e_1_3_2_33_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-004-4202-4"},{"key":"e_1_3_2_34_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(78)90029-5"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-16229","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/JIFS-16229","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-16229","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:39:03Z","timestamp":1777455543000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/JIFS-16229"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,24]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,2,24]]}},"alternative-id":["10.3233\/JIFS-16229"],"URL":"https:\/\/doi.org\/10.3233\/jifs-16229","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,2,24]]}}}