{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T15:08:37Z","timestamp":1769353717548,"version":"3.49.0"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,7,13]],"date-time":"2018-07-13T00:00:00Z","timestamp":1531440000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2018,9]]},"DOI":"10.1007\/s10957-018-1343-0","type":"journal-article","created":{"date-parts":[[2018,7,13]],"date-time":"2018-07-13T15:43:22Z","timestamp":1531496602000},"page":"998-1013","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["A Characterization of Nash Equilibrium for the Games with Random Payoffs"],"prefix":"10.1007","volume":"178","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2253-9528","authenticated-orcid":false,"given":"Vikas Vikram","family":"Singh","sequence":"first","affiliation":[]},{"given":"Abdel","family":"Lisser","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,7,13]]},"reference":[{"key":"1343_CR1","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/BF01448847","volume":"100","author":"J Neumann von","year":"1928","unstructured":"von Neumann, J.: On the theory of games. Math. Ann. 100, 295\u2013320 (1928)","journal-title":"Math. Ann."},{"key":"1343_CR2","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1073\/pnas.36.1.48","volume":"36","author":"JF Nash","year":"1950","unstructured":"Nash, J.F.: Equilibrium points in n-person games. Proc. Nat. Acad. Sci. 36, 48\u201349 (1950)","journal-title":"Proc. Nat. Acad. Sci."},{"key":"1343_CR3","doi-asserted-by":"publisher","first-page":"348","DOI":"10.1016\/0022-247X(64)90021-6","volume":"9","author":"OL Mangasarian","year":"1964","unstructured":"Mangasarian, O.L., Stone, H.: Two-person nonzero-sum games and quadratic programming. J. Math. Anal. Appl. 9, 348\u2013355 (1964)","journal-title":"J. Math. Anal. Appl."},{"key":"1343_CR4","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/s00500-007-0207-6","volume":"12","author":"WD Collins","year":"2008","unstructured":"Collins, W.D., Hu, C.: Studying interval valued matrix games with fuzzy logic. Soft. Comput. 12, 147\u2013155 (2008)","journal-title":"Soft. Comput."},{"key":"1343_CR5","doi-asserted-by":"crossref","unstructured":"Li, D.-F., Nan, J.X., Zhang, M.J.: Interval programming models for matrix games with interval payoffs. Optim. Methods Softw. 27, 1\u201316 (2012)","DOI":"10.1080\/10556781003796622"},{"key":"1343_CR6","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s10700-012-9145-6","volume":"12","author":"J Gao","year":"2013","unstructured":"Gao, J.: Uncertain bimatrix game with applications. Fuzzy Optim. Decis. Mak. 12, 65\u201378 (2013)","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"1343_CR7","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1016\/j.omega.2011.01.007","volume":"39","author":"DF Li","year":"2011","unstructured":"Li, D.F.: Linear programming approach to solve interval-valued matrix games. J. Omega 39, 655\u2013666 (2011)","journal-title":"J. Omega"},{"key":"1343_CR8","doi-asserted-by":"publisher","first-page":"1283","DOI":"10.1109\/TPWRS.2005.851954","volume":"20","author":"P Couchman","year":"2005","unstructured":"Couchman, P., Kouvaritakis, B., Cannon, M., Prashad, F.: Gaming strategy for electric power with random demand. IEEE Trans. Power Syst. 20, 1283\u20131292 (2005)","journal-title":"IEEE Trans. Power Syst."},{"key":"1343_CR9","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1002\/etep.650","volume":"23","author":"M Mazadi","year":"2013","unstructured":"Mazadi, M., Rosehart, W.D., Zareipour, H., Malik, O.P., Oloomi, M.: Impact of wind integration on electricity markets: a chance-constrained Nash-Cournot model. Int. Trans. Electr. Energy Syst. 23, 83\u201396 (2013)","journal-title":"Int. Trans. Electr. Energy Syst."},{"key":"1343_CR10","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1109\/TPWRS.2006.889142","volume":"22","author":"J Valenzuela","year":"2007","unstructured":"Valenzuela, J., Mazumdar, M.: Cournot prices considering generator availability and demand uncertainty. IEEE Trans. Power Syst. 22, 116\u2013125 (2007)","journal-title":"IEEE Trans. Power Syst."},{"key":"1343_CR11","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1287\/mnsc.43.2.190","volume":"43","author":"DD Wolf","year":"1997","unstructured":"Wolf, D.D., Smeers, Y.: A stochastic version of a Stackelberg-Nash-Cournot equilibrium model. Manag. Sci. 43, 190\u2013197 (1997)","journal-title":"Manag. Sci."},{"key":"1343_CR12","doi-asserted-by":"publisher","first-page":"548","DOI":"10.1137\/S1052623402420528","volume":"14","author":"D Dentcheva","year":"2003","unstructured":"Dentcheva, D., Ruszczy\u0144ski, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14, 548\u2013566 (2003)","journal-title":"SIAM J. Optim."},{"key":"1343_CR13","doi-asserted-by":"publisher","first-page":"1220","DOI":"10.1287\/opre.1080.0686","volume":"57","author":"V DeMiguel","year":"2009","unstructured":"DeMiguel, V., Xu, H.: A stochastic multiple leader Stackelberg model: analysis, computation, and application. Oper. Res. 57, 1220\u20131235 (2009)","journal-title":"Oper. Res."},{"key":"1343_CR14","doi-asserted-by":"publisher","first-page":"1050","DOI":"10.1007\/s10957-014-0673-9","volume":"165","author":"B Jadamba","year":"2015","unstructured":"Jadamba, B., Raciti, F.: Variational inequality approach to stochastic Nash equilibrium problems with an application to Cournot oligopoly. J. Optim. Theory Appl. 165, 1050\u20131070 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"1343_CR15","doi-asserted-by":"publisher","first-page":"1168","DOI":"10.1137\/100792644","volume":"21","author":"U Ravat","year":"2011","unstructured":"Ravat, U., Shanbhag, U.V.: On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games. SIAM J. Optim. 