{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T04:03:26Z","timestamp":1768881806107,"version":"3.49.0"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,1,19]],"date-time":"2022-01-19T00:00:00Z","timestamp":1642550400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,19]],"date-time":"2022-01-19T00:00:00Z","timestamp":1642550400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1007\/s11128-021-03395-4","type":"journal-article","created":{"date-parts":[[2022,1,19]],"date-time":"2022-01-19T08:03:43Z","timestamp":1642579423000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["The influence of entanglement on complex dynamics of a quantum Stackelberg duopoly with heterogeneous expectations"],"prefix":"10.1007","volume":"21","author":[{"given":"Xinli","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Chao","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Yihan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Shuning","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,1,19]]},"reference":[{"issue":"4","key":"3395_CR1","first-page":"1089","volume":"42","author":"SE Landsburg","year":"2011","unstructured":"Landsburg, S.E.: Quantum game theory. Not. AMS 42(4), 1089\u20131099 (2011)","journal-title":"Not. AMS"},{"issue":"5","key":"3395_CR2","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1103\/PhysRevLett.82.1052","volume":"82","author":"DA Meyer","year":"1999","unstructured":"Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82(5), 1052\u20131055 (1999)","journal-title":"Phys. Rev. Lett."},{"issue":"15","key":"3395_CR3","doi-asserted-by":"publisher","first-page":"3077","DOI":"10.1103\/PhysRevLett.83.3077","volume":"83","author":"J Eisert","year":"1999","unstructured":"Eisert, J., Wilkens, M., Lewenstein, M.: Quantum games and quantum strategies. Phys. Rev. Lett. 83(15), 3077\u20133080 (1999)","journal-title":"Phys. Rev. Lett."},{"issue":"5\u20136","key":"3395_CR4","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1016\/S0375-9601(00)00441-2","volume":"272","author":"L Marinatto","year":"2000","unstructured":"Marinatto, L., Weber, T.: A quantum approach to static games of complete information. Phys. Lett. A 272(5\u20136), 291\u2013303 (2000)","journal-title":"Phys. Lett. A"},{"issue":"1\u20132","key":"3395_CR5","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1016\/S0378-4371(02)00842-7","volume":"312","author":"EW Piotrowski","year":"2002","unstructured":"Piotrowski, E.W., Sladkowski, J.: Quantum market games. Phys. A 312(1\u20132), 208\u2013216 (2002)","journal-title":"Phys. A"},{"issue":"5","key":"3395_CR6","doi-asserted-by":"publisher","first-page":"1089","DOI":"10.1023\/A:1025443111388","volume":"42","author":"EW Piotrowski","year":"2003","unstructured":"Piotrowski, E.W., Sladkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys. 42(5), 1089\u20131099 (2003)","journal-title":"Int. J. Theor. Phys."},{"issue":"5","key":"3395_CR7","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1016\/S0370-1573(02)00266-1","volume":"369","author":"M Keyl","year":"2002","unstructured":"Keyl, M.: Fundamentals of information theory. Phys. Rep. 369(5), 431\u2013548 (2002)","journal-title":"Phys. Rep."},{"key":"3395_CR8","doi-asserted-by":"publisher","first-page":"3129","DOI":"10.1098\/rspa.1999.0443","volume":"455","author":"D Deutsch","year":"1999","unstructured":"Deutsch, D.: Quantum theory of probability and decisions. Proc. R. Soc. Lond. A 455, 3129\u20133137 (1999)","journal-title":"Proc. R. Soc. Lond. A"},{"issue":"1","key":"3395_CR9","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1016\/j.dss.2008.07.001","volume":"46","author":"H Guo","year":"2008","unstructured":"Guo, H., Zhang, J., Koehler, G.J.: A survey of quantum games. Decis. Support Syst. 46(1), 318\u2013332 (2008)","journal-title":"Decis. Support Syst."