{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T04:46:37Z","timestamp":1770698797040,"version":"3.49.0"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,7,28]],"date-time":"2018-07-28T00:00:00Z","timestamp":1532736000000},"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":["Dyn Games Appl"],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s13235-018-0277-3","type":"journal-article","created":{"date-parts":[[2018,7,27]],"date-time":"2018-07-27T22:52:40Z","timestamp":1532731960000},"page":"416-432","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Subgame Perfect Nash Equilibrium: A\u00a0Learning\u00a0Approach via Costs to Move"],"prefix":"10.1007","volume":"9","author":[{"given":"Francesco","family":"Caruso","sequence":"first","affiliation":[]},{"given":"Maria Carmela","family":"Ceparano","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6984-9617","authenticated-orcid":false,"given":"Jacqueline","family":"Morgan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,7,28]]},"reference":[{"key":"277_CR1","volume-title":"Variational convergence for functions and operators","author":"H Attouch","year":"1984","unstructured":"Attouch H (1984) Variational convergence for functions and operators. Pitman Advanced Publishing Program, Boston"},{"key":"277_CR2","first-page":"207","volume":"13","author":"H Attouch","year":"2006","unstructured":"Attouch H, Soubeyran A (2006) Inertia and reactivity in decision making as cognitive variational inequalities. J Convex Anal 13:207","journal-title":"J Convex Anal"},{"key":"277_CR3","doi-asserted-by":"publisher","first-page":"1061","DOI":"10.1137\/060657248","volume":"18","author":"H Attouch","year":"2007","unstructured":"Attouch H, Redont P, Soubeyran A (2007) A new class of alternating proximal minimization algorithms with costs-to-move. SIAM J Optim 18:1061\u20131081. \n                    https:\/\/doi.org\/10.1137\/060657248","journal-title":"SIAM J Optim"},{"key":"277_CR4","volume-title":"Dynamic noncooperative game theory","author":"T Ba\u015far","year":"1999","unstructured":"Ba\u015far T, Olsder G (1999) Dynamic noncooperative game theory. SIAM, University City"},{"key":"277_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex analysis and monotone operator theory in Hilbert spaces","author":"H Bauschke","year":"2011","unstructured":"Bauschke H, Combettes P (2011) Convex analysis and monotone operator theory in Hilbert spaces. Springer, Berlin"},{"key":"277_CR6","unstructured":"Bich P (2016) Prudent equilibria and strategic uncertainty in discontinuous games. \n                    https:\/\/halshs.archives-ouvertes.fr\/halshs-01337293\/file\/prudent-equilibria-preprint-24-juin.pdf\n                    \n                  . Accessed 24 June 2016 (preprint)"},{"key":"277_CR7","doi-asserted-by":"publisher","first-page":"3224","DOI":"10.1137\/100795450","volume":"50","author":"H Bonnel","year":"2012","unstructured":"Bonnel H, Morgan J (2012) Semivectorial bilevel convex optimal control problems: existence results. SIAM J Control Optim 50:3224\u20133241. \n                    https:\/\/doi.org\/10.1137\/100795450","journal-title":"SIAM J Control Optim"},{"key":"277_CR8","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/BF00939867","volume":"59","author":"M Breton","year":"1988","unstructured":"Breton M, Alj A, Haurie A (1988) Sequential Stackelberg equilibria in two-person games. J Optim Theory Appl 59:71\u201397. \n                    https:\/\/doi.org\/10.1007\/BF00939867","journal-title":"J Optim Theory Appl"},{"key":"277_CR9","doi-asserted-by":"publisher","first-page":"1208","DOI":"10.1016\/j.jmaa.2017.11.031","volume":"459","author":"F Caruso","year":"2018","unstructured":"Caruso F, Ceparano M, Morgan J (2018) Uniqueness of Nash equilibrium in continuous two-player weighted potential games. J Math Anal Appl 459:1208\u20131221. \n                    https:\/\/doi.org\/10.1016\/j.jmaa.2017.11.031","journal-title":"J Math Anal Appl"},{"key":"277_CR10","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1016\/j.jmateco.2017.02.011","volume":"70","author":"M Ceparano","year":"2017","unstructured":"Ceparano M, Quartieri F (2017) Nash equilibrium uniqueness in nice games with isotone best replies. J Math Econ 70:154\u2013165. \n                    https:\/\/doi.org\/10.1016\/j.jmateco.2017.02.011","journal-title":"J