21, 1168\u20131199 (2011)","journal-title":"SIAM J. Optim."},{"key":"1343_CR16","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1007\/s10589-013-9538-7","volume":"55","author":"H Xu","year":"2013","unstructured":"Xu, H., Zhang, D.: Stochastic Nash equilibrium problems: sample average approximation and applications. Comput. Optim. Appl. 55, 597\u2013645 (2013)","journal-title":"Comput. Optim. Appl."},{"key":"1343_CR17","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1007\/s10957-016-0888-z","volume":"171","author":"F Faraci","year":"2016","unstructured":"Faraci, F., Jadamba, B., Raciti, F.: On stochastic variational inequalities with mean value constraints. J. Optim. Theory Appl. 171, 675\u2013693 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"1343_CR18","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1287\/opre.11.1.18","volume":"11","author":"A Charnes","year":"1963","unstructured":"Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11, 18\u201339 (1963)","journal-title":"Oper. Res."},{"key":"1343_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-3087-7","volume-title":"Stochastic Programming","author":"A Pr\u00e9kopa","year":"1995","unstructured":"Pr\u00e9kopa, A.: Stochastic Programming. Springer, Netherlands (1995)"},{"key":"1343_CR20","doi-asserted-by":"publisher","first-page":"640","DOI":"10.1016\/j.orl.2016.07.013","volume":"44","author":"VV Singh","year":"2016","unstructured":"Singh, V.V., Jouini, O., Lisser, A.: Existence of Nash equilibrium for chance-constrained games. Oper. Res. Lett. 44, 640\u2013644 (2016)","journal-title":"Oper. Res. Lett."},{"key":"1343_CR21","doi-asserted-by":"publisher","first-page":"1385","DOI":"10.1007\/s11590-016-1077-6","volume":"11","author":"VV Singh","year":"2017","unstructured":"Singh, V.V., Jouini, O., Lisser, A.: Distributionally robust chance-constrained games: existence and characterization of Nash equilibrium. Optim. Lett. 11, 1385\u20131405 (2017)","journal-title":"Optim. Lett."},{"key":"1343_CR22","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.ejor.2015.12.024","volume":"252","author":"J Cheng","year":"2016","unstructured":"Cheng, J., Leung, J., Lisser, A.: Random-payoff two-person zero-sum game with joint chance constraints. Eur. J. Oper. Res. 252, 213\u2013219 (2016)","journal-title":"Eur. J. Oper. Res."},{"key":"1343_CR23","doi-asserted-by":"publisher","first-page":"1243","DOI":"10.1287\/opre.22.6.1243","volume":"22","author":"RA Blau","year":"1974","unstructured":"Blau, R.A.: Random-payoff two person zero-sum games. Oper. Res. 22, 1243\u20131251 (1974)","journal-title":"Oper. Res."},{"key":"1343_CR24","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1287\/mnsc.19.3.266","volume":"19","author":"RG Cassidy","year":"1972","unstructured":"Cassidy, R.G., Field, C.A., Kirby, M.J.L.: Solution of a satisficing model for random payoff games. Manag. Sci. 19, 266\u2013271 (1972)","journal-title":"Manag. Sci."},{"key":"1343_CR25","doi-asserted-by":"publisher","first-page":"628","DOI":"10.1137\/1113079","volume":"13","author":"A Charnes","year":"1968","unstructured":"Charnes, A., Kirby, M.J.L., Raike, W.M.: Zero-zero chance-constrained games. Theory Probab. Appl. 13, 628\u2013646 (1968)","journal-title":"Theory Probab. Appl."},{"key":"1343_CR26","doi-asserted-by":"crossref","unstructured":"Song, T.: Systems and Management Science by Extremal Methods, chap. On random payoff matrix games, pp. 291\u2013308. Springer, Berlin (1992)","DOI":"10.1007\/978-1-4615-3600-0_19"},{"key":"1343_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-2937-2","volume-title":"Symmetric Multivariate and Related Distributions","author":"KT Fang","year":"1990","unstructured":"Fang, K.T., Kotz, S., Ng, K.W.: Symmetric Multivariate and Related Distributions. Chapman and Hall, London (1990)"},{"key":"1343_CR28","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"Rockafellar R.T","year":"1970","unstructured":"R.T, Rockafellar: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"1343_CR29","doi-asserted-by":"publisher","DOI":"10.1002\/0471787779","volume-title":"Nonlinear Programming: Theory and Algorithms","author":"M Bazaraa","year":"2006","unstructured":"Bazaraa, M., Sherali, H., Shetty, C.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, USA (2006)","edition":"3"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-018-1343-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-018-1343-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-018-1343-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,12]],"date-time":"2019-07-12T19:15:25Z","timestamp":1562958925000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-018-1343-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,13]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,9]]}},"alternative-id":["1343"],"URL":"https:\/\/doi.org\/10.1007\/s10957-018-1343-0","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,13]]},"assertion":[{"value":"18 September 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 June 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 July 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}