},{"key":"3395_CR10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511997716","volume-title":"Quantum Models of Cognition and Decision","author":"JR Busemeyer","year":"2012","unstructured":"Busemeyer, J.R., Bruza, P.D.: Quantum Models of Cognition and Decision. Cambridge University Press, Cambridge (2012)"},{"issue":"2","key":"3395_CR11","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/S0375-9601(02)01628-6","volume":"306","author":"H Li","year":"2002","unstructured":"Li, H., Du, J.F., Massar, S.: Continuous-variable quantum games. Phys. Lett. A 306(2), 73\u201378 (2002)","journal-title":"Phys. Lett. A"},{"issue":"4\u20135","key":"3395_CR12","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1016\/j.physleta.2003.09.047","volume":"318","author":"CF Lo","year":"2003","unstructured":"Lo, C.F., Kiang, D.: Quantum Stackelberg duopoly. Phys. Lett. A 318(4\u20135), 333\u2013336 (2003)","journal-title":"Phys. Lett. A"},{"issue":"1\u20133","key":"3395_CR13","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.physleta.2005.07.055","volume":"346","author":"CF Lo","year":"2005","unstructured":"Lo, C.F., Kiang, D.: Quantum Stackelberg duopoly with incomplete information. Phys. Lett. A 346(1\u20133), 65\u201370 (2005)","journal-title":"Phys. Lett. A"},{"issue":"11","key":"3395_CR14","doi-asserted-by":"publisher","first-page":"3040","DOI":"10.1088\/0256-307X\/24\/11\/007","volume":"24","author":"X Wang","year":"2007","unstructured":"Wang, X., Yang, X.H., Miao, L., et al.: Quantum Stackelberg duopoly of continuous distributed asymmetric information. Chin. Phys. Lett. 24(11), 3040\u20133043 (2007)","journal-title":"Chin. Phys. Lett."},{"issue":"12","key":"3395_CR15","doi-asserted-by":"publisher","first-page":"120302","DOI":"10.1088\/0256-307X\/30\/12\/120302","volume":"30","author":"X Wang","year":"2013","unstructured":"Wang, X., Liu, D., Zhang, J.P.: Asymmetric model of the quantum Stackelberg duopoly. Chin. Phys. Lett. 30(12), 120302\u2013120306 (2013)","journal-title":"Chin. Phys. Lett."},{"issue":"9","key":"3395_CR16","first-page":"3643","volume":"8","author":"R Yu","year":"2012","unstructured":"Yu, R., Xiao, R.: Quantum Stackelberg duopoly with isoelastic demand function. J. Comput. Inform. Syst. 8(9), 3643\u20133650 (2012)","journal-title":"J. Comput. Inform. Syst."},{"issue":"43","key":"3395_CR17","doi-asserted-by":"publisher","first-page":"375301","DOI":"10.1088\/1751-8113\/43\/37\/375301","volume":"43","author":"S Khan","year":"2010","unstructured":"Khan, S., Ramzan, M., Khan, M.K.: Quantum Stackelberg duopoly in the presence of correlated noise. J. Phys. A: Math. Theor. 43(43), 375301\u2013375314 (2010)","journal-title":"J. Phys. A: Math. Theor."},{"issue":"7","key":"3395_CR18","doi-asserted-by":"publisher","first-page":"70202","DOI":"10.1088\/0256-307X\/28\/7\/070202","volume":"28","author":"S Khan","year":"2011","unstructured":"Khan, S., Khan, M.K.: Quantum Stackelberg duopoly in noninertial frame. Chin. Phys. Lett. 28(7), 70202\u201370203 (2011)","journal-title":"Chin. Phys. Lett."},{"issue":"48","key":"3395_CR19","doi-asserted-by":"publisher","first-page":"3463","DOI":"10.1016\/j.physleta.2018.10.002","volume":"382","author":"P Frackiewicz","year":"2018","unstructured":"Frackiewicz, P.: On subgame perfect equilibria in quantum Stackelberg duopoly with incompete information. Phys. Lett. A 382(48), 3463\u20133469 (2018)","journal-title":"Phys. Lett. A"},{"key":"3395_CR20","volume-title":"Game Theory: Strategies, Equilibria and Theorems","author":"NH Ingrid","year":"2009","unstructured":"Ingrid, N.H., Anna, S.N.: Game Theory: Strategies, Equilibria and Theorems. Nova Science Publishers, Inc, New York (2009)"},{"key":"3395_CR21","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s11128-018-2082-8","volume":"17","author":"FS Khan","year":"2018","unstructured":"Khan, F.S., Solmeyer, N., Balu, R., et al.: Quantum games: a review of the history, current state, and interpretation. Quantum Inf. Process. 17, 309 (2018). https:\/\/doi.org\/10.1007\/s11128-018-2082-8","journal-title":"Quantum Inf. Process."},{"key":"3395_CR22","doi-asserted-by":"publisher","DOI":"10.1142\/11681","volume-title":"Understanding Game Theory","author":"VN Kolokoltsov","year":"2020","unstructured":"Kolokoltsov, V.N., Malafeyev, O.A.: Understanding Game Theory. World Scientific, Singapore (2020)"},{"key":"3395_CR23","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1016\/0304-4068(78)90022-8","volume":"5","author":"D Rand","year":"1978","unstructured":"Rand, D.: Exotic phenomena in games and duopoly models. J. Math. Econ. 5, 173\u2013184 (1978)","journal-title":"J. Math. Econ."