{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T11:41:11Z","timestamp":1761824471266,"version":"3.37.3"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T00:00:00Z","timestamp":1677628800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"content-version":"vor","delay-in-days":12,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We enrich the theory of variational inequalities in the case of an aggregative structure by implementing recent results obtained by using the Selten\u2013Szidarovszky technique. We derive existence, semi-uniqueness and uniqueness results for solutions and provide a computational method. As an application we derive very powerful practical equilibrium results for Nash equilibria of sum-aggregative games and illustrate with Cournot oligopolies.\n<\/jats:p>","DOI":"10.1007\/s10957-023-02164-w","type":"journal-article","created":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T18:02:48Z","timestamp":1678730568000},"page":"1056-1092","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Aggregative Variational Inequalities"],"prefix":"10.1007","volume":"196","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5702-3340","authenticated-orcid":false,"given":"Peter Hubert Mathieu","family":"von Mouche","sequence":"first","affiliation":[]},{"given":"Ferenc","family":"Szidarovszky","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,13]]},"reference":[{"key":"2164_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02106-0","volume-title":"Nonlinear Oligopolies: Stability and Bifurcations","author":"G Bischi","year":"2010","unstructured":"Bischi, G., Chiarella, C., Kopel, M.O., Szidarovszky, F.: Nonlinear Oligopolies: Stability and Bifurcations. Springer, Heidelberg (2010)"},{"key":"2164_CR2","series-title":"Lecture Notes in Economics and Mathematical Systems","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22531-8","volume-title":"Theories of Imperfectly Competitive Markets","author":"LC Corch\u00f3n","year":"1996","unstructured":"Corch\u00f3n, L.C.: Theories of Imperfectly Competitive Markets. Lecture Notes in Economics and Mathematical Systems, vol. 442, 2nd edn. Springer, Berlin (1996)","edition":"2"},{"key":"2164_CR3","volume-title":"Well-Behaved Aggregative Games. Economic Discussion Paper May 24, School of Social Sciences","author":"R Cornes","year":"2011","unstructured":"Cornes, R., Hartley, R.: Well-Behaved Aggregative Games. Economic Discussion Paper May 24, School of Social Sciences. The University of Manchester, Manchester (2011)"},{"key":"2164_CR4","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/978-3-319-29254-0_5","volume-title":"Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi","author":"R Cornes","year":"2016","unstructured":"Cornes, R., Sato, T.: Existence and uniqueness of Nash equilibrium in aggregative games: an expository treatment. In: von Mouche, P.H.M., Quartieri, F. (eds.) Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi, pp. 47\u201361. Springer, Cham (2016)"},{"key":"2164_CR5","series-title":"Springer Series in Operational Research and Financial Engineering","volume-title":"Finite-Dimensional Variational Inequalities and Complementarity Problems","author":"F Facchinei","year":"2011","unstructured":"Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operational Research and Financial Engineering, Springer, Berlin (2011)"},{"key":"2164_CR6","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/BF01360282","volume":"159","author":"D Gale","year":"1965","unstructured":"Gale, D., Nikaido, H.: The Jacobian matrix and global univalence of mappings. Math. Ann. 159, 81\u201393 (1965)","journal-title":"Math. Ann."},{"issue":"2","key":"2164_CR7","doi-asserted-by":"publisher","first-page":"399","DOI":"10.2307\/2297975","volume":"58","author":"G Gaudet","year":"1991","unstructured":"Gaudet, G., Salant, S.W.: Uniqueness of Cournot equilibrium: new results from old methods. Rev. Econ. Stud. 58(2), 399\u2013404 (1991)","journal-title":"Rev. Econ. Stud."},{"issue":"2","key":"2164_CR8","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1515\/JAA.2003.261","volume":"9","author":"I Ginchev","year":"2003","unstructured":"Ginchev, I., Ivanov, V.I.: Second-order characterizations of convex and pseudoconvex functions. J. Appl. Anal. 9(2), 261\u2013273 (2003)","journal-title":"J. Appl. Anal."},{"key":"2164_CR9","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/BF02392210","volume":"115","author":"P Hartman","year":"1966","unstructured":"Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Math. 115, 271\u2013310 (1966)","journal-title":"Acta Math."},{"issue":"1","key":"2164_CR10","doi-asserted-by":"publisher","first-page":"1350,005","DOI":"10.1142\/S0219198913500059","volume":"15","author":"S Hirai","year":"2013","unstructured":"Hirai, S., Szidarovszky, F.: Existence and uniqueness of equilibrium in asymmetric contests with endogenous prizes. Int. Game Theory Rev. 15(1), 1350,005 (2013)","journal-title":"Int. Game Theory Rev."},{"key":"2164_CR11","doi-asserted-by":"publisher","first-page":"2033","DOI":"10.1007\/s11590-021-01780-7","volume":"16","author":"JI Itaya","year":"2022","unstructured":"Itaya, J.I., von Mouche, P.H.M.: Equilibrium uniqueness in aggregative games: very practical conditions. Optim. Lett. 16, 2033\u20132058 (2022)","journal-title":"Optim. Lett."},{"key":"2164_CR12","volume-title":"Spieltheorie","author":"C Kanzow","year":"2018","unstructured":"Kanzow, C., Schwartz, A.: Spieltheorie. Birkhauser, Basel (2018)"},{"key":"2164_CR13","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/BF00930577","volume":"4","author":"S Karamardian","year":"1969","unstructured":"Karamardian, S.: The nonlinear complementarity problem with applications, part 2. J. Optim. Theory Appl. 4, 167\u2013181 (1969)","journal-title":"J. Optim. Theory Appl."