Math Econ"},{"key":"277_CR11","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1109\/TAC.1972.1100179","volume":"17","author":"C Chen","year":"1972","unstructured":"Chen C, Cruz J (1972) Stackelberg solution for two-person games with biased information patterns. IEEE Trans Autom Control 17:791\u2013798. \n                    https:\/\/doi.org\/10.1109\/TAC.1972.1100179","journal-title":"IEEE Trans Autom Control"},{"key":"277_CR12","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/s10479-007-0176-2","volume":"153","author":"B Colson","year":"2007","unstructured":"Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153:235\u2013256. \n                    https:\/\/doi.org\/10.1007\/s10479-007-0176-2","journal-title":"Ann Oper Res"},{"key":"277_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0327-8","volume-title":"An introduction to $$\\Gamma $$ \u0393","author":"G Maso Dal","year":"1993","unstructured":"Dal Maso G (1993) An introduction to \n                    \n                      \n                    \n                    $$\\Gamma $$\n                    \n                      \n                        \u0393\n                      \n                    \n                  -convergence. Birkhauser, Basel"},{"key":"277_CR14","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1007\/s10957-008-9353-y","volume":"137","author":"G Marco De","year":"2008","unstructured":"De Marco G, Morgan J (2008) Slightly altruistic equilibria. J Optim Theory and Appl 137:347\u2013362. \n                    https:\/\/doi.org\/10.1007\/s10957-008-9353-y","journal-title":"J Optim Theory and Appl"},{"key":"277_CR15","volume-title":"Foundations of bilevel programming","author":"S Dempe","year":"2002","unstructured":"Dempe S (2002) Foundations of bilevel programming. Springer, New York"},{"issue":"4","key":"277_CR16","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1080\/02331934.2012.696641","volume":"63","author":"S Dempe","year":"2014","unstructured":"Dempe S, Mordukhovich B, Zemkoho A (2014) Necessary optimality conditions in pessimistic bilevel programming. Optimization 63(4):505\u2013533. \n                    https:\/\/doi.org\/10.1080\/02331934.2012.696641","journal-title":"Optimization"},{"key":"277_CR17","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/BF02614504","volume":"78","author":"S Fl\u00e5m","year":"1996","unstructured":"Fl\u00e5m S, Antipin A (1996) Equilibrium programming using proximal-like algorithms. Math Program 78:29\u201341. \n                    https:\/\/doi.org\/10.1007\/BF02614504","journal-title":"Math Program"},{"key":"#cr-split#-277_CR18.1","unstructured":"Fl\u00e5m S, Greco G (1992) Non-cooperative games"},{"key":"#cr-split#-277_CR18.2","unstructured":"methods of subgradient projection and proximal point. In: Oettli W, Pallaschke D (eds) Advances in optimization: proceedings of the 6th French-German colloquium on optimization held at Lambrecht, FRG, June 2-8, 1991. Springer, Berlin, pp 406-419"},{"key":"277_CR19","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/BF01737572","volume":"2","author":"J Harsanyi","year":"1973","unstructured":"Harsanyi J (1973) Oddness of the number of equilibrium points: a new proof. Int J Game Theory 2:235\u2013250","journal-title":"Int J Game Theory"},{"key":"277_CR20","doi-asserted-by":"publisher","DOI":"10.1142\/8442","volume-title":"Games and dynamic games","author":"A Haurie","year":"2012","unstructured":"Haurie A, Krawczyk J, Zaccour G (2012) Games and dynamic games. World Scientific Publishing Company, Singapore"},{"key":"277_CR21","doi-asserted-by":"publisher","first-page":"650","DOI":"10.1287\/moor.10.4.650","volume":"10","author":"M Kojima","year":"1985","unstructured":"Kojima M, Okada A, Shindoh S (1985) Strongly stable equilibrium points of n-person noncooperative games. Math Oper Res 10:650\u2013663","journal-title":"Math Oper Res"},{"key":"277_CR22","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1007\/BF00933155","volume":"26","author":"G Leitmann","year":"1978","unstructured":"Leitmann G (1978) On generalized Stackelberg strategies. J Optim Theory Appl 26:637\u2013643. \n                    https:\/\/doi.org\/10.1007\/BF00933155","journal-title":"J Optim Theory Appl"},{"key":"277_CR23","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF02191740","volume":"84","author":"M