},{"key":"3395_CR24","first-page":"1","volume":"2014","author":"L Shi","year":"2014","unstructured":"Shi, L., Le, Y., Sheng, Z.: Analysis of price Stackelberg duopoly game with bounded Rationality. Discrete Dyn. Nat. Soc. 2014, 1\u20138 (2014)","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"3395_CR25","first-page":"259","volume":"271","author":"Y Peng","year":"2015","unstructured":"Peng, Y., Lu, Q.: Complex dynamics analysis for a duopoly Stackelberg game model with bounded rationality. Appl. Math. Comput. 271, 259\u2013268 (2015)","journal-title":"Appl. Math. Comput."},{"key":"3395_CR26","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1016\/j.chaos.2016.01.024","volume":"85","author":"Y Peng","year":"2016","unstructured":"Peng, Y., Lu, Q., Xiao, Y.A.: dynamic Stackelberg duopoly model with different strategies. Chaos Solitons Fract. 85, 128\u2013134 (2016)","journal-title":"Chaos Solitons Fract."},{"key":"3395_CR27","doi-asserted-by":"publisher","first-page":"1980","DOI":"10.1016\/j.physa.2017.11.112","volume":"492","author":"Y Xiao","year":"2018","unstructured":"Xiao, Y., Peng, Y., Lu, Q., et al.: Chaotic dynamics in nonlinear duopoly Stackelberg game with heterogeneous players. Phys. A 492, 1980\u20131987 (2018)","journal-title":"Phys. A"},{"key":"3395_CR28","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1016\/j.chaos.2019.04.007","volume":"123","author":"XN Yang","year":"2019","unstructured":"Yang, X.N., Peng, Y., Xiao, Y., et al.: Nonlinear dynamics of a duopoly Stackelberg game with marginal costs. Chaos Solitons Fract. 123, 185\u2013191 (2019)","journal-title":"Chaos Solitons Fract."},{"key":"3395_CR29","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1016\/S0378-4371(02)01648-5","volume":"320","author":"HN Agiza","year":"2003","unstructured":"Agiza, H.N., Elsadany, A.A.: Nonlinear dynamics in the Cournot duopoly game with heterogeneous player. Phys. A 320, 512\u2013524 (2003)","journal-title":"Phys. A"},{"issue":"3","key":"3395_CR30","first-page":"843","volume":"49","author":"HN Agiza","year":"2004","unstructured":"Agiza, H.N., Elsadany, A.A.: Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Appl. Math. Comput. 49(3), 843\u2013860 (2004)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"3395_CR31","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.cnsns.2010.03.002","volume":"16","author":"T Dubiel-Teleszynski","year":"2011","unstructured":"Dubiel-Teleszynski, T.: Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale. Commun. Nonliear Sci. Numer. Simul. 16(1), 296\u2013308 (2011)","journal-title":"Commun. Nonliear Sci. Numer. Simul."},{"key":"3395_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24699-2","volume-title":"Attractors, Bifurcation and Chaos: Nonlinear Phenomena in Economics","author":"T Puu","year":"2003","unstructured":"Puu, T.: Attractors, Bifurcation and Chaos: Nonlinear Phenomena in Economics. Springer, Berlin (2003)"},{"key":"3395_CR33","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/BF01221359","volume":"67","author":"JL Kaplan","year":"1979","unstructured":"Kaplan, J.L., Yorke, Y.A.: A regime observed in a fluid flow model of Lorenz. Commun. Math. Phys. 67, 93\u2013108 (1979)","journal-title":"Commun. Math. Phys."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03395-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-021-03395-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03395-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,16]],"date-time":"2022-02-16T09:35:17Z","timestamp":1645004117000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-021-03395-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,19]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["3395"],"URL":"https:\/\/doi.org\/10.1007\/s11128-021-03395-4","relation":{},"ISSN":["1570-0755","1573-1332"],"issn-type":[{"value":"1570-0755","type":"print"},{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,19]]},"assertion":[{"value":"12 April 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 November 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"56"}}