},{"issue":"4","key":"2164_CR14","doi-asserted-by":"publisher","first-page":"681","DOI":"10.2307\/2297489","volume":"54","author":"CD Kolstad","year":"1987","unstructured":"Kolstad, C.D., Mathiesen, L.: Necessary and sufficient conditions for uniqueness of a Cournot equilibrium. Rev. Econ. Stud. 54(4), 681\u2013690 (1987)","journal-title":"Rev. Econ. Stud."},{"key":"2164_CR15","series-title":"Advances in Mechanics and Mathematics","volume-title":"Variational Inequalities with Applications","author":"A Matei","year":"2009","unstructured":"Matei, A., Sofonea, M.: Variational Inequalities with Applications. Advances in Mechanics and Mathematics, Springer, Berlin (2009)"},{"key":"2164_CR16","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/BF00935475","volume":"34","author":"A Namatame","year":"1981","unstructured":"Namatame, A., Tse, E.: Adaptive expectations and dynamic adjustments in non-cooperative games with incomplete information. J. Optim. Theory Appl. 34, 243\u2013261 (1981)","journal-title":"J. Optim. Theory Appl."},{"key":"2164_CR17","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1007\/978-3-030-84122-5_35","volume-title":"Approximation and Computation in Science and Engineering","author":"MA Noor","year":"2022","unstructured":"Noor, M.A., Noor, K.I., Mohsen, B., Rassias, M.T., Raigorodskii, A.: General preinvex functions and variational-like inequalities. In: Daras, N.J., Rassias, T.M. (eds.) Approximation and Computation in Science and Engineering, pp. 643\u2013666. Springer, Berlin (2022)"},{"key":"2164_CR18","doi-asserted-by":"publisher","first-page":"981","DOI":"10.1007\/s10440-020-00366-2","volume":"170","author":"MA Noor","year":"2020","unstructured":"Noor, M.A., Noor, K.I., Rassia, M.T.: New trends in general variational inequalities. Acta Appl. Math. 170, 981\u20131064 (2020)","journal-title":"Acta Appl. Math."},{"key":"2164_CR19","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/0165-1765(83)90123-4","volume":"12","author":"K Okuguchi","year":"1983","unstructured":"Okuguchi, K.: The Cournot oligopoly and competitive equilibria as solutions to non-linear complementarity problems. Econ. Lett. 12, 127\u2013133 (1983)","journal-title":"Econ. Lett."},{"key":"2164_CR20","volume-title":"The Theory of Oligopoly with Multi-product Firms","author":"K Okuguchi","year":"2012","unstructured":"Okuguchi, K., Szidarovszky, F.: The Theory of Oligopoly with Multi-product Firms, 2nd edn. Springer, Berlin (2012)","edition":"2"},{"key":"2164_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-48888-7","volume-title":"Preispolitik der Mehrproduktunternehmung in der Statischen Theorie","author":"R Selten","year":"1970","unstructured":"Selten, R.: Preispolitik der Mehrproduktunternehmung in der Statischen Theorie. Springer, Berlin (1970)"},{"key":"2164_CR22","unstructured":"Szidarovszky, F.: On the oligopol game. Technical Report 1970-1. Karl Marx University of Economics, Budapest (1970)"},{"key":"2164_CR23","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1006\/game.1997.0517","volume":"18","author":"F Szidarovszky","year":"1997","unstructured":"Szidarovszky, F., Okuguchi, K.: On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games Econ. Behav. 18, 135\u2013140 (1997)","journal-title":"Games Econ. Behav."},{"key":"2164_CR24","doi-asserted-by":"publisher","first-page":"787","DOI":"10.2307\/2525963","volume":"18","author":"F Szidarovszky","year":"1977","unstructured":"Szidarovszky, F., Yakowitz, S.: A new proof of the existence and uniqueness of the Cournot equilibrium. Int. Econ. Rev. 18, 787\u2013789 (1977)","journal-title":"Int. Econ. Rev."},{"key":"2164_CR25","volume-title":"Oligopoly Pricing: Old Ideas and New Tools","author":"X Vives","year":"2001","unstructured":"Vives, X.: Oligopoly Pricing: Old Ideas and New Tools. MIT Press, Cambridge (2001)"},{"key":"2164_CR26","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/978-3-319-43838-2_8","volume-title":"Recent Advances in Game Theory and Applications","author":"PHM von Mouche","year":"2016","unstructured":"von Mouche, P.H.M.: The Selten\u2013Szidarovszky technique: the transformation part. In: Petrosyan, L.A., Mazalov, V.V. (eds.) Recent Advances in Game Theory and Applications, pp. 147\u2013164. Birkh\u00e4user, Cham (2016)"},{"key":"2164_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-29254-0","volume-title":"Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi","author":"PHM von Mouche","year":"2016","unstructured":"von Mouche, P.H.M., Quartieri, F.: Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi. Springer, Cham (2016)"},{"issue":"4","key":"2164_CR28","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1080\/02331934.2017.1405954","volume":"67","author":"PHM von Mouche","year":"2017","unstructured":"von Mouche, P.H.M., Quartieri, F.: Cournot equilibrium uniqueness via demi-concavity. Optimization 67(4), 441\u2013455 (2017)","journal-title":"Optimization"},{"issue":"2","key":"2164_CR29","first-page":"353","volume":"16","author":"PHM von Mouche","year":"2015","unstructured":"von Mouche, P.H.M., Yamazaki, T.: Sufficient and necessary conditions for equilibrium uniqueness in aggregative games. J. Nonlinear Convex Anal. 16(2), 353\u2013364 (2015)","journal-title":"J. Nonlinear Convex Anal."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-023-02164-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-023-02164-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-023-02164-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T03:03:38Z","timestamp":1679886218000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-023-02164-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["2164"],"URL":"https:\/\/doi.org\/10.1007\/s10957-023-02164-w","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2023,3]]},"assertion":[{"value":"6 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}