Lignola","year":"1995","unstructured":"Lignola M, Morgan J (1995) Topological existence and stability for Stackelberg problems. J Optim Theory Appl 84:145\u2013169. \n                    https:\/\/doi.org\/10.1007\/BF02191740","journal-title":"J Optim Theory Appl"},{"key":"277_CR24","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1023\/A:1022695113803","volume":"93","author":"M Lignola","year":"1997","unstructured":"Lignola M, Morgan J (1997) Stability of regularized bilevel programming problems. J Optim Theory Appl 93:575\u2013596. \n                    https:\/\/doi.org\/10.1023\/A:1022695113803","journal-title":"J Optim Theory Appl"},{"key":"277_CR25","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/s10957-017-1085-4","volume":"173","author":"M Lignola","year":"2017","unstructured":"Lignola M, Morgan J (2017) Inner regularizations and viscosity solutions for pessimistic bilevel optimization problems. J Optim Theory Appl 173:183\u2013202. \n                    https:\/\/doi.org\/10.1007\/s10957-017-1085-4","journal-title":"J Optim Theory Appl"},{"key":"277_CR26","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/BF00121269","volume":"8","author":"P Loridan","year":"1996","unstructured":"Loridan P, Morgan J (1996) Weak via strong Stackelberg problem: new results. J Global Optim 8:263\u2013287. \n                    https:\/\/doi.org\/10.1007\/BF00121269","journal-title":"J Global Optim"},{"key":"277_CR27","first-page":"154","volume":"4","author":"B Martinet","year":"1970","unstructured":"Martinet B (1970) Br\u00e8ve communication r\u00e9gularisation d\u2019in\u00e9quations variationnelles par approximations successives. Mod\u00e9lisation Math\u00e9matique et Analyse Num\u00e9rique (ESAIM Math Model Numer Anal) 4:154\u2013158","journal-title":"Mod\u00e9lisation Math\u00e9matique et Analyse Num\u00e9rique (ESAIM Math Model Numer Anal)"},{"key":"277_CR28","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511794216","volume-title":"Game theory","author":"M Maschler","year":"2013","unstructured":"Maschler M, Solan E, Zamir S (2013) Game theory. Cambridge University Press, Cambridge. \n                    https:\/\/doi.org\/10.1017\/CBO9780511794216"},{"key":"277_CR29","doi-asserted-by":"publisher","first-page":"273","DOI":"10.24033\/bsmf.1625","volume":"93","author":"J Moreau","year":"1965","unstructured":"Moreau J (1965) Proximit\u00e9 et dualit\u00e9 dans un espace hilbertien. Bulletin de la Soci\u00e9t\u00e9 math\u00e9matique de France 93:273\u2013299","journal-title":"Bulletin de la Soci\u00e9t\u00e9 math\u00e9matique de France"},{"key":"277_CR30","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1007\/978-1-4757-6019-4_18","volume-title":"Nonsmooth optimization and related topics","author":"J Morgan","year":"1989","unstructured":"Morgan J (1989) Constrained well-posed two-level optimization problems. In: Clarke F, Dem\u2019yanov V, Giannessi F (eds) Nonsmooth optimization and related topics. Springer, Boston, pp 307\u2013325"},{"key":"277_CR31","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/0-8176-4501-2_12","volume-title":"Advances in dynamic games: applications to economics, management science, engineering, and environmental management","author":"J Morgan","year":"2006","unstructured":"Morgan J, Patrone F (2006) Stackelberg problems: subgame perfect equilibria via Tikhonov regularization. In: Haurie A, Muto S, Petrosjan L, Raghavan T (eds) Advances in dynamic games: applications to economics, management science, engineering, and environmental management. Birkh\u00e4user Boston, Boston, pp 209\u2013221"},{"key":"277_CR32","first-page":"91","volume":"15","author":"A Moudafi","year":"1999","unstructured":"Moudafi A (1999) Proximal point algorithm extended to equilibrium problems. J Nat Geom 15:91\u2013100","journal-title":"J Nat Geom"},{"key":"277_CR33","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/BF01753236","volume":"7","author":"R Myerson","year":"1978","unstructured":"Myerson R (1978) Refinements of Nash equilibrium concept. Int J Game Theory 7:73\u201380","journal-title":"Int J Game Theory"},{"key":"277_CR34","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/BF01769260","volume":"10","author":"A Okada","year":"1981","unstructured":"Okada A (1981) On stability of perfect equilibrium points. Int J Game Theory 10:67\u201373","journal-title":"Int J Game Theory"},{"key":"277_CR35","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1561\/2400000003","volume":"1","author":"N Parikh","year":"2014","unstructured":"Parikh N, Boyd S (2014) Proximal algorithms. Found Trends Optim 1:127\u2013239. \n                    https:\/\/doi.org\/10.1561\/2400000003","journal-title":"Found Trends Optim"},{"key":"277_CR36","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"R Rockafellar","year":"1976","unstructured":"Rockafellar R (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14:877\u2013898. \n                    https:\/\/doi.org\/10.1137\/0314056","journal-title":"SIAM J Control Optim"},{"key":"277_CR37","doi-asserted-by":"publisher","first-page":"520","DOI":"10.2307\/1911749","volume":"33","author":"J Rosen","year":"1965","unstructured":"Rosen J (1965) Existence and uniqueness of equilibrium points for concave \n                    \n                      \n                    \n                    $$n$$\n                    \n                      \n                        n\n                      \n                    \n                  -person games. Econometrica 33:520\u2013534","journal-title":"Econometrica"},{"key":"277_CR38","unstructured":"Selten R (1965) Spieltheoretische behandlung eines oligopolmodells mit nachfragetr\u00e4gheit. Zeitschrift f\u00fcr die gesamte Staatswissenschaft 301\u2013324, 667\u2013689"},{"key":"277_CR39","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/BF01766400","volume":"4","author":"R Selten","year":"1975","unstructured":"Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25\u201355","journal-title":"Int J Game Theory"},{"key":"277_CR40","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/BF00935665","volume":"11","author":"M Simaan","year":"1973","unstructured":"Simaan M, Cruz JB (1973) On the Stackelberg strategy in nonzero-sum games. J Optim Theory Appl 11:533\u2013555. \n                    https:\/\/doi.org\/10.1007\/BF00935665","journal-title":"J Optim Theory Appl"},{"key":"277_CR41","first-page":"1035","volume":"4","author":"A Tikhonov","year":"1963","unstructured":"Tikhonov A (1963) Solution of incorrectly formulated problems and the regularization method. Soviet Math Dokl 4:1035\u20131038","journal-title":"Soviet Math Dokl"},{"key":"277_CR42","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1007\/BF01096458","volume":"5","author":"L Vicente","year":"1994","unstructured":"Vicente L, Calamai P (1994) Bilevel and multilevel programming: a bibliography review. J Glob Optim 5:291\u2013306. \n                    https:\/\/doi.org\/10.1007\/BF01096458","journal-title":"J Glob Optim"},{"key":"277_CR43","volume-title":"The theory of the market economy","author":"H Stackelberg von","year":"1952","unstructured":"von Stackelberg H (1952) The theory of the market economy. Oxford University Press, Oxford"},{"key":"277_CR44","first-page":"1307","volume":"11","author":"W Wen-Tsun","year":"1962","unstructured":"Wen-Tsun W, Jia-He J (1962) Essential equilibrium points of n-person non-cooperative games. Sci Sin 11:1307\u20131322","journal-title":"Sci Sin"},{"key":"277_CR45","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1137\/120864015","volume":"23","author":"W Wiesemann","year":"2013","unstructured":"Wiesemann W, Tsoukalas A, Kleniati PM, Rustem B (2013) Pessimistic bilevel optimization. SIAM J Optim 23:353\u2013380. \n                    https:\/\/doi.org\/10.1137\/120864015","journal-title":"SIAM J Optim"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-018-0277-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s13235-018-0277-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-018-0277-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,27]],"date-time":"2019-07-27T19:07:30Z","timestamp":1564254450000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s13235-018-0277-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,28]]},"references-count":46,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["277"],"URL":"https:\/\/doi.org\/10.1007\/s13235-018-0277-3","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"value":"2153-0785","type":"print"},{"value":"2153-0793","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,28]]},"assertion":[{"value":"28 July 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}